An abstract structure is described which provides a general model
for the innermost core of object-oriented programming and modelling.
The structure is called basic structure of ϵ
and is
introduced in the signature
(O, ϵ, ϵ¯^{⁽*⁾}, r, .ec, .ɛɕ)
with a single sort O of objects.
•
ϵ is the object membership relation, an indirect counterpart to
set membership.
•
ϵ¯^{⁽*⁾} is a left-infinite sequence
…, ϵ¯^{-1}, ϵ¯^{0}, ϵ¯^{1} of relations, whose 0-th member
is
≤ – the inheritance relation, corresponding to inclusion between sets.
The 1-indexed member is the power membership relation ϵ¯
which, if .ec is total,
equals the composition
(.ec) ○ (≤)
(in general, ϵ¯ is an abstraction of this composition).
•
r is the inheritance root, a distinguished object containing all
objects, including r itself, as members in ϵ,
thus playing the
role of the universal set.
•
.ec is the powerclass partial map,
corresponding to a relativized powerset operator.
•
.ɛɕ is the primary singleton partial map,
corresponding to set membership between non-singleton sets and singleton sets.
As a key characteristics induced by the structure,
each object is assigned a rank between
0 and a fixed limit ordinal ϖ.
Objects with rank ϖ are unbounded,
the remaining ones are bounded.
For unbounded objects x, the images of {x} under
ϵ and ϵ¯ are coincident.
Therefore, object membership is formed as the union
(ϵ) = (∊) ∪ (ϵ¯)
where ∊ is the domain-restriction of ϵ to
bounded objects.
It is shown that every basic structure is a substructure of
an (ϖ+1)-superstructure
and thus can be embedded in the
von Neumann universe of sets.
The inheritance root r appears as the ϖ-th
cumulation of urelement-like sets.
The above equality for object membership is expressed as
x ϵ y iff
x ∈ y or ℙ(x) ∩r⊆ y.
(ℙ(x) stands for the powerset of x.)
However, the HTML version is considered primary and can be more up-to-date.
Warning
This document has been created without any prepublication review
except those made by the author himself.
Membership glyphs
This document uses HTML entities
to display mathematical symbols.
In particular, there are multiple glyphs
used for different relation symbols of membership.
Best readability is achieved when the sequences of symbols from the following
two lines look (almost) identically.
The standard
Zermelo-Fraenkel set theory with the axiom of choice (ZFC)
can be viewed as a one-sorted structure (𝕌, ∈)
where 𝕌 is the universal class of all sets
and
∈
is the relation symbol of set membership,
the only non-logical symbol of the original language of ZFC.
Virtually all texts about set theory introduce a definitional extension
that includes the following symbols
(with ∅ / ℙ / ∖ having alternatives,
e.g. 0 / ℘ / −):
⊆,
∅,
ℙ,
∖,
∪,
∪∩,
∩,
where ⊆ is a relation symbol of set inclusion,
∅ is a constant symbol for the empty set,
ℙ and ∪ are unary function symbols
for powerset and union, respectively,
∖, ∪ and ∩ are binary function symbols
for difference, union and intersection, respectively,
and ∩ is
a unary partial function symbol for set intersection.
There are also composite function symbols
for enumerated sets and tuples
that use curly brackets and parentheses,
respectively:
{ , … , } and ( , … , ).
In particular, {x} is the singleton of x and
(x,y) is the ordered pair of x and y
having x as the first coordinate.
All the additional symbols denote distinguished sets or relations between sets
(considering functions as right-unique relations)
that are implicitly given by ∈.
The uniqueness of the definition of ∅,
ℙ(x), x ∖ y, ∪x, …,
x ∩ y, {x} or (x,y) for given sets x, y
as well as the antisymmetry of ⊆
is ensured by the extensionality axiom which says that sets x, y
are equal if and only if they have the same members:
x = y↔
for every set a,
a ∈ x ↔ a ∈ y.
In another words, a set is identified by its members.
For instance, the set ℙ^{6}(∅)
(the 6-th application of the powerset operator to the empty set
which is the partial von Neumann universe of rank 6)
is given by its 2^{65536} members.
[]
Let us single out the four relations between sets
which are referred to by the symbols involved in the following two equivalences.
For every sets x, y,
The 2x2 core of set theory
x ∈ℙ(y)
↔
x ⊆ y,
x ∈ y
↔
{x} ⊆ y.
That is, the singled out relations are
(1) set membership (∈),
(2) set inclusion (⊆),
(3) the powerset map (x ↦ℙ(x)) and
(4) the singleton map (x ↦ {x}).
Note that (3) and (4) are functional subrelations of ∈.
The distinction of (1)–(4) can be made in two steps:
Single out ∈ and ⊆ as the two most fundamental relations
of set theory.
Single out ℙ and x ↦ {x} as the two fundamental maps
that provide a connection between ∈ and ⊆.
Observe that apart from the equality symbol =
(and apart from symbols based on
∈ or ⊆ like e.g. ∉, ∋ or ⊂)
there is no obvious third most fundamental relational symbol of set theory.
We can therefore regard (1)–(4) as a core definitional extension,
the 2x2 core of set theory.
This document shows that the above 2x2 core has an abstract counterpart
that can be regarded as the core of object technology.
Most object-oriented programming languages
(e.g. Java, C++, Python, CLOS or Perl)
or ontology languages (e.g. RDF Schema or OWL)
support just
∈ and ⊆
and usually call the
counterpart relations
instance-of and inheritance, respectively.
However, there are significant examples of programming languages
which support links between objects that correspond to the powerset map
or
to its inverse
[]:
in Smalltalk-80 and Objective-C:
class x↦ the implicit metaclass of x,
in Ruby:
object x↦ the eigenclass of x,
in JavaScript:
the prototype of y↤ constructor y.
There are also examples of languages that support the singleton map:
Dylan, Julia and Scala.
The objective
The objective of this document is to provide a connection between
object technology and set theory according to the following diagram:
The connection is established through the family of basic structures
(shown by the small region labelled with the ϵ symbol)
which encompass abstract counterparts of the four distinguished set-theoretic relations.
It is shown how
core parts of object models of established languages like Ruby or Python
arise as special cases of basic structures,
and, on the other hand, how
basic structures can be gradually completed to superstructures,
which can be considered a generalization of partial von Neumann universe
allowing more than one element in the ground stage.
Main correspondence
By simplification,
main correspondence between
the 2x2 core of set theory and
the core of object technology can be expressed as follows.
(Zermelo-Fraenkel) For every setsx, y
(set membership) ∈
↭
ϵ (object membership)
(set inclusion) ⊆
↭
≤ (inheritance)
(powerset) ℙ(x)
↭
x.ec (powerclass)
(singleton set) {x}
↭
x.ɛϲ (singleton)
x ∈ℙ(y)
↔
x ⊆ y,
x ∈ y
↔
{x} ⊆ y or ℙ(x) ⊆ y
↕
(Morse-Kelley) For every classesx, y
(Object technology) For every objectsx, y
x єℙ(y)
↔
x ⊆ y,
x є y
↔
{x} ⊆ y or ℙ(x) ⊆ y
↔
x ϵ y.ec
↔
x ≤ y,
x ϵ y
↔
x.ɛϲ ≤ y or x.ec ≤ y
That is,
ϵ, ≤, .ec and .ɛϲ
are the abstract counterparts of set membership, set inclusion,
powerset and singleton set maps, respectively.
The correspondence is obtained via the following steps.
Adjust the equivalence for x ∈ y
by adding or ℙ(x) ⊆ y on the right side.
Since in ZFC,
{x} ⊆ℙ(x) for every set x,
the adjusted equivalence is equivalent to the original one.
Switch to Morse-Kelley set theory
[]
–
change the universe of discourse from sets to classes. Accordingly,
the singleton map becomes strictly partial (⁎),
let the ℙ map be total by extending its semantics
from powerset to powerclass,
change the ∈ symbol to є in order to reflect
the semantic change of ℙ.
Change the universe from classes to objects and introduce
ϵ ,
≤,
.ec and
.ɛϲ as abstractions of
є,
⊆,
ℙ and
x ↦ {x},
respectively.
Notes:
(⁎)
In
[]
the singleton map is total
– {x} equals the universal class whenever x is a proper class.
Interestingly,
this extension would not change the definition of є.
We further introduce slight adjustment of the definition of
powerclass
as known from set theory
[][],
shifting the meaning from
the class of subsets of x
to
the class of non-empty subsets of x.
As a result, there will be no fixed point of .ec.
Basic structures from superstructures
Basic structures arise by abstraction of suitable partial universes of
well-founded sets.
A brief specification of basic structures is provided via
the following abstraction steps.
(a)
(b)
Start with an
(ϖ+1)-superstructure(O, ∊).
Let ϖ be a limit ordinal and
let ∊ be a well-founded relation
on a set O of objects
such that (1) r(O),
the rank of O w.r.t. ∊,
equals ϖ+1
and (2) for every non-empty subset X of
O
satisfying r(X) ≤ϖ
there is a unique object x whose pre-image
under ∊ equals X.
The
(a) diagram on the right partially shows a restriction
of an (ϖ+1)-superstructure to objects whose rank is
less than 3 and thus form the third stage.
Objects are depicted as circles and are aligned into columns according
to their rank. There are just two objects with zero rank
– the terminals
which form the first stage (ground stage).
The ∊ relation is represented by blue arrows.
Make a definitional extension of (O, ∊).
Define r, ≤, .ec and .ɛϲ
as follows.
Use X.∍ and x.∍
for the pre-image of sets X and {x}
of objects under ∊.
Let r be the inheritance root –
the unique object x such that
x.∍=O.∍.
Let ≤ be the inheritance relation
defined by
x ≤ y ↔
x = y or ∅≠ x.∍⊆ y.∍.
Let .ec be the powerclass map –
the unique map O→O such that
u ∊ x.ec
↔
u ≤ x and u ∈O.∍.
Let .ɛϲ be the singleton map –
the unique partial map O↷O
such that
x.ɛϲ = y
↔ {x} = y.∍.
In particular, .ɛϲ is a distinguished subrelation
of ∊
and so is the restriction of .ec to O.∍.
Objects from the range of .ec (resp. of .ɛϲ)
are powerclasses (resp. singletons).
In the (a) and (b) diagrams,
.ec is represented by horizontal blue arrows
and .ɛϲ by blue arrows pointing to ◉
which indicate singletons.
The (b) diagram shows the inheritance relation
(green arrows, upwards directed) in the reflexive transitive reduction.
The reduction of blue arrows in (b) is based on
the (∊) = (.ɛϲ) ○ (≤) equality
(i.e.
∊ equals
the composition of the singleton map with inheritance).
Define metaobject structures
as abstraction of (O, ≤, r, .ec, .ɛϲ).
Use the following definitional extension for the axiomatization.
Let ∊, ϵ¯, ϵ and ϵ^{-1}
be relations between objects such that
(∊)
=
(.ɛϲ) ○ (≤)
is the bounded membership,
(x ∊ y ↔ x.ɛϲ ≤ y)
(ϵ¯)
=
(.ec) ○ (≤)
is the power membership,
(x ϵ¯ y ↔ x.ec ≤ y)
(ϵ)
=
(∊) ∪ (ϵ¯)
is the (object) membership,
(ϵ^{-1})
=
(≤) ○ (.ce)
is the anti-membership,
with .ce denoting the inverse of .ec.
For every integer i, let
(ϵ^{i}) (resp. (ϵ¯^{i}))
be the i-th relational composition of ϵ
(resp. of ϵ¯) with itself
whenever i > 0,
(ϵ^{i}) = (ϵ¯^{i})
be the -i-th relational composition of ϵ^{-1} with itself
whenever i < 0, and let
(ϵ^{0}) = (ϵ¯^{0}) = (≤).
Define the
rank function .d from O to
ϖ+1
in terms of r and ϵ^{i}.
In general, .d differs from the ∊-rank.
Adapt the nomenclature of objects to .d.
Objects with zero rank are terminal(s),
the other non-terminal(s),
objects x whose rank is not maximal are bounded,
the other (such that x.d =ϖ) are unbounded.
As a particular consequence of the definition of .d,
objects that are non-well-founded in ϵ are unbounded.
In the axiomatization, assert that the singleton map .ɛϲ is defined
exactly for bounded objects.
Define basic structures
as abstraction of (O, ϵ, ϵ¯^{⁽*⁾}, r, .ec, .ɛɕ)
where
ϵ¯^{⁽*⁾} is the left-infinite sequence
{ ϵ¯^{i}| i ∈ℤ, i ≤ 1 }
and
.ɛɕ stands for the difference (.ɛϲ) ∖ (.ec).
Define the rank .d
of objects by the same prescription as for metaobject structures
and preserve the nomenclature.
Define ∊ as the domain restriction of ϵ to bounded objects.
Obtain .ɛϲ back from .ɛɕ so that
(a) (.ɛɕ) ⊆ (.ɛϲ),
(b) .ɛϲ.ɛϲ = .ɛϲ.ec and
(c) x.ɛϲ = x.ec for every terminal x.
In contrast to metaobject structures, allow .ec
and .ɛɕ to be arbitrarily partial, possibly empty.
As a consequence, a basic structure might contain only finitely many objects,
with O= {r} being the minimum case.
For a natural i,
let .ec(i) be the i-th composition of .ec
with itself, .ec(0) being the identity on O.
Subsequently, let .ec(-i) be the inverse of .ec(i).
In the axiomatization, assert the following:
ϵ¯ is a subrelation of ϵ.
Powers of ϵ¯ compose transitively:
the composition of ϵ¯^{i} with ϵ¯^{j} is a subrelation
of ϵ¯^{i+j}.
Powers of ϵ compose transitively if they are positive on the left side:
(ϵ) ○ (ϵ^{i}) is a subrelation
of ϵ^{1+i}.
Powers of ϵ¯ are antisymmetric.
The intersection of
ϵ¯^{i} with its inverse equals .ec(i).
Objects from O.ϵ (the range of ϵ)
are inheritance descendants of r.
Every object is in the domain of ϵ, i.e.
O=O.϶.
Singletons and terminals x
are minimal w.r.t. (ϵ^{-i}) ∖ (϶¯^{i})
for every natural i
and power members: x.ϵ= x.ϵ¯.
The .ɛɕ map is
an injective range-restriction of ϵ
and is disjoint with ϵ¯.
Moreover, if x.ɛɕ = y then
u ϵ^{-i} x
↔
u ϵ^{1-i} y for every object u and every natural i.
Reserved.
Every object x has a finite metalevel index,
i.e.
x is related to r
in only finitely many negative powers of ϵ.
Unbounded objects x are power members: x.ϵ= x.ϵ¯.
Complete structure
The above mentioned (ϖ+1)-superstructures,
in an appropriate definitional extension,
are themselves a special case of basic structures,
called complete structures.
Main characterization of basic structures w.r.t. set theory
can be therefore obtained by
comparing S= (O, ∊)
with
(𝕌, ∈).
The following observations can be made:
In contrast to 𝕌 which is a proper class,
O is a set.
This is because basic structures are primarily meant to provide
a general model of the core part of object technology
based on set theory.
There can be more than one object x in the ground stage
O∖O.∊ of S
(i.e. such that x.∍=∅).
As a consequence, S is only weakly extensional.
Ground stage objects are urelements.
Note:
Objects from the ground stage are exactly the terminal objects of S.
For complete structures, we tend to use this term less frequently
than in the general case.
There are unbounded objects, most notably r,
that have maximum rank, ϖ,
and therefore do not appear in the domain of ∊.
There is no counterpart to ∅, the empty set.
Ground stage objects are incomparable in the strict inheritance <
to any object.
On the other hand,
the inheritance root r being a universal container of all bounded objects
has no counterpart in
(𝕌, ∈).
The powerclass map .ec deviates from the powerset operator
ℙ in two respects.
.ec is not a subrelation of ∊
(a consequence of the existence of unbounded objects:
x ∊ x.ec ↔ x ∈O.∍).
.ec increases the metalevel index,
the length of the
shortest ∊-path from the ground stage.
The second property can be expressed as
x.mli + 1 = x.ec.mli
and is a consequence of the missing empty set.
In contrast, the length of the shortest ∈-path
from the ground stage of (𝕌, ∈)
to ℙ(x)
equals constantly 1
for every set x
(since ∅∈ℙ(x) by definition of ℙ
and
∅ is an element (the only element) of the ground stage).
The singleton map .ɛϲ is strictly partial,
having the same domain as ∊.
As a further consequence of the missing empty set,
.ɛϲ is partially coincident with .ec
(see the diagrams above
for the common part of .ec and .ɛϲ
in the third stage):
x.ɛϲ = x.ec↔x is terminal or a singleton.
(In (𝕌, ∈),
ℙ(x) = {x} ↔ x =∅.)
The observations can be summarized into the following characterization.
Smallness.
Although used for modelling of largeness,
basic structures are themselves small by having a set
(the set O of objects)
as the universe of discourse.
Similarly to club sets[],
largeness of objects is expressed via their unboundedness
relative to a limit ordinal.
Presence of unbounded objects.
There are objects that are not in the domain of ∊.
Presence of urelements (atoms, terminals).
There are possibly multiple objects that are not in the range of ∊.
Absence of an empty set object.
There is no distinguished object outside the range of ∊
that would correspond to the empty set.
Completion
A major part of this document consists of showing
that
every basic structure (as specified by the axioms)
is a substructure of some
complete basic structure.
That is,
every basic structure
can be faithfully embedded into a complete structure.
Or, concisely,
every basic structure has a completion.
The completion is established gradually in the following steps:
Rank pre-completion.
Attach a set X of ϖ new members
to each object x that is
(a) not well-founded in ϵ,
(b) not ∊-ranked
(i.e. the rank x.d differs from r_{∊}(x),
the ∊-rank of x),
and
(c) not a powerclass (i.e. x is primary).
The members are attached
in such a way that (X, ϵ, ≤) is isomorphic
to (ϖ, ∈, ⊆).
Powerclass completion.
Append an infinite powerclass chain to each object for which the powerclass
is not defined.
Singleton completion.
Append an infinite singleton chain to each bounded object for which the
singleton is not defined.
Technically, this is performed in two steps
by first adding just the missing primary singletons
and subsequently performing the powerclass completion.
Extensional pre-completion.
To every object x that does not satisfy certain consistency conditions
attach two powerclass chains each of which starts in a terminal object.
The resulting structure is pre-complete,
that is,
powerclass complete,
(every object x has a powerclass x.ec),
singleton complete
(every bounded object x has a singleton x.ɛϲ), and
∊-ranked
(for every object x, x.d = r_{∊}(x)).
In particular, the structure is fully determined by bounded membership
∊
using the prescriptions already introduced for complete structures
(see
Basic structures from superstructures).
Cumulative embedding into an
(ϖ+1)-superstructure.
Let S= (O, ∊) be the pre-complete structure
to be embedded.
Choose an (ϖ+1)-superstructure V= (V, ∊)
so that its ground stage V_{1}
has the same cardinality as the set T of terminal objects of S.
Then the requested embedding map .ν is obtained as a limit of
a transfinite sequence
.ν_{0}, .ν_{1}, …, .ν_{ϖ}= .ν
of maps from O to V
defined as follows:
The restriction of .ν_{i} to terminals is for every i
identical and forms a bijection
between T and V_{1}.
The restriction of .ν_{i} to the set
O.∊
of non-terminal objects x is recursively defined by
x.ν_{0}.∍
=
x.∍¯.ν_{0}.ec.∍∪ x.∍.ν_{0}
(where ∊¯ equals (∊) ∩ (ϵ¯)),
x.ν_{i}.∍
=
x.϶¯.ν_{i-1}.ec.∍∪ x.∍.ν_{0}
if i is a successor ordinal,
x.ν_{i}.∍
=
∪{ x.ν_{k}.∍| k < i }
if i is a limit ordinal.
Note that the definition of .ν_{0}
is by the well-founded recursion on (O, ∊),
whereas the definition of .ν_{i} for i > 0
uses transfinite recursion over i.
Embedding into the von Neumann universe
The final embedding of (ϖ+1)-superstructures
(and thus of basic structures)
into the von Neumann universe of well-founded sets
is established
via powerset cumulation.
For every ordinal number α
let
ℙ_{⋆}^{α}
be a map between sets defined using transfinite recursion by
Equivalently,
ℙ_{⋆}^{0}(x) = x,
ℙ_{⋆}^{1}(x) = x ∪ (ℙ(x) ∖ {∅}),
ℙ_{⋆}^{α+1}(x) =ℙ_{⋆}^{1}(ℙ_{⋆}^{α}(x)),
ℙ_{⋆}^{α}(x) =∪{ℙ_{⋆}^{β}(x) | β < α }
with the last equality being satisfied for every limit α.
Consequently,
partial von Neumann universes are cumulations of {∅}, i.e.
for every ordinal α,
𝕍_{1+α}=ℙ_{⋆}^{α}({∅})
(the set of
of well-founded pure sets of rank less than 1+α).
(Note that the 1+-shift only makes a difference for finite ordinals
α.)
Given an (ϖ+1)-superstructure (O,∊)
to be embedded into (𝕌,∈)
as V= (V,∈)
it is sufficient to choose the ground stage V_{1} of V
to be a set such that
(a) all elements of V_{1} are singletons,
(b) the cardinality of V_{1} equals the cardinality of
O∖O.∊,
and
(c)
V_{1} is a subset of
𝕍_{i+1}∖𝕍_{i}
for some suitably large ordinal i
(so that each element of V_{1} has equal rank, namely i).
The V set is then obtained by
V=ℙ_{⋆}^{ϖ+1}(V_{1}).
The inheritance root of V equals
ℙ_{⋆}^{ϖ}(V_{1})
and is simultaneously the set of bounded objects of V.
Correspondence between constituents of a basic structure
and their set-theoretic counterparts is provided
by the Set representation theorem.
Sample structure
The following diagram shows a sample
basic structure
of object membership.
The structure is built from 4 relations between objects:
≤, the inheritance relation,
is a partial order which is shown in its reflexive transitive reduction
by green arrows.
ϵ, the (object) membership relation,
is shown via both blue arrows and green
arrows. Specifically, ϵ
equals the composition
(→) ○ (≤)
where
→
is the (exact) relation indicated by blue arrows.
.ec, the (partial) powerclass map, is a subrelation of ϵ
indicated by horizontal blue arrows.
(That is, x.ec = y iff there is a horizontal blue arrow from
x to y.)
.ɛɕ, the (partial) primary singleton map,
is another subrelation of ϵ
– the range-restriction of ϵ to primary singletons,
indicated by orange circles
()
(That is, x.ɛɕ = y iff there is a blue arrow from
x to y and the y object is displayed as an orange circle.
There are just two such objects in the sample structure.)
The (→)
relation represented by blue arrows is also reduced.
It is the minimum relation R such that
R ○ (≤) = (ϵ) and (.ec) ⊆ R.
(The minimum relation S satisfying just
S ○ (≤) = (ϵ)
has two pairs less than (→):
the horizontal arrows starting at c and e.ec.)
Note that since
R ○ (≤) = (ϵ) for some relation R,
it follows, due transitivity and reflexivity of ≤, that
(ϵ) ○ (≤) = (ϵ)
(the subsumption rule).
x ≤ y
iff
x
→ …
→ y
x ϵ y
iff
x →→ …
→ y
If x → y
then:
x.ec = y
iff
→ is horizontal
x.ɛɕ = y
iff
y is a primary singleton
x.ɛϲ = y
iff
y is a singleton
x.↧= {a | a ≤ x}
…
descendants of x
x.϶= {a | a ϵ x}
…
members of x
x ϵ¯ y
iff
x.↧⊆ y.϶
x ∊ y
iff
x ϵ y and x is bounded
O
…
objects
=T⊎C⊎O.ec
⊎O.ɛɕ
O.ec
…
powerclasses
O.pr
…
primary objects
=T⊎C⊎O.ɛɕ
T
…
terminal objects
=O∖r.↧
C
…
classes
=O.pr ∖ (T∪O.ɛɕ)
O.∍
…
bounded objects
O.ɛɕ
…
primary singletons
O.ɛϲ
…
singletons
= (T.ec ∪O.ɛɕ).ec^{∗}
H
…
helix objects
=r.ϵ^{∗}
R
…
reduced helix
=r.ec^{∗}
r
…
inheritance root
Objects form a set denoted O.
Objects are either powerclasses or primary
according to whether they appear in the image of .ec or not,
respectively.
Components of .ec are powerclass chains.
Each powerclass chain starts in a primary object.
(A chain may consist of just its primary object.)
Correspondingly,
each object x has its primary object x.pr.
An object x is either unbounded or bounded
according to whether x appears in a cycle of ϵ or not,
respectively.
(Note:
This characterization of boundedness
is applicable due to the finiteness of the structure,
see
Speciality of the sample.)
That is, unbounded objects x are those such that
x ϵ^{i} x for some natural i > 0,
where ϵ^{i} is the i-th composition of ϵ with itself.
Boundedness is preserved along powerclass chains:
An object x is (un)bounded iff the primary object x.pr
is (un)bounded.
In the sample structure,
the primary unbounded objects are displayed in blue.
There is a distinguished unbounded object, denoted r and
called the inheritance root.
It is the highest unbounded object in inheritance.
In fact, r is the top of O.ϵ,
that is, the common ancestor of all objects that have any members.
Simultaneously, r is a common container of all objects,
r.϶=O, including itself, rϵr.
In addition, there are two distinguished sets of unbounded objects.
H,
the set of helix objects, is the set of all direct or indirect
containers of r,
i.e. x ∈H iff rϵ^{i} x for some
i.
R, the reduced helix,
is the powerclass chain whose primary object is r.
Note that R is linearly ordered by inheritance,
inversely to .ec.
Observe the following inclusion chain:
{r}
⊆R⊆H⊆O.ϵ.
Moreover, all 4 sets have the same descendants
– that is,
r.↧= … =O.ϵ.↧.
Objects that are not descendants of r are terminal(s).
All terminal objects are primary.
Objects that are neither terminal nor powerclasses nor primary singletons
are classes.
The sets of terminals and classes are denoted T and C,
respectively, so that we can express a fundamental partition of objects into 4 sets:
O=T⊎C⊎O.ec
⊎O.ɛɕ.
Power membership
ϵ¯
ϵ,
≤,
r,
.ec,
.ɛɕ
ϵ^{-1}
ϵ^{-2}
ϵ^{-3}
⋮
The four definitory relations
≤, ϵ, .ec and .ɛɕ
of the sample structure are among constituents of the
signature of basic structures
according to the diagram on the left.
In particular, the inheritance relation, ≤,
appears as the 0-th member of the infinite sequence
ϵ¯^{⁽*⁾}= { ϵ¯^{i}| i ∈ℤ, i ≤ 1 },
of relations between objects,
where (ϵ¯) = (ϵ¯^{1}) is the power membership relation,
and (≤) = (ϵ¯^{0}).
As a definitional extension, we let (ϵ^{i}) = (ϵ¯^{i})
for every i ≤ 0.
Moreover, for i > 0, we let
ϵ^{i} (resp. ϵ¯^{i}) be
equal to the i-th relational composition of ϵ
(resp. of ϵ¯) with itself.
In the particular case of the sample structure,
we assume that ϵ¯ and ϵ^{-i}, i ∈ℕ,
are derived from
≤, ϵ, .ec and .ɛɕ
as follows:
ϵ¯ is defined by:
x ϵ¯ y ↔ x.↧⊆ y.϶
(x is a power member of y if all descendants of x
are members of y),
(ϵ^{-1}) =
(≤) ○ ((.ce)∪(.ɕɛ)) ○ (≤),
(.ce / .ɕɛ denote the inverse of
.ec / .ɛɕ.), and
for every natural i > 1,
ϵ^{-i}
equals the i-th relational composition of ϵ^{-1} with itself.
In the
sample structure,
the difference (ϵ) ∖ (ϵ¯) contains exactly 8 membership pairs,
indicated by blue arrows with a highlighted background.
(For instance, (d,f) ∉ (ϵ¯)
since n.ec ∈ d.↧∖ f.϶.)
The ϵ-diamond
The diagram on the right shows significant subrelations of ϵ
(note that .ec and .ɛɕ are
another significant subrelations of ϵ):
∊, the bounded membership,
is the domain-restriction of ϵ to bounded objects,
that is,
x ∊ y ↔
x ϵ y and x is bounded.
Additionally, ∊¯ denotes the
bounded power membership, that is,
(∊¯) = (∊) ∩ (ϵ¯).
The point of the diagram is that the relations form a lattice w.r.t. inclusion, that is,
in addition to the (not much significant) definition of ∊¯ we also have
Although the sample structure appears to be fairly wild,
it does not expose all features allowed by the
general definition.
In particular,
the sample possesses the following special properties.
As described above,
ϵ¯ and ϵ^{-i}, i ∈ℕ,
are derived from
≤, ϵ, .ec and .ɛɕ.
There are only finitely many objects.
In particular, the set C of classes is finite.
This allowed us to define unboundedness of objects via their circularity in ϵ.
H=R.↥
– that is, helix objects are ancestors of R.
This simplifies the description of
metalevels below.
Powerclasses and singletons
In addition to the powerclass map, .ec,
there
is a second definitory constituent of the structure that is
a partial map between objects:
the primary singleton map, .ɛɕ.
This map in turn appears to be just a difference
(.ɛϲ) ∖ (.ec) where
x.ɛϲ = y
↔
(ⅰ) x.ɛɕ = y or
(ⅱ) x.ec = y and
x.pr∈T∪O.ɛɕ.
The derived partial map .ɛϲ is the singleton map.
Objects from the image O.ɛϲ are singletons.
The following table shows the main properties of .ec
and .ɛϲ in comparison.
y is the powerclass of x,
x.ec = y
y is the singleton of x, x.ɛϲ = y y is the primary singleton of x, x.ɛɕ = y
(a)
x is the highest member of y
x.↧= y.϶
{x} = y.϶
x is the only member of y
(a')
(b)
y is the lowest power container of x
x.ϵ¯= y.↥
x.ϵ= y.↥
y is the lowest container of x
(b')
Note that although conditions (a) and (b)
(resp. (a') and (b'))
define a one-to-one relation between objects,
the .ec (resp. .ɛɕ) map is given explicitly.
For example, in the sample structure,
the (q.ec, v) pair satisfies both (a) and (b) but
v is not a powerclass,
the (g, s) pair satisfies both (a') and (b') but
s is not a singleton.
Metalevels
In the diagram of the
sample structure,
objects are grouped into columns in a correspondence to metalevels.
The metalevel index of an object x
is denoted x.mli and equals
the number of ancestors of x that belong to R,
the reduced helix.
By (b~10),
x.mli is asserted to be finite even if R is
infinite.
The set T of terminal objects is the 0-th metalevel.
For i > 0, the top of the i-th metalevel is the i-th
element from R,
starting with r as the 1-st object.
Axioms of basic structures assert that
when going alongside ≤,
ϵ,
.ec or .ɛɕ there are the following limitations
about the possible increment of the metalevel index:
If x ≤ y
then y.mli - x.mli ≤ 0,
if x ϵ y
then y.mli - x.mli ≤ 1,
if x.ec = y
then y.mli - x.mli = 1,
if x.ɛɕ = y
then y.mli - x.mli = 1.
Rank
In addition to the metalevel index, each object x
has defined a rank,
denoted x.d.
For the purpose of the introductory sample,
we provide a simplified definition of .d.
Let E = {ℇ^{i}| i ∈ℤ}
be a system of relations between objects
generated by the following rules.
(ϵ) ⊆ (ℇ^{1}).
(≤) ⊆ (ℇ^{0}).
(.ce) ∪ (.ɕɛ) ⊆ (ℇ^{-1}).
(.ce / .ɕɛ denote the inverse of
.ec / .ɛɕ.)
(ℇ^{j}) ○ (ℇ^{k}) ⊆ (ℇ^{j+k}) for every
integers j, k.
Equivalently,
x ℇ^{i} y iff
x = x_{0}ϵ^{i1} x_{1}ϵ^{i2} ⋯
ϵ^{in} x_{n}= y
for some natural n > 0,
objects x_{0}, x_{1}, …, x_{n}
and
signs
i_{1}, …, i_{n}∈ {-1,0,1}
such that
i_{1} + ⋯ + i_{n}= i
and
where
(ϵ^{1}) = (ϵ),
(ϵ^{0}) = (≤), and
(ϵ^{-1}) =
(≤) ○ ((.ce)∪(.ɕɛ)) ○ (≤).
(See power membership.)
We might call the above sequence
x_{0}, x_{1}, …, x_{n} of objects an
ℇ^{i}-path.
If x ℇ^{i} y for an integer i,
then x is an i-path member of y.
Due to the limitations for the possible increment of
metalevel index
we obtain the following implication for every integer i:
if u ℇ^{i} x then
u.mli + i ≥ x.mli.
Now the rank of an object x is defined by
x.d = sup {u.mli + i | u ℇ^{i} x, i ∈ℤ}.
Informally, the rank maximalizes the metalevel increment along (inverted)
ℇ^{i}-paths.
The diagram on the right shows such a maximalizing path for the a object
from the sample structure.
The values (in brown) show the contribution of each arrow
(ϵ or ≤) to the rank.
The overall increment is 7 so that
a.d = a.mli + 7 = 9.
Note:
In general, it is possible to have maximalizing paths with some
horizontal (i.e. .ec) arrows oriented in opposite direction.
Observe that
since x ℇ^{0} x, the rank is at least so big as the metalevel index.
Since (ϵ^{i}) ⊆ (ℇ^{i}) for every i > 0,
the rank of every unbounded object x
(such that x ϵ^{i} x for some i > 0)
is infinite, that is, x.d = ω.
This is in contrast to the metalevel index which is always finite.
Moreover, we have the following classification of objects according to their rank:
Unbounded objects are those with infinite rank.
Bounded objects are those with finite rank.
Terminal objects are those with zero rank.
Similarly to the metalevel index,
the powerclas map increases the rank of bounded objects exactly by 1.
Preliminaries
Some familiarity with elementary algebra, order theory and set theory is assumed.
This involves such notions as
structure,
substructure,
sort,
generating set/structure,
(partial) function,
relation,
domain,
range,
restriction,
extension,
injectivity,
reflexivity,
transitivity,
antisymmetry,
monotonicity,
isomorphism, or
closure operator.
Relations are regarded as sets (or in some cases as set-theoretic classes)
of ordered pairs (x,y).
The domain (resp. range) of a relation R
is the set of all x (resp. y) such that (x,y)
is from R for some y (resp. x).
Functions (maps) are regarded as functional (left-unique) relations.
We speak about a total (resp. partial) function on a set X
if the function's domain equals X (resp. is a subset of X).
Main notational conventions
Most set-theoretic symbols have been described in the
introduction.
We also use the ⊎ symbol for disjoint union of sets,
i.e.
x ⊎ y = z iff x ∪ y = z and x ∩ y =∅.
We use set-builder notation with a vertical bar
so that e.g.
{x | x ≤ y} is the set of all x such that x ≤ y.
The ○ symbol is used for relational composition.
Inscribed triangle indicates the left-to-right direction,
so that R ○ S is the set of all pairs (x,z) such that
there is a y such that
(x,y) ∈ R and (y,z) ∈ S.
We mostly use dot notation for function application.
The expression x.f refers to the application of f to x
(the value of f at x)
whenever f is a map and it is asserted that
x is from the domain of f.
In contrast,
the expression X.f
(with an uppercase X on the left side instead of the lowercase x)
refers to the image of the set X under f, i.e.
X.f = { y | there exists x ∈ X such that (x,y) ∈ f }.
We let the dot symbol . be the initial character
of map names whenever the dot notation is used for the map application.
According this rule,
the f function above would be referred to by .f.
If .f and .g are functions then .f.g refers to the
composition (.f) ○ (.g).
Furthermore, we let the dot notation be applicable for images of relations.
If ϵ is a relation then X.ϵ is the image
of X under ϵ
and x.ϵ is a shorthand of {x}.ϵ.
If in addition,
.f is a function then
x.ϵ.f refers to the image of x under (ϵ) ○ (.f).
Let On denote the proper class of ordinal numbers
(shortly ordinals).
According to the standard definition,
α belongs to On iff α is
a transitive set strictly well-ordered by ∈.
Following standard conventions we will use
special symbols for set operations on ordinals.
In particular, for ordinals α, β and a set X of ordinals,
α + 1 = α ∪ {α}
(the successor of α),
α ≤ β iff α ⊆ β,
α < β
iff α ∈ β
iff α ≤ β and α ≠ β,
α ∧ β = α ∩ β
(the minimum between α and β),
similarly,
α ∨ β = α ∪ β,
sup(X) =∨X=∪X.
(The supremum of X.
We will mostly prefer the sup(X) expression.)
If β is a successor ordinal then β-1 refers to
the unique ordinal of which β is a successor.
We will also sometimes use the ≤ symbol
(and its variants <, ≥ and >)
for relations between sets of ordinal numbers
(cf. Polars of ≤).
If X, Y and {α} are sets of ordinals then
X ≤ Y
iff
α ≤ β for every
α ∈ X and β ∈ Y,
α ≤ X
iff
{α} ≤ X.
Note that this introduces ambiguity of the meaning of X ≤ Y
since X or Y
may themselves be ordinals.
A disambiguation should be clear from the context.
Cardinal numbers
We assume axiom of choice and use the von Neumann cardinal assignment.
The cardinality of a set X (notation: card(X))
is the least ordinal α
such that there is a bijection between X and α.
Cardinal numbers (shortly cardinals)
are ordinal numbers α such that card(α) = α.
Natural numbers
We regard natural numbers as the finite ordinal numbers
0, 1, 2, ….
We let the set of natural numbers be denoted by either ℕ
or by ω according to which symbol is considered
appropriate in the given context.
Integers
We denote ℤ the set of integer numbers
and
let the set ℕ of natural numbers be coincident with
non-negative integers.
The binary operation of addition is extended by
i + ω = ω for every integer i.
Well-foundedness
For a relation ϵ on a set X, an element x ∈ X is
well-founded in ϵ
if x is not a member of an infinite descending chain in ϵ, i.e.
if there is no infinite chain of the form
… x_{2}ϵ x_{1}ϵ x_{0}= x.
A relation ϵ on a set X is well-founded
if all elements x ∈ X are well-founded in ϵ.
Assuming the axiom of choice, this is equivalent to the condition that
every non-empty subset Y ⊆ X contains an element y that is
minimal in (Y,ϵ),
i.e.
there is no u from Y such that u ϵ y.
Rank
For a well-founded relation ∊ on a set X,
the rank function of ∊
(alternatively, the ∊-rank)
is a map r() from X to ordinal numbers such that for every
x ∈ X,
r(x) = sup {r(a) + 1 | a ∊ x}.
By well-founded recursion, there is exactly one such map.
Obviously,
r(x) = 0 ↔ x is minimal in ∊.
Moreover,
let the ∊-rank of a subset Y of X be
sup {r(a) + 1 | a ∈ Y},
let the rank of ∊ be the ∊-rank of X.
Limited rank
Assume that ϵ is a (not necessarily well-founded) relation on a set X
and let ϖ be a limit ordinal.
Define recursively a function r_{ϵ} : X → ϖ+1 by
r_{ϵ}(x) = ϖ ∧
sup {r_{ϵ}(a) + 1 | a ϵ x }
if x is well-founded in ϵ,
r_{ϵ}(x) = ϖ
otherwise.
For x ∈ X, we call r_{ϵ}(x)
the ϖ-limited rank of x (w.r.t. ϵ).
Fixing a limit ordinal ϖ
If not stated otherwise we
will further assume that a fixed limit ordinal ϖ is given in the context.
This ordinal number will be the highest rank of objects under consideration.
The choice of the symbol indicates that the first limit ordinal
ω is considered to be sufficient with respect to object technology.
Ordinal numbers less than ϖ are bounded,
the remaining (including ϖ itself) are unbounded.
More generally,
a set X of ordinal numbers is bounded or unbounded according to
whether sup(X) is bounded or unbounded, respectively.
Basic structure
This section provides a formal definition of basic structures.
Before stating the axioms we first introduce
the family of
ϵ5-structures
which defines the language of basic structures.
Note:
The family of basic structures can be viewed as a generalization
of metaobject structures
introduced later.
The generalization consists of
allowing .ec and .ɛϲ to be incomplete,
abstraction from ϵ, ϵ¯ and ϵ^{-i}
for every natural i.
Moreover,
because of the incompleteness,
the singleton map .ɛϲ is not taken as the definitory
constituent.
Instead, the primary singleton map, .ɛɕ, is introduced,
which stands for the difference (.ɛϲ) ∖ (.ec).
As a consequence, the existence of x.ec and that of x.ɛɕ
for a given object x are independent of each other.
ϵ5-structure
⋮
⋮
ϵ^{2}
ϵ¯^{2}
ϵ
ϵ¯
≤
ϵ^{-1}
ϵ^{-2}
⋮
By an ϵ5-structure
we mean a structure
S= (O, ϵ^{⁽*⁾}, ϵ¯^{⁽*⁾}, r, .ec, .ɛɕ)
where
O is a set of objects,
ϵ^{⁽*⁾} and
ϵ¯^{⁽*⁾}
are bi-infinite sequences of relations between objects such that
(ϵ^{i}) = (ϵ¯^{i}) for i ≤ 0 and where
(ϵ) = (ϵ^{1}) is the (object) membership relation,
(ϵ¯) = (ϵ¯^{1}) is the power membership relation,
(≤) = (ϵ^{0}) is the inheritance relation
(with .↧ / .↥ used for preimages / images under ≤),
(ϵ^{-1}) is the anti-membership relation,
r
is the inheritance root, a distinguished object.
The remaining two definitory constituents are partial
maps O↷O
(i.e. functional relations between objects):
.ec
is the powerclass map
(objects from O.ec are powerclasses),
.ɛɕ
is the primary singleton map
(objects from O.ɛɕ are primary singletons).
The only additional constraint which are ϵ5-structures subject to
is the following condition:
(b~0)
For every positive natural i,
(a)
(ϵ) ○ (ϵ^{i}) = (ϵ^{1+i}),
(b)
(ϵ¯) ○ (ϵ¯^{i}) = (ϵ¯^{1+i}).
That is, for every positive natural i, ϵ^{i}
(resp. ϵ¯^{i})
is the i-th relational composition of ϵ
(resp. of ϵ¯)
with itself.
Consequently,
we consider the bi-infinite sequence ϵ¯^{⁽*⁾}
to be a definitional extension of the left-infinite sequence
{ ϵ¯^{i}| i ∈ℤ, i ≤ 1 }
and
the bi-infinite sequence ϵ^{⁽*⁾}
to be obtainable by a definitional extension of
(ϵ, ϵ¯^{⁽*⁾}),
so that we usually drop the wildcard superscript from ϵ^{⁽*⁾}
in the signature.
The following definitions are (almost) sufficient to state the axioms of
basic structures.
For every integer i, ϶^{i} (resp. ϶¯^{i})
denotes the inverse of
ϵ^{i} (resp. of ϵ¯^{i}).
For a natural i,
let .ec(i)
be the i-th composition
of .ec with itself,
with .ec(0) being the identity on O.
Let .ec(-i) be the inverse of .ec(i).
Let
T=O∖O.ϵ.↧
be the set of terminal objects (or terminals).
The metalevel index, x.mli,
and rank, x.d, of an object x are
defined by
where the definition of .d only applies
to the special case ϖ= ω and
under additional assumptions which are consequences of
(b~1)–(b~10).
The general definition
of .d is provided in the next section.
Basic structure
By a basic structure (of ϵ)
we mean an
ϵ5-structureS=(O, ϵ, ϵ¯^{⁽*⁾}, r, .ec, .ɛɕ)
satisfying the following axioms:
First observations:
The (b~2) condition has three important special cases:
For i = j = 0 the transitivity of ≤:
(≤) ○ (≤) ⊆ (≤).
For i = 0 and j = 1 the monotonicity of ϵ¯:
(≤) ○ (ϵ¯) ⊆ (ϵ¯).
For i = 1 and j = 0 the subsumption of ϵ¯:
(ϵ¯) ○ (≤) ⊆ (ϵ¯).
Using
(b~0)(b~1) and
(b~3)
it follows that
(b~2) is equivalent to:
(b~2)'
(a)
(ϵ¯) ○ (≤) ⊆ (ϵ¯),
(b)
(ϵ¯^{-i}) ○ (ϵ¯^{1-j}) ⊆ (ϵ¯^{1-i-j})
for every natural i, j.
Condition (b~3)
has the subsumption of ϵ as an important case:
(ϵ) ○ (≤) ⊆ (ϵ).
Moreover,
it follows from
(b~0)
that
(b~3)
can be equivalently stated as
any of (ⅰ) or (ⅱ):
(ϵ) ○ (ϵ^{-i}) ⊆ (ϵ^{1-i})
for every natural i.
(ϵ^{i}) ○ (ϵ^{j}) ⊆ (ϵ^{i+j})
for every natural i > 0 and every integer j.
For i = 0 in (b~4) we obtain
the antisymmetry and reflexivity of ≤:
(≤) ∩ (≥) = (O, =).
It follows that
≤ is a partial order on O.
The case i = 1 shows that .ec is given by ϵ¯^{⁽*⁾}.
By (b~5)
the inheritance root r is given by (O, ϵ, ≤)
as the unique object x such that x.↧=O.ϵ.↧.
That is,
O=T⊎r.↧.
Moreover, since
rϵ x ≤r for some object x
it follows by subsumption of ϵ that rϵr.
Since r is non-well-founded in ϵ it follows by
(b~11) that rϵ¯r.
Consequently, for every non-terminal x,
x ϵ¯r
by monotonicity of ϵ¯
and
x ϵr since (ϵ¯) ⊆ (ϵ).
Since it is already asserted by
(b~1)–(b~5)
and (b~11) that
x ϵ¯r for every non-terminal x,
the (b~6) condition can be stated just for
terminal x,
i.e. as
T⊆O.϶.
Moreover, since (b~7)(b) (with i = 0) asserts
x.ϵ= x.ϵ¯ for every terminal x
it follows that
every object has a (power) container and
r is a universal (power) container:
O.϶=O.϶¯=O=r.϶¯=r.϶.
(However, r is not necessarily a unique object with this property.)
Axiom
(b~7)(a) can be stated as
any of (ⅰ)–(ⅳ),
using
additional definitions.
For every x from T∪O.ɛɕ,
x.϶¯^{-i}⊆ x.ϵ¯^{i}
for every natural i,
x.϶¯^{-∗}⊆ x.ϵ¯^{∗}
(where
(ϵ¯^{∗}) =∪{ϵ^{i}| i ∈ℕ}
and
(ϵ¯^{-∗}) =∪{ϵ^{-i}| i ∈ℕ}),
x.϶¯^{-∗}= x.ec^{∗}
(where .ec^{∗} is the reflexive transitive closure of
.ec),
For (ⅲ), use well-foundedness of x
and
apply x.ec(i) ϵ^{-j} x
→ i = j, see
observation B3.
For (ⅳ),
use x.϶¯=∅
which is asserted by
(b~8).
Axiom
(b~7)(b) can be stated as
the equality
x.ϵ= x.ϵ¯ for every
x from (T∪O.ɛɕ).ec^{∗}
(which is exactly the set of terminals and
singletons).
In (b~8)
we obtain x.↥= y.ϵ^{-1} for i = 0 in (b)
and thus x ϶^{-1} y.
It follows that
.ec and .ɛɕ
are disjoint subrelations of ϶^{-1}:
(.ec)
=
(϶^{-1}) ∩ (ϵ¯),
(.ɛɕ)
⊆
((϶^{-1}) ∖ (ϵ¯)) ∩ (ϵ)
(where (.ɛɕ) ⊆ (ϵ) is by (b~8)(a)
or also by (b~8)(b) for i = 1).
In in the (b) condition,
= can be replaced by ⊆
(the inverse inclusion follows by
(b~3)).
Moreover, the
(c) condition (i.e. (x,y) ∉ (ϵ¯)) can be
stated as any of
(ⅰ)–(ⅲ):
(ⅰ)
y.϶¯=∅,
(ⅱ)
(x,y) ∉ (.ec),
(ⅲ)
y ∉O.ec,
using (b~8)(a) for
(ⅰ)
and
(b~4)
for
(ⅱ) and (ⅲ).
In particular,
(ⅲ) means
O.ec ∩O.ɛɕ =∅
–
powerclasses are disjoint with primary singletons.
Axioms (b~11) and (b~1)
can be equivalently stated as the single equality
(ϵ) = (∊) ∪ (ϵ¯)
where ∊ is the domain-restriction of ϵ
to bounded objects –
objects x such that x.d < ϖ.
A minimum basic structure is such that O= {r}.
In such a case,
.ec = .ɛɕ =∅= (ϵ¯^{i})
for i < 0
and
(ϵ¯^{i}) = (ϵ) = {(r, r)} for i ≥ 0.
Observations about .ec
Observations A:
Assume (b~1)–(b~4).
For every objects x, y, if x.ec = y then
(a)
x.↧= y.϶= y.϶¯,
(b)
x.ϵ¯= y.↥,
(c)
x.↥= y.ϵ¯^{-1},
(d)
x.϶^{-1}= y.↧.
It follows from x.↧= y.϶ that .ec is injective.
If x.ec(i) = y for an integer i then
(a)–(d)
from the previous statement hold with
ϵ^{-1}/ϵ¯/ϵ replaced by
ϵ^{-i}/ϵ¯^{i}/ϵ^{i}.
Corollary:
In a
powerclass complete
(pre-)basic structure,
powers of ϵ¯ are given by ≤ and .ec:
(ϵ¯^{i})=(≤) ○ .ec(i) ○ (≤)
for every integer i.
For every objects x, y
and every integers i, j, k,
if both x.ec(i) and y.ec(j) are defined then
x.ec(i) ϵ¯^{k} y.ec(j)
↔
x ϵ¯^{k+i-j} y,
x.ec(i) ϵ^{k} y.ec(j)
→
x ϵ^{k+i-j} y,
.ec(i)x ϵ^{k} y.ec(j)
↔
x ϵ^{k-j} y.
Corollary (the case i = j = 1, k = 0
in (ⅰ)):
The .ec map is an order embedding
of (O.ce,≤)
into (O, ≤):
x ≤ y
↔
x.ec ≤ y.ec
whenever both x.ec and y.ec are defined.
If x.ec = y then:
x is well-founded in ϵ↔y is well-founded in ϵ.
If y is well-founded in ϵ then
y.ec(-i) is only defined for finitely many natural i
so that
y.pr exists.
Proof:
(Apply:
(a)
x.↧= y.϶^{i} ↔
y.϶^{i}⊆ x.↧⊆ y.϶^{i},
(b)
x.ϵ¯^{i}= y.↥ ↔
y.↥⊆ x.ϵ¯^{i}⊆ y.↥.)
Assume that x.ec(i) = y, i ∈ℤ. Then
x.↧= y.϶^{i} follows from:
.ec(i) ○ (϶^{i}) ⊆ (≥)
and
.ec(-i) ○ (≥) ⊆ (϶^{i}).
(The same holds with ϶¯^{i} instead of ϶^{i}.)
Cases (c) and (d) correspond to (b) and (a), respectively.
Observations B:
In addition, assume (b~10).
(This condition is not asserted in
pre-basic structures.)
.ec is a well-founded map:
If x.ec(-i) exists then
(by composition of .ec(-i) and ϵ¯)
x ϵ¯^{1-i}r.
It follows that there can only be finitely many natural i such that
x.ec(-i) exists.
Note:
Well-foundedness of .ec is a consequence
of well-foundedness of ϶^{-1},
see anti-membership.
If x ϵ^{k} x then k ≥ 0.
(The case k < 0 is again disallowed by (b~10).)
If x.ec(i) ϵ^{-j} x for natural i, j then
x ϵ^{i-j} x
and j ≤ i.
If x is well-founded then i = j.
Definitions in detail
This section provides detailed definitions
for
ϵ5-structures
and thus
for basic structures.
The already introduced definitions are repeated.
In many cases, the definitions make sense only
together with additional assumptions.
A distinguished collection of assumptions is formed
by axioms of pre-basic structures.
Note:
Definitions considered to be closely related to
consistency conditions
are provided in the next section.
Powerclass chains
Let .ec^{∗}
be the reflexive transitive closure of .ec.
For a positive natural i,
.ec(i) is the i-th
composition of .ec with itself,
.ec(0) is the identity on O,
.ec(-i) is the inverse of .ec(i).
Let .ce, .ce^{∗},
.ce(i)
be the the inverse of .ec,
.ec^{∗},
.ec(i),
respectively.
Let (.ec^{∗-∗}) = (.ec^{∗}) ∪
(.ce^{∗}).
For a set Y of objects, Y.ce denotes the image of
Y under .ce.
For an object y (rather than a set of objects)
we consider y.ce
to be defined and equal to x iff {x} = {y}.ce.
Similarly with .ec(i) or .ce(i).
Let .pr be the map between objects such that:
x = y.pr
↔
{x} = y.ce^{∗}∖O.ec.
For an object x, (in each case assume that the value is defined)
x.ec is the powerclass of x,
x.ec^{∗} is the powerclass chain of x,
x.ce is the powerclass predecessor of x,
x.pr is the primary object of x,
x.eci is the powerclass index of x,
defined
by x.eci = sup {i | {x}.ce(i) ≠∅}.
If x.pr exists then: x.eci = i
↔
x.pr.ec(i) = x.
S is powerclass complete
if .ec is a total map between objects,
i.e. O=O.ce.
Object membership
ϵ
∊
ϵ¯
∊¯
In the definitions below we
consider that the first two definitory constituents of an ϵ5-structure
are of the form (ϵ, ϵ¯^{⁽*⁾})
where
ϵ¯^{⁽*⁾} is only left-infinite
((ϵ¯^{⁽*⁾}={ ϵ¯^{i}| i ∈ℤ, i ≤ 1 }).
There are four membership relations between objects:
ϵ, the (object) membership,
is the first constituent of an ϵ5-structure,
ϵ¯, the power membership,
equals ϵ¯^{1} from the definitory sequence
ϵ¯^{⁽*⁾}.
∊, the bounded membership,
is defined by:
x ∊ y ↔
x ϵ y and x.d < ϖ.
∊¯ denotes the intersection
(∊) ∩ (ϵ¯)
(the bounded power membership).
For an integer i,
the i-th power of
ϵ / ϵ¯ / ∊ / ∊¯
is denoted
ϵ^{i} / ϵ¯^{i} / ∊^{i} / ∊¯^{i}
and defined as follows.
For i > 0,
ϵ^{i} / ϵ¯^{i} / ∊^{i} / ∊¯^{i}
is
the i-th relational composition of
ϵ / ϵ¯ / ∊ / ∊¯
with itself.
(E.g., (∊^{1}) = (∊),
(∊^{2}) = (∊) ○ (∊).)
For i ≤ 0,
ϵ¯^{i} is the i-th member (item) of
the definitory sequence ϵ¯^{⁽*⁾},
(ϵ^{i}) = (ϵ¯^{i}),
∊^{i} is defined by:
x ∊^{i} y ↔
x ϵ^{i} y and x.d < ϖ,
(∊¯^{i}) = (∊^{i}).
There are one-way closures and two-way closures of membership:
(ϵ^{∗}) =∪{ϵ^{i}| i ∈ℕ},
similarly for ϵ¯ / ∊ / ∊¯,
(note that due (b~0),
ϵ^{∗} is the transitive closure of
(ϵ^{0}) ∪ (ϵ^{1})),
(ϵ^{-∗}) =∪{ϵ^{-i}| i ∈ℕ},
(ϵ¯^{-∗}) = (ϵ^{-∗})
and
(ϵ¯^{∗-∗}) = (ϵ¯^{∗}) ∪ (ϵ¯^{-∗}).
The respective inverses of the above relations are denoted via reversed symbols:
ϵ / ϵ¯ / ∊ / ∊¯
⇢
϶ / ϶¯ / ∍ / ∍¯.
We use dot notation for images under the above relations,
both for objects and sets of objects.
For an object x,
x.϶ is the set of members of x,
x.϶¯ is the set of power members of x,
x.∍ is the set of bounded members of x,
x.ϵ is the set of containers of x,
x.ϵ¯ is the set of power containers of x.
An object x is said to be a power member
(resp. power container) if x.ϵ= x.ϵ¯≠∅
(resp. x.϶= x.϶¯≠∅).
Inheritance
The inheritance relation,
≤, is the 0-th power of ϵ:
(≤) = (ϵ^{0}) = (ϵ¯^{0}),
that is,
≤
is a special notation for the
0-th member of the definitory sequence ϵ¯^{⁽*⁾}.
As usual, we use < for the strict inheritance:
x < y ↔
x ≤ y and x ≠ y.
Similarly, let ≥ and > be the inverses of
≤ and <, respectively.
There is also the bounded inheritance relation,
∊^{0},
for which no special symbol is introduced.
By the definitions made so far,
∊^{0} is the domain-restriction of ≤:
x ∊^{0} y ↔
x ≤ y and x.d < ϖ.
We let .↥ and .↧ denote the
image and preimage operators for ≤.
We shall use these operators both for objects and sets of objects.
For images of a single object
under < and >,
the polar maps.⋀ and .⋁ are used, respectively.
For an object x,
x.↧ is the set of descendants of x,
x.⋁ is the set of strict descendants of x,
x.∍^{0} is the set of bounded descendants of x,
x.∍^{0}= x.↧∩O.∍,
(assuming O=O.϶ for the equality)
x.↥ is the set of ancestors of x,
x.⋀ is the set of strict ancestors of x,
x.⋀= x.↥∖ {x}.
Polars of ≤
We also introduce notation for lower and upper bounds in ≤.
First we extend the meaning of ≤ for relationship between sets of objects.
For an object a and sets X, Y of objects,
X ≤ Y iff
x ≤ y for every x ∈ X and y ∈ Y,
a ≤ X iff {a} ≤ X.
Similarly with <, ≥ and >.
For a set X of objects, X.▽ denotes
the set of lower bounds of X, whereas X.⋁ is the set
of strict lower bounds of X, similarly for upper bounds.
That is,
X.▽= { x | x ≤ X },
X.△= { x | x ≥ X }
(.▽ and .△ are the polar maps of ≤),
X.⋁= { x | x < X },
X.⋀= { x | x > X }
(.⋁ and .⋀ are the polar maps of <).
Anti-membership
There is an anti-relation of ϵ¯:
ϵ^{-1} is the anti-membership relation.
(We might also consider
∊^{-1} to be the bounded anti-membership.)
For an object x,
x.϶^{-1} is the set of anti-members of x,
and
x.ϵ^{-1} is the set of anti-containers of x.
Recall the
(b~10) axiom:
For every object x, x.mli
(= sup { i | x ϵ^{1-i}r, i ∈ℕ })
is finite.
(b~10)→϶^{-1} is a well-founded relation in which every object has a finite rank.
If ϵ^{-i} equals the i-th relational composition of ϵ^{-1}
for every i > 0,
then
↔ is satisfied in the previous observation.
O.϶^{-1}=r.϶^{-1}∪T.϶^{-1}.
Distinguished sets of objects
In an ϵ5-structure
(even if axioms of basic structures are assumed),
there is only one distinguished object
whose existence is asserted –
the inheritance root r.
However, there are several distinguished sets of objects:
The set O is the set of all objects.
The set O.∍ is the set of bounded objects.
(Again, we assume O=O.϶ for this expression.)
The set O.ec consists of powerclasses.
The set O.pr =O∖O.ec
is the set of primary objects.
The set T of terminal objects
(or terminals)
is defined as O∖O.ϵ.↧.
The set R=r.ec^{∗}
(the powerclass chain of r)
is the reduced helix.
The set H=r.ϵ^{∗}
is the set of helix objects.
The set O.ɛɕ
is the set of primary singletons.
The set O.ɛϲ
is the set of singletons.
(See below
for the definition of .ɛϲ.)
The set C of classes
is defined as
O∖ (T∪O.ec ∪O.ɛɕ).
Recall the last definitory constituent of an
ϵ5-structure:
.ɛɕ is a partial map between objects,
x.ɛɕ (if defined) is the primary singleton of x.
The (derived) singleton map is denoted .ɛϲ
and is defined as a partial map between objects by
x.ɛϲ = y
↔
{x} = y.϶ and y is from
(T.ec ∪O.ɛɕ).ec^{∗}.
Objects from the image O.ɛϲ are singletons.
We say that S is
primary singleton complete
if x.ɛɕ is defined for every object x from
O.∍∖
(T∪O.ɛϲ),
singleton complete
if x.ɛϲ is defined for every object x from
O.∍.
We call the sets
O.∍∖ (T∪O.ɛϲ)
and
O.∍
the potential domain of .ɛɕ and .ɛϲ,
respectively.
We let the integer powers, inverses and transitive closures of
.ɛɕ and .ɛϲ
be denoted and defined
in a similar way to that of .ec.
The 0-th power of .ɛɕ and .ɛϲ
is defined as the identity on the respective potential domain.
In particular,
.ɕɛ is the inverse of .ɛɕ,
O.ɕɛ is the set of objects with a defined primary singleton,
O.ɛɕ(0)
(=O.∍∖ (T∪O.ɛϲ))
is the potential domain of .ɛɕ.
The i-th metalevel, i ∈ℕ, equals
r.϶^{1-i}∖r.϶^{-i}.
(A consequence of r.϶^{-i}⊆r.϶^{1-i}.)
If t =r.ec(i) then
t.϶∖ t.↧ is the i-th metalevel,
(with t.ce the top if i > 0)
t.↧∖ t.϶^{-1} is the (i+1)-th metalevel
with t as the top.
(Recall that i + ω = ω
for every integer i.)
If x ϵ^{i} y for an integer i
then i + x.mli ≥ y.mli.
(A consequence of r.϶^{1-k}.϶^{i}⊆r.϶^{i+1-k},
k ∈ℕ.)
In particular,
x ≤ y
→
x.mli ≥ y.mli,
x ϵ y
→
1 + x.mli ≥ y.mli,
x.ec = y
→
1 + x.mli = y.mli.
(A consequence of (.ec) ⊆ (ϵ) ∩ (϶^{-1}).)
As a consequence, if x.ec = y then:
x.mli = ω ↔ y.mli = ω.
If, in addition,
(.ɛϲ) ⊆ (ϵ) ∩ (϶^{-1}) (as asserted by
(b~8))
then
x.ɛϲ = y
→
1 + x.mli = y.mli.
For every object x,
if x.mli < ω then x.pr exists.
Helix number
The
helix numberh
of an ϵ5-structure is defined by
(Recal that H denotes the set r.ϵ^{∗} of helix
objects.
Also recall
that i + ω = ω
for every integer i.)
For every natural i,
i < h ↔
r.ϵ^{i-1}≠H ↔
r.ϵ^{i}≠r.ϵ^{i-1}.
The helix number is at least 1. Moreover,
h= 1
↔
r.ϵ= {r} =H,
h is finite
↔
r.ϵ^{i}=H for some natural i.
The following less elegant definition of h
can be used to avoid a reference to r.ϵ^{-1}.
This can be useful in
ϵ-based monotonic structures
where ϵ^{-k}, k > 0,
are derived from ϵ and ≤.
h
= 1
if H= {r},
h
= sup {i + 2 |r.ϵ^{i}≠H, i ∈ℕ}
otherwise.
Rank
The rank of an object x
is denoted x.d
and defined to be an ordinal number
at most equal to a
fixed limit ordinalϖ
according to the following prescription.
Let W be the set of all objects z
such that
(a) z is well-founded in ϵ
and
(b)
for every a from z.϶^{∗}.϶^{-∗},
the metalevel index a.mli is finite.
Then .d is defined recursively using well-foundedness
of (W, ϵ).
For every object x,
For every object x, the following sets are identical:
(Therefore, (ⅱ) can be used for the definition of
.d.)
{i-j + a.mli |
a ∈ x.϶^{i}.϶^{-j}, i,j ∈ℕ},
{i+j + a.mli |
a ∈ x.϶^{i}.϶^{j}, i,j ∈ℤ}.
For every object x satisfying
x.϶^{∗}.϶^{-∗}.mli < ω
the following set is identical to
(ⅰ):
(ⅰ')
{i-j + a.mli |
a ∈ x.϶^{i}.϶^{-j}∩O.pr, i,j ∈ℕ}.
Similarly, the additional condition of a being a primary object
can be added to (ⅱ).
If ϖ= ω then
x.d = sup {i-j + a.mli |
a ∈ x.϶^{i}.϶^{-j}, i,j ∈ℕ}.
Boundedness
Boundedness of objects is based on their rank.
Objects x such that x.d < ϖ are bounded,
the remaining objects x
(such that x.d =ϖ)
are unbounded.
A set X of objects is bounded (resp. unbounded) if
sup(X.d) < ϖ
(resp. sup(X.d) =ϖ).
The already introduced bounded membership relation, ∊, is
therefore the domain restriction of ϵ to bounded objects.
If O=O.϶ (as is asserted in basic structures)
then O.∍ is the set of bounded objects.
Note in particular, that every object that is non-well-founded in ϵ
is unbounded.
Power instance-of and the .class map
Recall that the set C of classes
is defined as
O∖ (T∪O.ec ∪O.ɛɕ).
The .class map is the (partial) map between objects given by
x.class = y
iff
y is the (unique) bottom of x.ϵ¯∩C,
that is, x.class equals the least power-container of x
that is a class
(whenever such an object exists).
If x.class = y then we say that
y is the class of x.
For an integer i we denote .class(i) the i-th power of
.class defined in the usual way. That is,
for i > 0,
.class(i) is the i-th composition of .class with itself,
for i = 0, .class(i) is
the identity map between objects,
for i < 0,
.class(i) is the inverse of .class(-i).
The power instance-of relation is the range-restriction of ϵ¯
to the set C of classes.
That is, for every objects x, y,
x is a power instance of y
↔
x ϵ¯ y ∈C.
As a subrelation of the power instance-of relation,
.class is also called the direct power instance-of relation.
For an object y,
{y}.class(-1) is the set of direct power instances of y.
The instance-of relation is the range-restriction of ϵ
to C.
Note:
The above definition of .class is tailored to the case of
monotonic structures
in which (ϵ¯) = (ϵ).
We did not introduce the non-monotonic class map correspondent to direct instance-of.
Observations:
Assume axioms of pre-basic structures.
Every object is a (power) instance of the inheritance root r.
The .class map is monotonic.
That is, for every objects x, y
such that both x.class and y.class are defined,
x ≤ y
→
x.class ≤ y.class.
Consistency and completeness
This section is related to miscelaneous consistency conditions
which express whether
an
ϵ5-structure
resembles
an (ϖ+1)-superstructure
(that is, a complete structure of ϵ)
in a particular respect.
ϵ-rank and ∊-rank
The following auxiliary rank functions are based solely on ϵ / ∊.
For an object x, let
r_{ϵ}(x) be the ϵ-rank of x
defined recursively by
r_{ϵ}(x) =ϖ∧
sup {r_{ϵ}(a) + 1 | a ϵ x }
if x is well-founded in ϵ,
r_{ϵ}(x) =ϖ
otherwise,
r_{∊}(x) be the ∊-rank of x
defined recursively by
r_{∊}(x) = sup {r_{∊}(a) + 1 | a ∊ x }.
That is,
r_{ϵ}(x) is the
ϖ-limited rank of x
w.r.t. ϵ,
and
r_{∊}(x) is the rank
of x
w.r.t. the well-founded relation ∊.
Moreover, for a set X of objects we let
r_{∊}(X) be the ∊-rank of X
defined as
sup {r_{∊}(a) + 1 | a ∈ X }.
We say that an object x is ∊-ranked
(resp. ϵ-ranked)
if x.d = r_{∊}(x)
(resp. x.d = r_{ϵ}(x)).
An ϵ5-structure is ∊-ranked
(resp. ϵ-ranked)
if so is every its object.
Observations:
For every object x,
r_{∊}(x) ≤ r_{ϵ}(x) ≤ x.d.
(In particular, being ∊-ranked implies being ϵ-ranked.)
If
r is ∊-ranked
then r_{∊}(O) =ϖ+1.
Groundedness
An object x is said to be
ϵ-grounded
if x ∈T.ϵ^{∗},
that is, u ϵ^{i} x for some terminal u and natural i,
∊-grounded
if x ∈T.∊^{∗},
that is, u ∊^{i} x for some terminal u and natural i.
The whole ϵ5-structure S is
ϵ-grounded if T.ϵ^{∗}=O
and
∊-grounded if T.∊^{∗}=O.
Proposition A:
Assume
(a) (ϵ) = (∊) ∪ (ϵ¯),
(b) (≤) ○ (ϵ¯) ⊆ (ϵ¯), and
(c) T⊆O.∍.
(All of (a)–(c) are satisfied in basic structures.)
Assume in addition that
(d) (∊) = (∊) ○ (∊^{0}).
For every natural i,
∊^{i} is the domain-restriction of ϵ^{i} to bounded objects.
Corollary: If S is ϵ-grounded
then
S is ∊-grounded.
(More specifically, if S is ϵ-grounded
then
T.ϵ^{i}=T.∊^{i} for every natural i.)
Proof:
For i equal 0 or 1 the statement holds by definition.
We further proceed by induction.
Assume that i > 0, x ϵ^{i} y ϵ z and x is bounded.
We should prove that x ∊^{i+1} z,
that is, x.∊^{i}∩ z.∍ is non-empty.
By induction assumption, x ∊^{i} y, so that
x ∊^{i-1} a ∊ y for some a.
By (c) there exist bounded b such that
a ∊ b ≤ y.
If y is bounded then y ∈ x.∊^{i}∩ z.∍.
If y is unbounded then, by (a), y ϵ¯ z, therefore
b ∊ z, and consequently,
b ∈ x.∊^{i}∩ z.∍.
Proposition B:
Assume
(b~2),
(b~3)
and O=r.϶=r.϶¯.
Then for every object x
the following are satisfied:
x is ϵ-grounded
→
x.mli < ω.
Corollary:
S is ϵ-grounded
→
S satisfies (b~10).
For and every natural i,
x is ϵ-grounded
and
i < x.mli
→
∅≠ x.϶^{i}≤r.
x is ϵ-grounded
and
i ≤ x.mli
→
∅≠ x.϶^{i}.
Proof:
Let x be and object.
We show that: x.mli = ω →
x ∉T.ϵ^{∗}.
Observe that
x.mli = ω↔
for every natural j, x ϵ^{1-j}r,
for every object u and every natural i, j:
i < j and u ϵ^{i} x ϵ^{1-j}r
→
u ϵ^{1+i-j}r
→
u ≤r.
As a consequence,
if x.mli = ω then
x.϶^{i}≤r for every natural i, i.e.
x ∉T.ϵ^{∗}.
To show (a),
assume that x is ϵ-grounded.
If i < x.mli then x ϵ¯^{-i}r
and therefore x.϶^{i}≤r
(by (ϵ^{i}) ○ (ϵ¯^{-i}) ⊆ (≤)).
Since x.϶^{∗}≰r by
ϵ-groundedness of x
it follows that
x.϶^{i} must be non-empty.
The (b) statement follows from (a).
Extensional consistency
An object x is said to be extensionally consistent
if for every object y,
∅≠ x.∍⊆ y.∍
→
x ≤ y.
The whole structure is extensionally consistent if so is every its object.
That is, assuming the subsumption rule (∊) ○ (≤) ⊆ (∊),
S is extensionally consistent iff
the following equivalence is satisfied for every object x, y:
x ≤ y↔x = y or ∅≠ x.∍⊆ y.∍.
∊-levelling
We say that an object x is
∊-levelled if x is
∊-grounded
(i.e. x ∈T.∊^{∗}) and
x.mli =
min { i | x ∈T.∊^{i}, i ∈ℕ },
that is, (assuming T=T.∊^{0})
the metalevel index of x
equals the length of the shortest ∊-path from
T to x.
The whole structure is
∊-levelled if all its objects are
∊-levelled.
Observations:
In a basic structure, an object x is ∊-levelled
↔
x ∈T.∊^{i}
where i = x.mli.
For a basic structure S the following are equivalent:
S is ∊-levelled.
Every non-terminal object z
has a bounded member u
such that u.mli + 1 = z.mli.
(That is, for every non-terminal z,
z.mli ≰ z.∍.mli.)
For every natural i,
the i-th metalevel equals T.∊^{i}∩r.϶^{1-i}.
Proposition:
If S is
basic structure that is
extensionally consistent
and powerclass complete
then
S is ∊-levelled.
Proof:
Assume that S is as in the antecedent of the proposition.
Since the reduced helixR
is complete it follows that for every natural i and every object x,
x.mli ≥ i+1↔x ≤r.ec(i).
Assume for a contradiction that z is a non-terminal object
such that
z.mli ≤ z.∍.mli.
Let y be the top of the (z.mli+1)-th metalevel,
i.e.
y =r.ec(i) where i = z.mli.
Then for every bounded object u,
u ∈ y.∍ ↔ z.mli ≤ u.mli.
Therefore,
z.∍⊆ y.∍.
By extensional consistency, z ≤ y,
so that z.mli ≥ z.mli+1
– a contradiction.
The diagram on the right shows a powerclass complete structure in which
the z object (or any object from z.ec^{∗})
is not ∊-levelled.
The dashed green arrows indicate pairs (x,y) for which the
condition of extensional consistency is violated.
An object x is said to be powerclass consistent if
x is powerclass-like
→x is a powerclass.
The whole structure S is powerclass consistent if so is every its object.
Example.
The diagram on the right shows a basic structure
that is (a) powerclass complete and (b)
extensionally consistent
but
in which the m object is not powerclass consistent.
(Horizontal lines indicate the .ec map, with implicit left-to-right
orientation.)
For every natural i, z_{i}.↧
(= z_{i+1}.϶= z_{i+1}.∍)
is the set of objects x from m.϶
such that x.d ≤ i+1.
Therefore, if X is a bounded subset of m.϶
and i is such that X.d ≤ i then
X ≤ z_{i+1}ϵ m,
thus X.△∩ m.϶ is non-empty.
Observations:
Assume that S is a basic structure.
If S is extensionally consistent and powerclass complete then:
x.϶= x.϶¯ ↔
x.϶ is a downset w.r.t. ≤.
(Thus, (ⅰ) and (ⅱ)
say that x.϶ is a boundedly complete ideal.)
Every singleton is powerclass consistent (and so is every terminal object).
If S is powerclass complete and extensionally consistent
(as is the case of the above example)
then every bounded object is powerclass consistent.
Combined consistency
For convenience, we introduce terminological shorthands.
For every object x,
Similarly, an ϵ5-structure S is e+p consistent
if so is every its object.
Free leaf
We define a partial map .ԏ between objects by
x.ԏ= y
↔
(a)
x is terminal,
(b)
y is a primary object whose metalevel index
k = y.mli is finite, and (y is unique such that:)
(c)
x.ϵ¯^{i}= y.ϵ¯^{i-k}⊎ {x}.ec(i)
for every natural i.
If x.ԏ= y then we say that
x is a direct free leaf of y.
We use the .ԏ(-1) for the inverse of .ԏ
so that {y}.ԏ(-1) is the set of direct free leaves of y.
An object x for which x.ԏ is defined is called
a free leaf.
Proposition A:
Assume axioms of pre-basic structures and O.mli < ω.
For every terminal object x there exists at most one pair (y,k)
from O.pr ×ℕ
such that (c) is satisfied.
Moreover, if (y,k) is such a pair then k = y.mli.
Proof:
Assume that x is a terminal object,
(y,k) is from O.pr ×ℕ and
satisfies (c) and
that (z,ℓ) is an alternative pair to (y,k).
Then it follows from
x ϵ¯^{k} y that z ϵ¯^{k-ℓ} y.
Similarly,
x ϵ¯^{ℓ} z → y ϵ¯^{ℓ-k} z
so that (y,z) ∈ (ϵ¯^{ℓ-k}) ∩ (϶¯^{k-ℓ}) = .ec(ℓ-k).
Since both y and z are primary it follows that
(y,k) = (z,ℓ).
To show that y.mli = k observe that for every integer i,
y.mli > k-i
↔ y ϵ¯^{i-k}r
↔ x ϵ¯^{i}r
↔ i > 0.
The first equivalence is by definition of .mli,
the second one is by (c) and the last one is by x being terminal.
Proposition B:
Assume axioms of basic structures.
If x.ԏ= y and k = y.mli then
x.ec(k-1) is defined
(in particular, y is non-terminal).
If x.ԏ= y, k = y.mli
and n is a natural number such that both
a = x.ec(k+n-1)
and
z = y.ec(n) exist then
a.∊= z.↥⊎ {a}.ec.
For every object z, if ℓ = z.mli
and z.pr.ԏ(-1).ec(ℓ-1) has at least 2 elements
then
z is e+p consistent
(that is, z is both extensionally consistent and powerclass consistent).
Proof:
Assume x.ԏ= y and denote k = y.mli.
Observe first that k ≠ 0 (i.e. y is non-terminal)
since otherwise x.↥= y.↥⊎ {x} = {x} ⊎ {x},
a contradiction.
It follows that x ϵ¯^{k} y for some k > 0
and thus x ϵ¯^{k-1} u ϵ¯ y for some object u.
By definition of .ԏ,
x.ϵ¯^{k-1}= y.ϵ^{-1}⊎ {x}.ec(k-1)
so that either u ∈ y.ϵ^{-1}∩ y.϶¯ or u = x.ec(k-1).
The former case is equivalent to u.ec = y which is disallowed by
the definition of .ԏ.
Let x, y, z, a, k and n be as in the
antecedent of the proposition.
It follows by boundedness of a and by
(b~7)(b) that
a.∊= a.ϵ= a.ϵ¯
and thus
a.∊= x.ϵ¯^{k+n}= y.ϵ¯^{n}⊎ {a}.ec
= z.↥⊎ {a}.ec.
Let z and ℓ be as in the
antecedent of the proposition
and let a and b be two different objects from
z.pr.ԏ(-1).ec(ℓ-1).
By the previous proposition,
a.∊= z.↥⊎ {a}.ec and
b.∊= z.↥⊎ {b}.ec.
It follows that for every object u,
z.∍⊆ u.∍→{a,b} ⊆ u.∍→z ≤ u.
This shows that z is extensionally consistent.
To show the powerclass consistency of z,
assume that z is primary and denote X = {a,b}.
Since a.↥= z.ϵ^{-1}⊎ {a}
and
b.↥= z.ϵ^{-1}⊎ {b} it follows
that
X.△= a.↥∩ b.↥= z.ϵ^{-1}
and thus
X.△∩ z.϶¯= z.ϵ^{-1}∩ z.϶¯=∅,
i.e.
X has no upper bound in z.϶
so that z is not powerclass-like.
Completeness
(Assume that S is an ϵ5-structure such that
O=O.϶.)
We say that S is
powerclass complete
if x.ec is defined for every object x,
singleton complete
if x.ɛϲ is defined for every object x from
O.∍,
metaobject complete
if S is both powerclass complete and singleton complete,
pre-complete
if S is a (ⅰ) basic structure
that is
(ⅱ) metaobject complete,
(ⅲ) extensionally consistent,
(ⅳ) powerclass consistent, and
(ⅴ)
∊-ranked,
extensionally complete
if
for every subset X of O.∍ there is an object x
such that x.∍= X,
complete
if S is pre-complete and extensionally complete.
Pre-basic structure
Many important properties of basic structures
are consequences of a weaker collection of conditions than
(b~1)–(b~11).
Such a weaker collection is singled out in this section to form
the family of pre-basic structures.
Pre-basic structure
By a pre-basic structure (of ϵ)
we mean an
ϵ5-structureS=(O, ϵ, ϵ¯^{⁽*⁾}, r, .ec, .ɛɕ)
satisfying the following axioms:
(ϐ~1)
(ϵ¯) ⊆ (ϵ).
(ϐ~2)
(ϵ¯^{i}) ○ (ϵ¯^{j}) ⊆ (ϵ¯^{i+j})
for every integer i, j.
(ϐ~3)
(ϵ) ○ (ϵ^{i}) ⊆ (ϵ^{1+i})
for every integer i.
(ϐ~4)
(ϵ¯^{i}) ∩ (϶¯^{-i}) = .ec(i)
for every integer i.
(ϐ~5)
O.ϵ≤r.
(ϐ~6)
O.϶¯=O.
(ϐ~7)
T.϶^{-i}∩r.϶^{-i}=∅
for every natural i.
Observations:
rϵ¯r,
O=r.϶=r.϶¯.
Every basic structure is a pre-basic structure.
Terminal objects are those that have rank 0.
Proof:
For every terminal x,
x.϶=∅ and
(as a consequence)
x.϶^{∗}.϶^{-∗}= x.϶^{-∗}
so that
x.d = sup {a.mli - i | a ∈ x.϶^{-i} }.
Since by (ϐ~7),
a ϵ^{-i} x → a.mli ≤ i,
it follows that x.d = x.mli - 0 = 0.
Conversely, if x is non-terminal then x.mli > 0,
and (since z.mli ≤ z.d for every object z)
thus x.d > 0.
Groundedness vs ϵ-rank
Proposition:
In a pre-basic structure S, the following are equivalent:
S is ϵ-ranked.
(I.e. x.d = r_{ϵ}(x) for every object x.)
For every object x that is well-founded in ϵ,
x.mli < ω (x has finite metalevel index),
and
x ∈T.ϵ^{∗}
(x is ϵ-grounded).
Proof:
ⅰ→ⅱ.
Assume (ⅰ) and
let
x be an object that is well-founded in ϵ.
By definition of well-foundedness,
there is an object a from x.϶^{∗} such
that r_{ϵ}(a) = 0.
Since
r_{ϵ}(a) = 0
↔
a.d = 0
↔
a ∈T
(the latter equality is by the observation made for pre-basic structures)
it follows that
x ∈T.ϵ^{∗}.
The
x.mli < ω condition then
follows by
proposition B1.
ⅰ←ⅱ.
Assume (ⅱ).
By simple observation,
(in any pre-basic structure)
the set of objects that are well-founded in ϵ
is closed w.r.t. .϶^{∗}.϶^{-∗},
so that (a) is equivalent to
(a')
a.mli < ω for every
a ∈ x.϶^{∗}.϶^{-∗}.
This simplifies the
definition of .d so that
for every object x,
x.d = r_{ϵ}(x)
=ϖ
if x is non-well-founded in ϵ.
Otherwise (if x is well-founded):
x.d
=ϖ∧
(sup {a.d + 1 | a ϵ x}
∨
sup {a.mli + i-j |
a ∈ x.϶^{i}.϶^{-j}, i,j ∈ℕ}),
r_{ϵ}(x)
=ϖ∧
sup {r_{ϵ}(a) + 1 | a ϵ x}.
Let
x be an object that is well-founded in ϵ
and denote A, B and A' the respective sets
over which the suprema in the definitions of
x.d and r_{ϵ}(x) are taken, thus
x.d
=ϖ∧ (sup(A) ∨ sup(B)),
r_{ϵ}(x)
=ϖ∧ sup(A').
By well-founded recursion we can assume that
a.d = r_{ϵ}(a)
for every
a ϵ x,
so that A = A'.
As a consequence, r_{ϵ}(x) ≤ x.d.
It remains to show that r_{ϵ}(x) ≥ sup(B),
that is,
for every m ∈ B,
(∗)
there exists a pair (b,n) from O×ℕ
such that
(a) b ϵ^{n} x and (b) n ≥ m.
Let m be from B and
let (a,i,j) be a corresponding triple
from O×ℕ×ℕ
such that
a ∈ x.϶^{i}.϶^{-j} and m =a.mli + i-j.
Since a is ϵ-grounded,
there is a pair (b,k) from T×ℕ
such that b ϵ^{k} a.
Denote n = k + i-j.
Then the pair (b,n) is the requested pair
from (∗):
b ϵ^{n} x
(since k ≥ j – a consequence of b ∈T,
so that
b.ϵ^{k}.ϵ^{-j}.ϵ^{i}⊆ b.ϵ^{k-j}.ϵ^{i}⊆ b.ϵ^{k-j+i}),
The diagram on the right shows a basic structure that is metaobject complete
–
the powerclass map .ec (shown by horizontal blue arrows) is total
and the singleton map .ɛϲ (shown by blue arrows pointing to a circle
which indicates a singleton)
is defined on the set O.∍ of bounded objects.
We have already observed that (a) in a powerclass complete basic structure,
all powers of ϵ¯ are given by
and that (b) in a singleton complete basic structure the bounded membership ∊
is given by
(∊) = (.ɛϲ) ○ (≤).
Subsequently, using the last axiom of basic structures, the object membership
is given by (ϵ) = (∊) ∪ (ϵ¯).
Since (.ɛɕ) = (.ɛϲ) ∖ (.ec)
it follows that a metaobject complete basic structure is fully determined by
≤, .ec and .ɛϲ.
The following subsection provides an axiomatization based on these three constituents.
Metaobject structure
By a metaobject structure
we mean a structure
S= (O, ≤, r, .ec, .ɛϲ)
where
O is a set of objects,
≤
is the inheritance relation between objects,
r
is the inheritance root, a distinguished object,
.ec
is the powerclass map O→O
(objects from O.ec are powerclasses),
.ɛϲ
is the singleton (partial) map O↷O
(objects from O.ɛϲ are singletons).
Denote T= { x | x ≰ r } the set of terminal
objects
and let .ec^{∗} denote the reflexive transitive closure of
.ec.
The structure is subject to the following axioms (⁎):
(mo~1)
Inheritance, ≤, is a partial order.
(mo~2)
The powerclass map, .ec, is an order-embedding of
(O, ≤) into itself.
(mo~3)
Objects from T.ec^{∗}
are minimal in ≤.
(mo~4)
Every powerclass is a descendant of r.
(mo~5)
The set r.ec^{∗}
has no lower bound in ≤.
(mo~6)
The singleton map, .ɛϲ, is injective.
(mo~7)
Objects from O.ɛϲ.ec^{∗}
are minimal in ≤.
(mo~8)
For every objects x, y such that x.ɛϲ is defined,
x.ɛϲ ≤ y.ec↔x ≤ y.
(mo~9)
For every object x,
x.ɛϲ is defined ↔
x.d < ϖ.
(⁎)
The definitions introduced before the axioms are sufficient to
state all axioms except the last one.
The definition of the rank function, .d,
used in the last axiom
is provided below. Assume that
(mo~1)–(mo~8) are satisfied.
Let ∊, ϵ¯ and ϵ be relations between objects
with the following definition and terminology:
(∊)
=
(.ɛϲ) ○ (≤)
is the bounded membership,
(ϵ¯)
=
(.ec) ○ (≤)
is the power membership,
(ϵ)
=
(∊) ∪ (ϵ¯)
is the (object) membership.
For an integer i,
let .ec(i) be the i-th composition
of .ec with itself if i ≥ 0
(with .ec(0) being the identity on O)
and the -i-th composition of the inverse of .ec otherwise.
Similarly with .ɛϲ(i),
but
with .ɛϲ(0) being the identity on O.∍.
Subsequently, let ∊^{i}, ϵ¯^{i} and ϵ^{i}
be the i-th power of
∊, ϵ¯ and ϵ, respectively,
defined as follows:
(∊^{i})
=
.ɛϲ(i) ○ (≤),
(for i < 0 we let
(∊^{i}) = (∊^{0}) ○ .ec(i))
(ϵ¯^{i})
=
(≤) ○ .ec(i) ○ (≤),
(ϵ^{i})
=
(∊^{i}) ∪ (ϵ¯^{i}).
The metalevel index, x.mli,
and rank, x.d, of an object x are then
defined like in basic structures by
x.mli
= sup { i | x ϵ^{1-i}r, i ∈ℕ },
x.d
=ϖ
if x is non-well-founded in ϵ,
x.d
=ϖ∧
(sup {a.d + 1 | a ϵ x}
∨
sup {a.mli + i-j |
a ∈ x.϶^{i}.϶^{-j}, i,j ∈ℕ})
If S= (O, ≤, r, .ec, .ɛϲ)
is a metaobject structure then
the ϵ5-structure
S' = (O, ϵ, ϵ¯^{⁽*⁾}, r, .ec, .ɛɕ)
derived from S is basic structure that is
metaobject complete.
The new constituents of S' are defined as follows:
(ϵ)
=
((.ec) ∪ (.ɛϲ)) ○ (≤),
(= (ϵ¯) ∪ ((.ɛϲ) ○ (≤)), see the proof below)
(ϵ¯^{i})
=
(≤) ○ .ec(i) ○ (≤)
for every integer i ≤ 1,
((∗) and also for i > 1, see the proof below)
(.ɛɕ)
=
(.ɛϲ) ∖ (.ec).
If S= (O, ϵ, ϵ¯^{⁽*⁾}, r, .ec, .ɛɕ)
is a metaobject complete basic structure then
S' = (O, ≤, r, .ec, .ɛϲ)
is a metaobject structure.
The new constituents of S' are defined as follows:
(≤)
=
(ϵ¯^{0}),
(.ɛϲ)
=
(ϵ) ∩ (O×
(T∪O.ɛɕ).ec^{∗}).
(We can equivalently write T.ec instead of
T.)
Proof:
Let
S and
S'
be as in (Ⅰ).
Since (≤) ○ (.ec) = (.ec) ○ (≤)
by
(mo~2),
it follows that for every natural i,
(≤) ○ .ec(i) ○ (≤) is the i-th relational composition
of ϵ¯ with itself
(which equals ϵ¯^{i} by definitional extension of ϵ5-structures)
so that
(∗)
is satisfied for every integer i.
Moreover, it follows that for every natural i,
(ϵ¯^{i})
=
(≤) ○ .ec(i)
=
.ec(i) ○ (≤),
(ϵ¯^{-i})
=
(≤) ○ .ec(-i).
Subsequently,
(b~1)–(b~11)
are verified as follows.
(b~1) (i.e. (ϵ¯) ⊆ (ϵ))
follows by the definition of ϵ.
(b~2)
is satisfied as a consequence of transitivity of
≤ and compositional commutativity of ≤ and .ec
so that
(ϵ¯^{i}) ○ (ϵ¯^{j}) = (ϵ¯^{i+j})
for every integer i, j.
To prove (b~3) it is sufficient
(using the equality above) to prove that
(a)
(ϵ) ○ (≤) ⊆ (ϵ) and
(b)
(ϵ) ○ (ϵ^{-1}) ⊆ (≤).
(a) follows by transitivity of ≤.
To show (b) apply
definitions of ϵ and ϵ^{-1}
and (mo~8):
(ϵ) ○ (ϵ^{-1})
= ((ϵ¯) ∪ ((.ɛϲ) ○ (≤))) ○ (ϵ^{-1})
= ((ϵ¯) ○ (ϵ¯^{-1})) ∪ ((.ɛϲ) ○ (≤) ○ (ϵ^{-1}))
= (≤) ∪ ((.ɛϲ) ○ (≤) ○ .ec(-1))
= (≤)
(since by (mo~8),
{x}.ɛϲ.↥.ec(-1) ⊆ x.↥ for every object x).
Moreover, since (ϵ) ○ (ϵ^{-1}) = (ϵ) ○ .ec(-1)
it follows that
(≥) = (.ec) ○ (϶).
Axiom (b~4) follows by antisymmetry and reflexivity
of ≤.
For every natural i,
O.ec.↥≤r.
(Since for O.ec ∋ x < t ≰r,
t would be a non-minimal terminal object (⁎), violating
(mo~3).)
O.ɛϲ.↥≤r.
(Since O.ɛϲ ⊆O.ec.↧
by (mo~8).)
O.ϵ≤r.
(Since by definition of ϵ,
O.ϵ=O.ec.↥∪O.ɛϲ.↥.)
In (⁎), we have used the term terminal object
according to the definition in the metaobject structure S.
However,
the definitions of
the set T of
terminal objects in S and S'
are coincidient:
Since rϵr (as consequence of r.ec ≤r)
it follows that O.ϵ.↧=r.↧.
Axiom (b~6) (O.϶=O)
follows by x ϵ¯r for every object x
(a consequence of x.ec ≤r
asserted by (mo~4)).
To prove (b~7)(a)
(x.϶^{-i}= {x}.ec(i)
for every x from T∪O.ɛɕ and
every natural i),
assume that x ∈T∪O.ɛɕ.
Then for every natural i,
(since (.ɛɕ) ⊆ (.ɛϲ)
and thus O.ɛɕ ⊆O.ɛϲ
so that x.ec(i) is from
(T∪O.ɛϲ).ec^{∗}
and therefore
x.ec(i) is minimal in ≤ due to
(mo~3) and (mo~7)).
Let us prove (b~7)(b),
i.e.
if X denotes the set
(T∪O.ɛɕ).ec^{∗}
then
x.ϵ= x.ϵ¯ for every x ∈ X.
That is, we have to show that
{x}.ɛϲ.↥⊆ x.ec.↥
for every x ∈ X.
Since {x}.ɛϲ ≤ {x}.ec for every object x
(by (mo~8)),
it follows that
{x}.ɛϲ.↥⊈ x.ec.↥
↔
x.ɛϲ is defined and x.ɛϲ < x.ec.
By (mo~3) and (mo~7),
the last strict inequality
is disallowed for objects x from
(T∪O.ɛϲ).ec^{∗}
and
thus for all x from X.
Let us prove (b~8), that is,
assume that x, y are objects such that x.ɛɕ = y
and show that
(a)
{x} = y.϶,
(b)
x.ϵ^{i}= y.ϵ^{i-1} for every i ≤ 1,
(c)
(x,y) ∉ (.ec).
The (c) property
follows by definition of .ɛɕ.
Since (.ɛɕ) ⊆ (.ɛϲ),
it follows in the first place that x.ɛϲ = y.
By definition, x.ϵ= {x}.ɛϲ.↥∪ x.ec.↥.
Since x.ɛϲ is defined and x.ɛϲ ≤ x.ec
it follows that
(b) is satisfied for i = 1.
For i ≤ 0, use
x ≤ u.ec(-i)↔x.ɛϲ ≤ u.ec(-i).ec
which holds by (mo~8) for every object u.
To show (a), assume that u is an object such that u ϵ y,
that is, (by definition of ϵ)
u.ɛϲ ≤ y or u.ec ≤ y.
By
(mo~7),
y is minimal in ≤ so that u.ɛϲ = y or
u.ec = y.
The latter case is impossible because
u.ec = y ϶ x implies x ≤ u
(by (≥) = (.ec) ○ (϶) proved before)
so that either
x = u and thus (x,y) ∈ (.ɛɕ) ∩ (.ec)
– which contradicts the definition
(.ɛɕ) = (.ɛϲ) ∖ (.ec), or
x < u which would imply x.ec < u.ec = y
– which would in turn violate the minimality of y.
It follows
that u.ɛϲ = y and therefore,
since (mo~6) asserts the injectivity of .ɛϲ,
u = x.
That is, (a) is satisfied.
To show (b~10)
(x.mli is finite for every object x),
apply (϶¯^{1-i}) = .ec(i-1) ○ (≥)
to the definition of .mli:
x.mli = sup {i | x ≤ {r}.ec(i-1) }.
Since {r}.ec(i-1) ≤ {r}.ec(k-1)
for every natural k ≤ i
(by rϵr and .ec being an order embedding)
it follows that
x.mli is finite
↔
x ≰r.ec^{∗}.
That is, (b~10) is asserted by
(mo~5).
It remains to verify that (b~11) is satisfied.
Note that the rank functions in S and S'
are identical since they have identical prescriptions based
on ϵ and ϵ¯^{⁽*⁾}.
By (mo~9), for every object x,
x.ɛϲ is undefined
↔
x.d =ϖ.
Therefore,
if x.d =ϖ then
x.ϵ= x.ec.↥= x.ϵ¯.
Finally, it also follows that (.ɛϲ) ○ (≤) is the
domain-restriction of ϵ to objects x such that
x.d < ϖ,
so that the bounded membership relation ∊ in S
is coincident with that in S'.
Assume that
S= (O, ϵ, …) is a metaobject complete basic structure
and let
S' = (O, ≤, r, .ec, .ɛϲ)
be the correspondent reduct of a definitional extension of S.
Then
except for
(mo~5),
all of
(mo~1)–(mo~9)
either directly follow from the definition of a basic structure
or are obtained as observations that have already been made.
The (b~10)↔(mo~5)
correspondence
is established like in the proof of (Ⅰ) using
x.mli is finite
↔
x ≰r.ec^{∗}.
Grounded metaobject structure
Because metaobject structures are definitionally equivalent to
metaobject complete basic structures,
the definitions introduced for
ϵ5-structures apply to metaobject structures.
In particular, a metaobject structure S is grounded
iff T.ϵ^{∗}=O.
Since
(∊) = (∊) ○ (∊^{0})
(a consequence of (∊) = (.ɛϲ) ○ (≤)),
there is no distinction between
ϵ-groundedness and ∊-groundedness:
T.ϵ^{∗}=O
↔
T.∊^{∗}=O
(see proposition A2).
Moreover,
by
proposition B1,
the groundedness condition
makes
(mo~5)
(R.▽=∅)
redundant, and
by groundedness in pre-basic structures,
every grounded basic structure is ϵ-ranked.
As a consequence,
grounded metaobject structures can be axiomatized with
(mo~5)
and
(mo~9) replaced as follows:
(mo~5)'
T.ϵ^{∗}=O.
(mo~9)'
For every object x,
x.ɛϲ is defined ↔
r_{ϵ}(x) < ϖ.
Monotonic structures
Basic structures in which (ϵ¯) = (ϵ)
are monotonic.
Most object models in object oriented programming
have core parts that
can be considered to be specializations of monotonic basic structures.
The (ϵ¯) = (ϵ) equality yields the monotonicity of ϵ:
(≤) ○ (ϵ) ⊆ (ϵ),
that is,
for every objects x, y,
x ≤ y → x.ϵ⊇ y.ϵ.
If every object x has a least container
(as is the case of object models in OOP),
i.e. x.ϵ= x.lc.↥ for a map .lc,
then the monotonicity condition further translates to
x ≤ y → x.lc ≤ y.lc.
There are two distinguished subfamilies of monotonic structures
given by whether .ec
is empty or total:
Notes:
In
[]
and [],
core parts of objects models of
Ruby, Python, Java, Scala, Smalltalk-80, Objective-C, CLOS, Perl and JavaScript
are considered to
be specializations of monotonic structures.
However, only Ruby and Python can be regarded as fully conformant to
the description provided below.
In Smalltalk-80 and CLOS, there are built-in monotonicity breaks.
In languages with a partial support of .ec
(Smalltalk-80, Objective-C and Scala),
there is no notational or terminological distinction between
.ec and .class.
In Java and Scala
(which are not generally considered to belong to dynamic programming languages)
there is no established consensus about whether
these languages support the classes are objects paradigm.
To capture the core structure of JavaScript
[],
one needs to introduce prototypes as additional
objects on the 0th metalevel which are
powerclass predecessors
of objects from the 1st metalevel.
where ϵ is exactly what can be detected by the
isa introspection method.
As a consequence, every non-terminal object x is circular: x ϵ x.
However, there is no evidence (known to the author)
whether the above equality between
membership and inheritance is a designed feature of the language.
Objective-C supports multiple inheritance roots
(Object, NSObject and NSProxy).
As of Pharo 1.3,
Smalltalk-80 contains a subsidiary inheritance root (PseudoContext).
Monotonic structure
ϵ
≤
ϵ^{-1}
ϵ^{-2}
⋮
By a monotonic structure (of ϵ)
we mean a structure
S= (O, ϵ^{⁽*⁾}, r, .ec)
where
O is a set of objects,
ϵ^{⁽*⁾}
is a sequence { ϵ^{i}| i ∈ℤ, i ≤ 1 }
of relations between objects,
with ϵ^{0} and ϵ^{1} distinguished:
(≤) = (ϵ^{0}) is the inheritance relation
(with .↧ / .↥ used for preimages / images under ≤),
(ϵ) = (ϵ^{1}) is the (object) membership relation,
r
is the inheritance root, a distinguished object,
.ec
is the (partial) powerclass map O↷O
(objects from O.ec are powerclasses).
(Alternative terminology:
.ec is the eigenclass map
and
objects from O.ec are eigenclasses.)
For every natural i > 0, let
ϵ^{i} be the i-th relational composition of ϵ
with itself,
.ec(i) be the i-th composition of .ec with itself,
and
.ec(-i) be the inverse of .ec(i).
Let .ec(0) be the identity on O.
Let
T=O∖O.ϵ.↧
be the set of terminal objects.
Let the metalevel index of an object x
be denoted and defined like in basic structures, i.e.
x.mli = sup {i | x ϵ^{1-i}r, i ∈ℕ}.
The structure is subject to the following axioms:
(m~1)
(ϵ^{i}) ○ (ϵ^{j}) ⊆ (ϵ^{i+j})
for every integer i, j.
(m~2)
(ϵ^{i}) ∩ (϶^{-i}) = .ec(i)
for every integer i.
(m~3)
O.ϵ≤r.
(m~4)
O=O.϶.
(m~5)
For every object x from T
and every natural i,
{x}.ec(i) = x.϶^{-i}.
(m~6)
For every object x,
the metalevel index x.mli is finite.
The correspondence
Proposition:
If S= (O, ϵ, ϵ¯^{⁽*⁾}, r, .ec, .ɛɕ)
is a basic structure then its reduct
S' = (O, ϵ¯^{⁽*⁾}, r, .ec)
is a monotonic structure of ϵ.
If S= (O, ϵ¯^{⁽*⁾}, r, .ec)
is a monotonic structure of ϵ then
S' = (O, ϵ, ϵ¯^{⁽*⁾}, r, .ec, .ɛɕ),
where (ϵ) = (ϵ¯) and (.ɛɕ) =∅,
is a basic structure.
Corollary:
There is a one-to-one correspondence between
monotonic structures of ϵ and
monotonic basic structures
– i.e. basic structures in which (ϵ) = (ϵ¯).
(Consequently, .ɛɕ is empty.)
Monotonic primary structure
By a monotonic primary structure of ϵ we mean a
monotonic structure in which every object is primary,
i.e. in which the powerclass map .ec is empty.
Such a structure can be axiomatized in the signature
(O, ϵ^{⁽*⁾}, r)
as follows:
(mp~1)
The same as (m~1).
(mp~2)
(a) ≤ is reflexive and antisymmetric and
(b) (ϵ^{i}) ∩ (϶^{-i}) =∅ for every i > 0.
(mp~3)
The same as (m~3).
(mp~4)
The same as (m~4).
(mp~5)
Terminal objects are minimal in ≤.
(mp~6)
The same as (m~6).
Observation:
There is a one-to-one correspondence between
By a membership-based monotonic structure
(alternatively, ϵ-based m. s.)
we mean a monotonic primary structure in which the negative powers of ϵ
are given by ϵ and ≤.
We offer three different prescriptions for ϵ^{-k},
k > 0.
Recall that H=r.ϵ^{∗} is the set of
helix objects.
(mp-α)
x ϵ^{-k} y
↔
x < y.ϵ^{k} and
y.ϵ^{k}≠ y.ϵ^{k-1},
(mp-β)
x ϵ^{-k} y
↔
x < y.ϵ^{k} and
y.ϵ^{k}≠ y.ϵ^{k-1} and
y ∈H,
(mp-γ)
x ϵ^{-k} y
↔
x < r.ϵ^{k+i} and
r.ϵ^{k+i}≠r.ϵ^{k+i-1} and
rϵ^{i} y for some natural i.
Observations:
Recall that
h is the
helix number
defined by
h= sup {i + 1 |r.ϵ^{i}≠r.ϵ^{i-1}, i ∈ℕ}
and that Y.⋁ is the set of
strict lower bounds of Y.
Assume that either of
(mp-α)–(mp-γ)
is imposed.
Condition (mp~2)(b),
(϶^{-k}) ∩ (ϵ^{k}) =∅ for every k > 0,
can be left out since it is implicitly satisfied.
If h is finite (in particular, if H is finite)
then
x.mli is finite by definition of ϵ^{-k}
so that
(mp~6) can be left out.
Let X be the i-th metalevel for a natural i > 1.
Then
X
=r.ϵ^{i-1}.⋁∖r.ϵ^{i}.⋁
if i < h,
X
=H.⋁
if i =h,
X
=∅
if i > h.
Prescriptions
(mp-β) and (mp-γ)
only allow helix objects in the range of
ϵ^{-k}
for every k > 0.
Example
The left diagram below shows an ϵ-based monotonic structure
S_{0}= (O_{0}, ϵ▫, ≤_{0}, r)
in which all helix objects (shown in blue) have metalevel index 1.
Negative powers of ϵ are given by any of
(mp-α)–(mp-γ).
Since
b < r.ϵ^{2}= {r,u,v} ≠ {r,u} =r.ϵ^{1}
and r.ϵ^{3}=r.ϵ^{2}
it follows that the metalevel index of b equals 3.
The right diagram shows a monotonic basic structure
S= (O, ϵ, …)
that is a powerclass extension of a basic structure definitionally
equivalent to S.
Negative powers of ϵ in S are powerclass based,
i.e.
(ϵ^{-i}) = (≤) ○ .ec(-i)
for every positive natural i.
Extensions of ϵ▫ and ≤_{0} is
to ϵ and ≤, respectively, are given by
a.ec(i) ϵ b.ec(j)
iff
a ϵ▫^{1+i-j} b,
a.ec(i) ≤ b.ec(j)
iff
a ϵ▫^{i-j} b,
where a, b are (primary) objects from O
and i, j are natural numbers such that
a.ec(i) and b.ec(j) are defined
(see powerclass completion
of basic structures).
Observations:
There can be gaps in metalevels.
(Consider the left diagram without the a object.)
The ϵ^{-2} relation is not decomposable in S_{0}
since
b ϵ^{-2}r but there is no x such that
b ϵ^{-1} x ϵ^{-1}r.
The (mp-γ) prescription
Since the
helix number h
is such that
r.ϵ^{j}≠r.ϵ^{j-1} ↔
j < h
for every natural j,
the (mp-γ) prescription can be stated as:
x ϵ^{-k} y
↔
there is a natural i such that:
x < r.ϵ^{k+i} and
k+i < h and
rϵ^{i} y.
(mp-γ~1)
≤ is a partial order.
(mp-γ~2)
(ϵ) ○ (≤) ○ (ϵ) = (ϵ).
(mp-γ~3)
O.ϵ≤r.
(mp-γ~4)
O=O.϶.
(mp-γ~5)
Objects from T are minimal.
(mp-γ~6)
If h= ω then H.▽=∅.
(mp-γ~7)
r.϶^{-k}.϶⊆r.϶^{1-k}
for every k > 0.
The proposition below shows that
if (mp-γ) is imposed then
(mp~1) and
(mp~2) can be equivalently replaced by
a conjunction of the following conditions:
≤ is a partial order,
(ϵ) ○ (≤) ○ (ϵ) = (ϵ)
(subsumption and monotonicity of ϵ),
x.ϵ.mli ≤ x.mli + 1 for every object x.
(The metalevel increment along ϵ it at most 1.)
The resulting axiomatization of structures
(O, ϵ, ≤, r)
is shown in the box on the right.
(Use the usual definitions of
ϵ^{i} for i ≥ 0,
ϵ^{∗},
T,
H,
and .▽.
For negative powers of ϵ, use
the original (mp-γ) prescription
and then apply the definition of h which refers to
r.ϵ^{-1}.)
Proposition:
Assume that the prescription (mp-γ) applies.
The (m~1) axiom,
(ϵ^{i}) ○ (ϵ^{j}) ⊆ (ϵ^{i+j})
for every integer i, j,
is asserted by the following conditions:
(≤) ○ (≤) ⊆ (≤)
(transitivity of ≤),
(ϵ) ○ (≤) ⊆ (ϵ)
(subsumption of ϵ),
(≤) ○ (ϵ) ⊆ (ϵ)
(monotonicity of ϵ),
(◈)
(ϵ) ○ (ϵ^{-i}) ⊆ (ϵ^{1-i})
for every positive natural i.
The (◈) condition
can be equivalently replaced by any of the following:
(ⅰ)
For every object x,
x.ϵ.mli ≤ x.mli + 1.
(That is, x ϵ y →
y.mli ≤ x.mli + 1.)
(ⅱ)
For every natural k,
r.϶^{-k}.϶⊆r.϶^{1-k}.
(That is, x ϵ y ϵ^{-k}r →
x ϵ^{1-k}r.)
Proof:
Let k, ℓ be positive natural numbers.
We show that
Let x, y, z be such that x ≤ y ϵ^{-k} z,
i.e.
for some natural i such that i + k < h,
x ≤ y < r.ϵ^{i+k}
and rϵ^{i} z.
Then x < r.ϵ^{i+k}, so that x ϵ^{-k} z.
Let x, y, z be such that x ϵ^{-k} y ϵ z,
i.e.
for some natural i such that i + k < h,
x < r.ϵ^{i+k},
and rϵ^{i} y ϵ z.
Then (i+1) is a natural number such that
(α) rϵ^{(i+1)} z,
(β) x < r.ϵ^{(i+1)+(k-1)}, and
(γ) (i+1) + (k-1) < h.
As a consequence, x ϵ^{1-k} z.
(For k = 1 it follows from (α) and (β)
that x < z.)
Let x, y, z be such that x ϵ^{-k} y ≤ z,
i.e.
for some natural i such that i + k < h,
x < r.ϵ^{i+k}
and rϵ^{i} y ≤ z.
Since (ϵ^{i}) ○ (≤) equals (ϵ^{i}),
we obtain
rϵ^{i} z
so that
x ϵ^{-k} z.
Let x, y, z be such that
x ϶^{-k} y ϶^{-ℓ} z,
i.e.
for some natural i, j such that
i + k < h and
j + ℓ < h,
rϵ^{i} x,
rϵ^{j} y < r.ϵ^{i+k},
and
z < r.ϵ^{j+ℓ}.
As a consequence of the underlined condition,
i+k < j so that z < r.ϵ^{i+k+ℓ}.
By definition of the metalevel index,
u.mli > i ↔ u ϵ^{-i}r
for every natural i,
so that for every objects x, y,
y.mli ≤ x.mli+1↔
for every k, if y ϵ^{-k}r then x ϵ^{1-k}r.
This shows (ⅰ)↔(ⅱ).
Since
(◈)→(ⅱ) trivially it remains to show
that (◈)←(ⅱ).
Assume therefore that (ⅱ) is satisfied and
let x, y, z be such that x ϵ y ϵ^{-k} z,
i.e.
for some natural i such that i + k < h,
x ϵ y < r.ϵ^{i+k}
and rϵ^{i} z.
Then y ϵ^{-i-k}r so that by (ⅱ)
x ϵ^{1-i-k}r
and thus
x < r.ϵ^{i+k-1}
and therefore
x ϵ^{1-k} z.
Structures with a canonical helix
In canonical primary structures
[]
the helix structure (H, ϵ, ≤, r)
looks like in the diagram on the right.
That is, helix classes are
(a) totally ordered by ≤,
(b) members of each other, and
(c) at least two in number.
As a consequence, r.ϵ^{0}≠r.ϵ^{1}=r.ϵ^{2}
– the helix number h equals 2.
It follows 2 is the highest possible metalevel index of an object,
whenever
any of
(mp-α)–(mp-γ)
is used
for the definition of ϵ^{-k}, k > 0.
Condition (mp-γ~7)
about the possible metalevel increment along ϵ
then reduces to
T.ϵ∩H.⋁=∅.
(That is, members of objects from the 2nd metalevel are non-terminal.)
However, the actual condition used in canonical primary structures reads
T.ϵ∩H.▽=∅.
(That is, metaclasses cannot have terminal objects as members.)
This is because the least helix class c,
the metaclass root,
(whose existence is asserted by an additional condition of finiteness of
C)
is considered to be extensionally equivalent to
r.ec, the powerclass of r in a powerclass extension.
Monotonic eigenclass structure
There are of course no problems with the definition of ϵ^{-k},
k > 0
in the case opposite to that of the previous subsection.
If .ec is total,
then (ϵ^{i}) = (≤) ○ .ec(i) ○ (≤)
for every integer i.
This simplifies the axiomatization.
By a monotonic eigenclass structure
we mean a structure
S= (O, ≤, r, .ec)
such that
O is a set of objects,
.ec
is the powerclass / eigenclass map O→O,
≤
is the inheritance relation between objects,
r
is the inheritance root, a distinguished object,
(≤) = (.ec) ○ (≤) ○ (.ce),
where .ce is the inverse of .ec.
(e~3)
(O×T.ec^{∗}) ∩ (<) =∅,
where T=O∖r.↧
and
.ec^{∗} is the reflexive transitive closure of .ec.
(e~4)
O.ec ≤r.
(e~5)
r.ec^{∗}.▽=∅.
(The set r.ec^{∗} has no lower bounds w.r.t. ≤.)
Proposition:
There is a one-to-one correspondence between
monotonic eigenclass structures and
monotonic basic structures that are powerclass complete.
Powerclass-based structures
Another family of basic structures which we consider as distinguished
is that which has no primary singletons
(and thus T.ec.ec^{∗} are the only singletons)
and in which power membership and anti-membership are based
on .ec:
(ϵ¯)
=
(≤) ○ (.ec) ○ (≤) ∪
(ϵ) ∩ (T×O),
(ϵ^{-1})
=
(≤) ○ (.ce) ○ (≤)
and for i > 1, (ϵ^{-i})
is the i-th relational composition of ϵ^{-1} with itself.
Powerclass-based structure
By a powerclass-based structure (of ϵ)
we mean a structure
(O, ϵ, ≤, r, .ec)
where
O is a set of objects,
ϵ
is the membership relation between objects,
≤
is the inheritance relation between objects,
(with .↧ / .↥ used for preimages / images under ≤),
r
is the inheritance root, a distinguished object,
.ec
is the powerclass map, a partial map between objects.
Let .ec^{∗} be the
reflexive transitive closure of .ec,
let .ce be the inverse of .ec.
Let
T=O∖O.ϵ.↧
be the set of terminal objects.
The structure is subject to the following axioms.
The definitions of .mli, .d and ϵ¯
used in the last two axioms
are provided subsequently.
The inheritance root r is the top of
O.ϵ, w.r.t. ≤.
(pb~5)
Every object has a container, O=O.϶.
(pb~6)
Objects from T.ec^{∗} are minimal in ≤.
(pb~7)
For every object x,
the metalevel index x.mli is finite.
(pb~8)
For every object x,
x.d =ϖ
→
x.ϵ= x.ϵ¯.
The power membership, ϵ¯, and its -1-st power,
ϵ¯^{-1}, are relations between objects defined by
(ϵ¯)
=
(≤) ○ (.ec) ○ (≤) ∪
(ϵ) ∩ (T×O),
(ϵ¯^{-1})
=
(≤) ○ (.ce) ○ (≤).
Subsequently, for every integer i, the i-th power of ϵ¯
is defined as follows.
For i > 0,
ϵ¯^{i}
equals
the i-th relational composition of ϵ¯ with itself.
ϵ¯^{0}
equals
≤.
For i < 0,
ϵ¯^{i}
equals
the -i-th relational composition of ϵ¯^{-1} with itself.
Let ϵ^{i}, for an integer i,
be i-th power of ϵ,
defined to be equal to the
i-th relational composition of ϵ with itself if i > 0
and
equal to ϵ¯^{i} for i ≤ 0.
The metalevel index and rank functions
.mli and x.d
are defined like in metaobject structures:
x.mli
= sup { i | x ϵ^{1-i}r, i ∈ℕ },
x.d
=ϖ
if x is non-well-founded in ϵ,
x.d
=ϖ∧
(sup {a.d + 1 | a ϵ x}
∨
sup {a.mli + i-j |
a ∈ x.϶^{i}.϶^{-j}, i,j ∈ℕ})
if x is well-founded in ϵ.
Observations:
r.ec must be defined.
r.ec.ec can be undefined.
Moreover,
O= {r, r.ec}
in the minimum powerclass-based structure.
If (pb~5)
is replaced by T⊂O.ce
(every terminal has a powerclass)
then we obtain more pure definition since
we could simply define
(ϵ¯) = (≤) ○ (.ec) ○ (≤).
Proof:
Since rϵr (by the same arguments as for basic structures)
it follows by
(pb~8) that
r.ϵ=r.ϵ¯.
Since r.↥= {r} it follows
that r.↥.ec.↥= {r}.ec.↥.
Since
r.↥.ec.↥=r.ϵ¯
and this set is non-empty (by rϵ¯r)
it follows that {r}.ec is non-empty.
((≤) ○ (.ce) ○ (≤))^{i}
for every natural i > 0 in the usual sense of relational
composition, and
(.ɛɕ)
=
∅.
If S= (O, ϵ, ≤, r, .ec)
is a powerclass-based structure then
the ϵ5-structure
S' = (O, ϵ, …)
derived from S by the above prescription (∗)
(with (ϵ¯^{0}) = (≤))
is a basic structure.
If S= (O, ϵ, …)
is basic structure satisfying (∗)
then
S' = (O, ϵ, ≤, r, .ec)
is a powerclass-based structure.
Proof:
Proceed similarly to the case of
metaobject structures.
(b~2) and (b~3)
are shown by the following inclusions:
The following definition provides an intermediate family of
structures
between basic structures and
abstract power type systems
introduced in the
specialized document[].
By a powertype-based structure (of ϵ)
we mean a structure
(O, ϵ, .ec, r)
where
O is a set of objects,
.ϵ
is the membership relation between objects,
.ec
is the powerclass map, a partial map between objects,
r
is the inheritance root, a distinguished object.
Let .ec^{∗} be the
reflexive transitive closure of .ec,
let .ce be the inverse of .ec.
Let ≤ be the inheritance relation between objects defined by
u ≤ x iff
u = x or u ϵ x.ec.
The structure is subject to the following axioms.
(The definition of .d is again postponed.)
(ptb~1)
O.ce =r.↧.
(The powerclass map .ec
is defined exactly for descendants of r.)
(ptb~2)
O.ϵ≤r.
(Every container is a descendant of r.)
(ptb~3)
O=O.϶.
(Every object has a container.)
(ptb~4)
(.ec) ⊆ (ϵ).
(Powerclass-of is a special case of container-of.)
(ptb~5)
(ϵ) ○ (≤) ⊆ (ϵ).
(The subsumption rule.)
(ptb~6)
(.ce) ○ (≤) ○ (.ec) ⊆ (≤).
(Monotonicity of .ec.)
(ptb~7)
(<) ∩ (>) =∅.
(Antisymmetry of ≤.)
(ptb~8)
r.ec^{∗}.▽=∅.
(Every object has a finite metalevel index.)
(ptb~9)
For every object x,
x.d =ϖ
→
x.ϵ= x.ec.↥.
The definitions of ϵ¯,
ϵ¯^{i},
ϵ^{i}
(i ∈ℤ),
.mli,
and .d are identical to those introduced for
powerclass-based structures.
Observations:
(Definitional correspondence.)
For a structure S= (O, ϵ, ≤, r, .ec)
the following are equivalent:
(Powerclass completion correspondence.)
There is a one-to-one correspondence between
powertype-based structures, and
basic structures that
(a)
are powerclass complete (O=O.ce)
and (b)
have no primary singletons
((.ɛɕ) =∅).
Extensions
Let
S= (O, ϵ, ϵ¯^{⁽*⁾}, r, .ec, .ɛɕ) and
S_{0}= (O_{0}, …)
be ϵ5-structures.
We say that
S is an extension of S_{0}
if S_{0} is a restriction of S
(see below for the precise meaning),
S is a powerclass extension of S_{0}
if S_{0} is a restriction of S
with the same set of primary objects,
S is a primary extension of S_{0}
if S_{0} is a restriction of S
with the same set of powerclasses.
If S is an extension of S_{0} then the following notation
and terminology apply.
We call objects from O_{0}old and
objects from O∖O_{0}new.
We will use the subscript to distinguish between symbols for S_{0}
and S,
as has already been applied for
O.
In particular, .ec_{0} and .ɛɕ_{0}
are the powerclass and primary singleton maps in S_{0}, respectively,
T_{0} is the set of terminal objects in S_{0}.
Similarly with other symbols that are used to denote either a set of objects,
or a (partial) map on objects, e.g.
.pr_{0}, .mli_{0} or .d_{0}.
For ϵ and similar symbols, we use a special (ligature) marker:
▫,
so that
ϵ▫ / ϵ▫¯ / ∊▫
denote the membership / power membership / bounded membership in S_{0}.
Similarly with
ϵ▫^{i} / ϵ▫¯^{i} / ∊▫^{i} for an integer i.
We can therefore express
S_{0}=
(O_{0}, ϵ▫, ϵ▫¯^{⁽*⁾}, r, .ec_{0}, .ɛɕ_{0}).
An ϵ5-structure
S= (O, ϵ, …) is an extension of S_{0} iff
O_{0}⊆O,
the inheritance roots in S and S_{0} are coincident,
for every old objects x, y and every integer i ≤ 1,
x ϵ▫ y ↔ x ϵ y,
x ϵ▫¯^{i} y ↔ x ϵ¯^{i} y,
x.ec_{0}= y ↔ x.ec = y,
x.ɛɕ_{0}= y ↔ x.ɛɕ = y.
S is a powerclass (resp. primary) extension of S_{0}
iff
O_{0}.pr_{0}=O.pr
(resp. O_{0}.ec_{0}=O.ec).
Faithful extension
Let
S= (O, ϵ, …) and
S_{0}= (O_{0}, ϵ▫, …)
be ϵ5-structures.
We say that S is a faithful extension of S_{0}
if S is an extension of S_{0} and
the following additional conditions are satisfied:
(A)
For every old object x,
{x}.pr = {x}.pr_{0}.
(That is,
O_{0}.pr_{0}⊆O.pr.)
(B)
For every natural i,
ϵ▫^{i} equals the restriction of ϵ^{i} to the set
O_{0} of old objects.
(C)
For every natural i,
ϵ▫¯^{i} equals the restriction of ϵ¯^{i} to the set
O_{0} of old objects.
(D)
For every old object x,
x.d = x.d_{0}.
Observe that for powerclass extensions as well as for primary extensions,
(A) is satisfied implicitly.
Embedding
For ϵ5-structures,
S_{1}= (O_{1}, ϵ, …) and
S_{2}= (O_{2}, ϵ, …)
a map .ν from O_{1} to O_{2}
is an embedding of S_{1} into S_{2}
if it is an isomorphism between S_{1} and
the restriction of S_{2} to O_{1}.ν,
that is,
r_{1}.ν =r_{2} and
for every objects x, y from O_{1}
and every integer i ≤ 1,
x ϵ y ↔ x.ν ϵ y.ν,
x ϵ¯^{i} y ↔ x.ν ϵ¯^{i} y.ν,
x.ec = y ↔ x.ν.ec = y.ν,
x.ɛɕ = y ↔ x.ν.ɛɕ = y.ν.
If S_{1}.ν denotes the restriction of S_{2} to
O_{1}.ν then
S_{2} is an extension of S_{1}.ν.
If this extension is faithful then .ν is said to be faithful.
Except for the ranking product, every listed extension is a completion
in the sense of idempotency of the extension.
Extensions 2,3,4 and 8 are true completions in the sense of least
extension.
If S_{0} is a basic structure (resp. pre-complete structure in the case 8),
and S is the respective extension of S_{0} then
S is the least extension of S_{0} that is
a basic structure that is
(2)
powerclass complete
(x.ec is defined for every x ∈O),
(3)
primary singleton complete
(x.ɛɕ is defined for every
x ∈O.∍∖ (T∪O.ɛϲ)),
(4)
singleton complete
(x.ɛϲ is defined for every x ∈O.∍),
(8)
complete
(S is an (ϖ+1)-superstructure).
Ranking product
This section describes embedding of a
pre-basic structureS
into an ϵ-ranked pre-basic structure S' called
the ranking product of S.
Such an embedding allows to express the rank function .d in
S
via the (simpler) ϵ-rank function r_{ϵ}() in S'.
The set Ƣ of objects of S' consists of
indexed objects –
pairs (x,-i)
where
x is an object of S and,
i is a natural number less than or equal to the metalevel index of x.
(We also impose the i = 0 condition whenever x is a powerclass.
This simplifies the diagrams of S'.)
Embedded objects are then those indexed by 0.
The desired embedding of the rank function is expressed by
x.d = r_{ϵ}(x,0) = (x,0).d
for every object x of S.
Example
The diagram on the right shows the ranking product S' = (Ƣ, …)
of a basic structure S= (O, …).
Negatively indexed objects
are displayed in khaki color.
The original structure S can be identified with the restriction
of S' to zero-indexed objects.
Observations:
S' is a basic structure of ϵ.
For every (x,-i) from Ƣ,
the metalevel index of (x,-i) equals x.mli - i.
Informally,
each primary object is equipped with
an ϵ-chain to reach the 0-th metalevel.
(The old objects have zero index,
the new objects are those with a negative index.)
Ranking product
For an ϵ5-structure
S= (O, ϵ, ϵ¯^{⁽*⁾}, r, .ec, .ɛɕ)
the ranking product
of S is an ϵ5-structure
S' =(Ƣ, ϵ, ϵ¯^{⁽*⁾}, ṙ, .ec, .ɛɕ)
defined as follows.
Ƣ is the subset of O×ℤ
such that for every object x and every integer i,
(x,i) ∈Ƣ
↔
-x.mli ≤ i ≤ 0
and
x is primary (⁎)
or i = 0
(and x is an arbitrary object).
For every (x,i), (y,j) from Ƣ and
every integer k ≤ 1,
(x,i) ϵ (y,j)
↔
(a)
(x,i+1) = (y,j)and j < 0, or
(b)
x ϵ¯^{1+i} y and j = 0, or
(c)
x ϵ y and i = j = 0,
(x,i) ϵ¯^{k} (y,j)
↔
(a)
(x,i+k) = (y,j)
and i ≤ j < 0, or
(b)
x ϵ¯^{k+i} y and j = 0,
ṙ= (r, 0),
(x,i).ec = (y,j)
↔
x.ec = y and i = j = 0,
(x,i).ɛɕ = (y,j)
↔
x.ɛɕ = y and i = j = 0.
Notes:
If x.mli = ω then
-x.mli ≤ i is understood as a void condition.
Assuming reflexivity of ≤ in S,
the grayed condition j < 0 can be left out.
(⁎)
Since there is no constraint for primary singletons x,
the definition does not preserve basic structures
– if S is a basic structure in which
.ɛɕ is non-empty then
S' is not a basic structure.
This can be presumably resolved by
disallowing pairs (x,i) with i ≠ 0
for primary objects x such that
{u} = x.϶ for some u from x.ϵ^{-1} and
x.϶¯^{-k}= {x}.ec(k)
for every natural k.
However, we only need embedding of pre-basic structures so
that we avoid such a complication.
Observations:
For every (x,-i) from Ƣ such that i ≠ 0
the following is satisfied:
(x,-i).϶ equals either {(x,-i-1)} or ∅
(the latter case occurs iff x.mli = i).
(x,-i) is well-founded in (Ƣ, ϵ)
↔
x.mli is finite.
If x.mli is finite then the ϵ-rank of
(x,-i)
equals x.mli - i.
Assume (ϵ¯) ⊆ (ϵ) in S.
The map x ↦ (x,0) is an embedding
of S into S'.
Embedding of pre-basic structures
Proposition:
Let S' be the ranking product
of a pre-basic structureS
and denote
.ν the map x ↦ (x,0) from O to Ƣ.
Then
For every (x,-i) from Ƣ,
(x,-i) ≤ṙ
↔
i < x.mli.
Corollary:
The set Ṫ of terminals of S'
consists of pairs (x,-i) such that x.mli = i.
T.ν ⊆Ṫ.
(ϐ~5) is satisfied:
Ƣ.ϵ≤ṙ.
(ϐ~6) is satisfied:
Ƣ.϶¯=Ƣ.
(ϐ~7) is satisfied:
For every natural k,
Ṫ.϶^{-k}∩ṙ.϶^{-k}=∅.
Proof:
This follows directly from the prescription for ϵ and ϵ¯^{1}
in S'.
Assume that for every natural k > 1, the ϵ¯^{k} relation
in S' is defined the same way as for k ≤ 1.
It is then sufficient to prove that for every
ẋ, ẏ, ż from Ƣ,
and
for every integers m, n and natural k
the following is satisfied:
(b~0):
k > 1 and ẋ ϵ¯^{k} ẏ→ẋ ϵ¯^{p} ṡ ϵ¯^{q} ẏ for some ṡ ∈Ƣ
and natural
p, q > 0 such that p + q = k.
(b~2):
ẋ ϵ¯^{m} ẏ ϵ¯^{n} ż→ẋ ϵ¯^{m+n} ż.
(b~3):
ẋ ϵ ẏ ϵ^{-k} ż→ẋ ϵ^{1-k} ż.
(b~4):
ẋ ϵ¯^{m} ẏ ϶¯^{-m} ẋ↔ẋ.ec(m) = ẏ.
To prove (b~0), let
ẋ = (x,i) ϵ¯^{k} (y,j) = ẏ.
Then the requested ṡ and p, q are obtained according
to the following:
If k+i > 0 then (x,i) ϵ¯^{-i} ṡ ϵ¯^{k+i} (y,0)
where ṡ = (x,0).
If
k+i ≤ 0 then (x,i) ϵ¯^{k-1} ṡ ϵ¯ (y,j)
where ṡ = (x,i+k-1).
To prove (b~2),
assume
(x,i) ϵ¯^{m} (y,j) ϵ¯^{n} (z,ℓ).
Then
(x,i) ϵ¯^{m+n} (z,ℓ) is a consequence of the following:
If ℓ < 0 then i ≤ j < ℓ
and (x,i+m) = (y,j) = (z,ℓ-n).
If j < ℓ = 0
then
(x,i+m) = (y,j) and
y ϵ¯^{n+j} z and thus x ϵ¯^{m+n+i} z.
If j = ℓ = 0
then
x ϵ¯^{m+i} y ϵ¯^{n} z
and thus x ϵ¯^{m+n+i} z.
To prove (b~3),
assume
(x,i) ϵ (y,j) ϵ^{-k} (z,ℓ).
If (x,i) ϵ¯ (y,j) then (b~2) applies.
We can therefore assume
further that ((x,i),(y,j)) ∉ (Ƣ, ϵ¯) so that condition (c)
in the definition of (Ƣ, ϵ) is satisfied:
x ϵ y and i = j = 0.
Consequently, ℓ = 0,
so that (x,0) ϵ^{1-k} (z,0) follows by the embedding of ϵ
and its powers.
To prove (b~4), assume
(x,i) ϵ¯^{m} (y,j) ϵ¯^{-m} (x,i).
If i < 0 or j < 0 then
(by definition of (Ƣ, ϵ¯^{⁽*⁾})) (x,i) = (y,j)
and m = 0.
Otherwise (i.e. if i = j = 0), x ϵ¯^{m} y ϵ¯^{-m} x
and thus (by (b~4) in S)
x.ec(m) = y.
It follows by the definition of (Ƣ, .ec) that
(x,i).ec(m) = (y,j).
This shows the → direction in the
(b~4) equivalence.
The reverse direction is verified similarly.
This is a consequence of:
(x,-i) ≤ (r,0)
↔
x ϵ¯^{-i}r
↔
i < x.mli.
This is a consequence of:
(x,-i) ϵ¯ (r,0)
↔
x ϵ¯^{1-i}r
↔
i ≤ x.mli.
Let (y,j) be from Ṫ and k a natural number.
We should show that
(∗)
(y,j).϶^{-k}∩ (r,0).϶^{-k}=∅.
If j ≠ 0 then
(y.j).϶^{-k}≠∅
↔
k = 0
so that
(∗) follows by Ṫ∩ṙ.↧=∅.
If j = 0 then y is terminal
so that for every (x,-i) from Ƣ,
(x,-i) ϵ¯^{-k} (r,0)
↔
x ϵ¯^{-i-k}r,
(x,-i) ϵ¯^{-k} (y,0)
↔
x ϵ¯^{-i-k} y.
Since by (ϐ~7) in S
there is no object x such that
x ϵ¯^{-i-k}r and x ϵ¯^{-i-k} y,
the equality (∗) follows.
Claim B:
For every (x,-i) from Ƣ,
i + (x,-i).mli = x.mli.
In particular,
(x,-i).mli < ω
↔
x.mli < ω.
(x,i) is primary
↔x is primary.
S' is ϵ-ranked.
Proof:
(a) follows by the equivalence
(x,-i) ϵ¯^{-k} (r,0)
↔
x ϵ¯^{-i-k}r.
(b)
follows by definition of (Ƣ, .ec).
ẋ.pr.mli < ω and
ẋ.pr ∈Ṫ.ϵ^{∗}
(since
(∗) applies to ẋ.pr),
therefore
ẋ.mli < ω and
ẋ ∈Ṫ.ϵ^{∗}
(since
ẋ.pr.mli < ω
↔
ẋ.mli < ω
and
ẋ.pr.ϵ^{∗}∋ ẋ).
Claim C:
For every object x from O the following are equivalent:
(a) x is well-founded in ϵ
and
(b) for every a from x.϶^{∗}.϶^{-∗},
the metalevel index a.mli is finite.
(x,0) is well-founded in (Ƣ, ϵ).
The ranking product S' preserves the rank:
For every object x from O,
(x,0).d = x.d.
S is a faithfully embedded in S'.
Proof:
The ¬(ⅰ)→¬(ⅱ)
implication follows by the
x ↦ (x,0) embedding.
To show ¬(ⅰ)←¬(ⅱ),
assume that x is an object such that
a.mli is finite for every a from
x.϶^{∗}.϶^{-∗}
and (x,0)
is non-well-founded in (Ƣ, ϵ), i.e.
there is an infinite chain
in (Ƣ, ϵ).
Since (y,-j).϶^{y.mli+1}=∅
for
every (y,-j) from Ƣ such that j ≠ 0
and y.mli is finite,
it follows that i_{k}= 0 for every natural k.
As a consequence, x is non-well-founded in (O, ϵ).
Denote W
the set of all x from O such that
(ⅰ) from C1 is satisfied,
and
Ẇ the set of ẋ
from Ƣ that are well-founded in (Ƣ, ϵ).
Define recursively functions r_{1}() / r_{2}()
on W / Ẇ
by
r_{1}(x) = sup { r_{1}(a) + 1 | a ϵ x }
∧
sup {a.mli + i-j |
a ∈ x.϶^{i}.϶^{-j}, i,j ∈ℕ}
(so that x.d = r_{1}(x) ∧ϖ),
r_{2}(ẋ) = sup { r_{2}(ȧ) + 1
| ȧ ϵ ẋ }
(so that r_{ϵ}(x) = r_{2}(x) ∧ϖ).
We show that r_{1}(x) = r_{2}(x,0) for every x from W
so that, consequently,
for every object x from O,
x.d = r_{ϵ}(x,0) = (x,0).d
(the latter equality is by B2).
Let x be from W and
let ẋ = (x,0).
Denote
A = { r_{1}(a) + 1 | a ϵ x },
B = { a.mli + i-j |
a ∈ x.϶^{i}.϶^{-j}, i,j ∈ℕ},
A' = { r_{2}(ȧ) + 1 | ȧ ϵ ẋ,
ȧ ∈O× {0} },
B' = { r_{2}(ȧ) + 1 | ȧ ϵ ẋ } ∖ A',
so that
r_{1}(x) = sup(A ∪ B) and
r_{2}(ẋ) = sup(A' ∪ B').
It is therefore sufficient to show that A ∪ B = A' ∪ B'.
By well-founded recursion, we can assume that A = A'.
To show B ⊆ A' ∪ B',
let a ϵ^{-j} y ϵ^{i} x
for some natural i, j
and denote m = a.mli (so that m+i-j ∈ B).
By
observations about .d,
we can assume that a is primary.
Since j ≤ a.mli it follows that (a,-j) ∈Ƣ. Subsequently,
(a,-m) ϵ^{m-j} (a,-j) ≤ (y,0) ϵ^{i} (x,0)
and thus (a,-m) ϵ^{m+i-j} (x,0)
which shows that m+i-j ∈ A' ∪ B'.
To show B' ⊆ B,
let ȧ = (a,-1-j) ϵ (x,0),
j ∈ℕ
so that r_{2}(ȧ) = a.mli -1-j
and a ϵ^{-j} x.
Subsequently,
r_{2}(ȧ) + 1 = a.mli - j
and
a ∈ x.϶^{0}.϶^{-j},
therefore r_{2}(ȧ)+1 ∈ B.
Embedding of constituents of the signature has already been observed.
Embedding of .pr
follows by definition of (Ƣ, .ec).
Embedding of positive powers of ϵ / ϵ¯
follows from
the closedness of the set
O.ν of embedded object w.r.t. .ϵ^{∗}.
Finally embedding of .d has been proved in C2.
Rank in pre-basic structures
The following proposition provides a summary of main properties
of the rank function .d.
Properties (3)–(6) are obtained as consequences of
the embedding into the ranking product.
Proposition:
In a pre-basic structure S the following are satisfied.
For every object x,
0 ≤ x.mli ≤ x.d ≤ϖ.
For every object x,
x.d = 0 ↔ x ∈T.
For every objects x, y,
x ≤ y → x.d ≤ y.d.
(That is, .d is monotone w.r.t. ≤.)
For every objects x, y,
x ∊ y → x.d < y.d.
For every object x such that x.ec is defined,
x.ec.d =ϖ∧ (x.d + 1), i.e.
In this section we show that every basic structure
can be faithfully embedded into a powerclass complete basic structure.
Powerclass completion
Let
S= (O, ϵ, …)
be an ϵ5-structure
and
S_{0}= (O_{0}, ϵ▫, …)
a pre-basic structure.
We say that
S is a powerclass completion of S_{0} if
S is an extension of S_{0} that is created in the following steps:
Extend (O_{0}, .ec) to (O, .ec)
so that
(a) .ec is injective well-founded map on O, and
(b) O.pr =O_{0}.pr_{0}.
(That is, S is a powerclass extension of S_{0}
– every new object is a powerclass.)
There is obviously exactly one such extension, up to isomorphism.
Extend ϵ▫ and ϵ▫¯^{⁽*⁾} to
ϵ and ϵ¯^{⁽*⁾}
according to the following.
For every primary objects
a, b
and every natural i, j such that
at least one of a.ec(i) or b.ec(j) is a new object,
the following holds:
(α)
a.ec(i) ϵ b.ec(j)
iff
a ϵ▫¯^{1+i-j} b,
(β)
a.ec(i) ϵ¯^{k} b.ec(j)
iff
a ϵ▫¯^{k+i-j} b
for every integer k.
Since this definition asserts the existence
of a unique, up to isomorphism, ϵ5-structure S
for any given pre-basic structure S_{0}
we can also speak about the powerclass completion of S_{0}.
Observations:
In (Ⅱ),
primary can be replaced by old
and natural i, j by integer i, j.
For a pre-basic structure S_{0} and an ϵ5-structure S
the following are equivalent:
S is a powerclass completion of S_{0}.
S is the least extension of S_{0} that is powerclass complete.
The powerclass completion theorem
Proposition:
The powerclass completion S of a basic structure S_{0}
is a powerclass complete basic structure
that is a faithful extension
of S_{0}.
Proof:
The proof consists in verifying that
S
satisfies (b~1)–(b~11)
and is a faithful extension of S_{0}.
This is done in a series of claims below.
Claim A:
Axioms
(b~2)–(b~4)
are satisfied.
For every integers m, n,
(ϵ¯^{m}) ○ (ϵ¯^{n}) ⊆ (ϵ¯^{m+n}),
(ϵ) ○ (ϵ^{n}) ⊆ (ϵ^{1+n}),
(ϵ¯^{m}) ∩ (϶¯^{-m}) = .ec(m).
Every new object x is both a power member and power container:
x.ϵ= x.ϵ¯ and x.϶= x.϶¯.
Corollary:
For every old object x,
if x.ϵ▫= x.ϵ▫¯ then x.ϵ= x.ϵ¯.
(Similarly in the inverse.)
(a)
(O∖O_{0}) ≤r,
(b)
O.ϵ=O.ϵ▫∪ (O∖O_{0}),
(c)
O.ϵ▫=O.ϵ∩O_{0}.
Corollary:
(b~5),
(b~6)
and
(b~7)(b)
are satisfied.
For every old object x,
if x.ec_{0}^{∗-∗}= x.▫϶¯^{∗-∗}
then
x.ec^{∗-∗}= x.϶¯^{∗-∗}.
Corollary: (b~7)(a) is satisfied.
Axiom (b~8) is satisfied:
If x.ɛɕ = y
then:
(a)
{x} = y.϶,
(b)
x.ϵ^{i}⊆ y.ϵ^{i-1} for every i ≤ 1,
(c)
(x,y) ∉ (ϵ¯).
Axiom (b~10) is satisfied:
For every object x, x.mli < ω.
Proof:
Let a, b, c be primary objects,
i, j, k natural numbers and
m, n integers.
a.ec(i) ϵ¯^{m} b.ec(j) ϵ¯^{n} c.ec(k)
↔
a ϵ▫¯^{m+i-j} b ϵ▫¯^{n+j-k} c
→
a ϵ▫¯^{m+n +i-k} c.
a.ec(i) ϵ b.ec(j) ϵ¯^{n} c.ec(k)
↔
a ϵ▫^{1+i-j} b ϵ▫¯^{n+j-k} c
→
a ϵ▫^{1+n +i-k} c.
a.ec(i) ϵ¯^{m} b.ec(j) ϵ¯^{-m} a.ec(i)
↔
a ϵ▫¯^{m+i-j} b ϵ▫¯^{-m+j-i} a
↔
a.ec(m+i-j) = b
↔
a.ec(i).ec(m) = b.ec(j).
This follows directly from the prescriptions
(α) and (β) (k = 1)
for ϵ¯ and ϵ.
Let x be a new object,
x = a.ec(i) for a primary a
and a natural i > 0. Then
(a) a ϵ▫¯^{i}r (since i > 0) and
thus x ≤r,
(b) x ϵ¯r
(since x ≤rϵ¯r)
and thus x ϵr
(since x.ϵ= x.ϵ¯).
To show (c), assume that x ϵ y and y is old.
(That is y ∈O.ϵ∩O_{0}.)
Then
a ϵ▫¯^{1+i} y
→
a ϵ▫^{1+i} y
→
O_{0}.ϵ▫.ϵ▫^{∗}∋ y
→
O_{0}.ϵ▫∋ y.
Assume that x is an old object such that
x.ec_{0}^{∗-∗}= x.▫϶^{∗-∗}
and let y be from x.▫϶^{∗-∗}.
That is,
b.ec(j) ϵ¯^{i} x,
for some integer i,
where b is a primary object and j a natural number such
that y = b.ec(j).
By definition of ϵ¯^{i},
b ϵ▫¯^{j+i} x
→
b ∈ x.▫϶¯^{∗-∗}
→
b ∈ x.ec_{0}^{∗-∗}
→
y ∈ x.ec^{∗-∗}.
Assume x.ɛɕ = y.
Then both x, y are old (since .ɛɕ_{0}= .ɛɕ).
Therefore, (a) and (c) are satisfied.
If i is an integer, i ≤ 1,
and z is a new object
from x.ϵ^{i} then
z ∈ x.ϵ¯^{i}⊆ y.ϵ¯^{-1}.ϵ¯^{i}⊆ y.ϵ¯^{i-1},
which shows that also (b) is satisfied.
This is a consequence of the following equivalence: For every object x,
and every natural j,
x ϵ^{1-j}r
↔
a ϵ▫^{1+i-j}r
where a is a primary object and i a
natural number such that x = a.ec(i).
(Therefore,
x.mli = sup { j | x ϵ^{1-j}r, j ∈ℕ }
= sup { j | a ϵ▫^{1+i-j}r, j ∈ℕ }
= i + a.mli_{0}.)
Claim B:
For every old objects x, y
and every natural i,
(a)
x ϵ^{i} y ↔ x ϵ▫^{i} y,
(b)
x ϵ¯^{i} y ↔ x ϵ▫¯^{i} y.
For every old object y, (a)
y.϶⊆ y.▫϶.↧,
(b)
y.϶¯= y.▫϶¯.↧.
Note:
We let ϵ¯^{k} be defined in
(β)
just for k ≤ 1.
Proof:
To prove (a), assume n > 1 and
let x, y be old objects such that x ϵ^{n} y.
That is, there are old objects
x = x_{0}, x_{1}, …, x_{n-1}, x_{n}= y
and
natural numbers
0 = i_{0}, i_{1}, …, i_{n-1}, i_{n}= 0
such that
We can assume that all of
x_{1}.ec(i_{1}), …, x_{n-1}.ec(i_{n-1})
are new objects
–
otherwise the proof follows by induction over n.
We can therefore apply the definition (α)
of (ϵ) ∖ (ϵ▫)
and write
To prove (b), assume n > 0 and
let x, y be old objects such that x ϵ¯^{n} y,
that is, x.ec(n) ≤ y.
If x.ec(n) is old then (x,y) ∈ .ec_{0}(n-1) ○ (ϵ▫¯).
If x.ec(n) is new then the definition of ϵ¯^{k}
aplies (use the adjusted
definition according to the observation.)
Assume that y is an old object.
Let a be an old object and i a natural number such that
x = a.ec(i) is a new object from y.϶
so that a ϵ▫¯^{1+i} y by definition of ϵ.
In particular, there exists an old object u such that
a ϵ▫¯^{i} u ϵ▫¯ y.
It follows from
x.ec(-i) = a ϵ¯^{i} u that x ≤ u.
This shows (a) y.϶⊆ y.▫϶.↧
and also y.϶¯⊆ y.▫϶¯.↧.
The y.϶¯⊇ y.▫϶¯.϶¯^{0} inclusion follows by
(b~2).
Claim C:
S preserves the rank of old objects,
i.e.
for every old object x, x.d = x.d_{0}.
Axiom (b~11) is satisfied:
x.ϵ= x.ϵ¯
for every
x from O∖O.∍.
Proof:
Consider the ranking products(Ƣ_{0}^{}, ϵ▫, …)
and
(Ƣ, ϵ, …)
of
S_{0} and S, respectively.
Since S is already known to be a pre-basic structure,
(Ƣ, ϵ, …) is an extension of
(Ƣ_{0}^{}, ϵ▫, …), in particular w.r.t. ϵ and ≤.
It is then sufficient to show that
(y,-j).϶⊆ (y,-j).▫϶.↧
for every (y,-j) from Ƣ_{0}^{}.
Assume that (y,-j) is from Ƣ_{0}^{}
and let (x,-i) be from (y,-j).϶.
We should prove that (x,-i) ≤ (u,k)
for some (u,k) from Ƣ_{0}^{}.
If i ≠ 0 or j ≠ 0 then put (u,k) = (x,-i).
(If j ≠ 0 then (x,1-i) = (y,-j).
If i ≠ 0 then x is primary, therefore old.)
If i = j = 0 then x ϵ y so that claim B2(a) applies:
there is an old object u such that x ≤ u ϵ y
and therefore (x,0) ≤ (u,0) ϵ (y,0).
Apply claims A2 and C1.
Singleton completion
The diagram on the right shows an extension of
a basic structure S_{0}= (O_{0}, …)
to a
basic structure S= (O, …)
by primary singletons.
New objects are those from {a,b,e}.ɛɕ
(indicated by orange circles
().
The difference between (O, ϵ, ≤) and
(O_{0}, ϵ, ≤)
is indicated by dashed arrows.
Moreover, the (ϵ) ∖ (ϵ¯) difference is
indicated by blue arrows with a highlighted background,
similarly as with the introductory sample.
In S_{0}, the difference is {(a,w)}.
Since
{a,b,e} =O.∍∖ (T∪O.ɛϲ),
the resulting extension S is primary singleton complete
–
every bounded object that is not terminal or a singleton has a singleton.
A subsequent partial powerclass completion of S,
applied just for the set T∪O.ɛϲ,
makes S singleton complete,
so that all bounded objects have a singleton.
Primary singleton completion
Let
S= (O, ϵ, …)
be an ϵ5-structure
and
S_{0}= (O_{0}, ϵ▫, …)
a basic structure.
We say that
S is a primary singleton completion of S_{0} if
S is a primary extension of S_{0}
such that
S is primary singleton complete,
every new object is a primary singleton, and
if x.ɛɕ = y and y is new then
y.϶= {x} and y.ϵ= x.ϵ▫.ϵ▫¯,
y.϶¯^{i}= {y}.ec(-i)
and
y.ϵ¯^{i}=
x.ϵ▫.ϵ▫¯^{i}∪
x.ϵ▫¯^{1+i}∪ {y}.ec(i)
for every integer i,
i.e.
y.϶¯=∅ and y.ϵ¯= x.ϵ▫.ϵ▫¯,
y.↧= {y} and y.↥= x.ϵ▫∪ {y},
y.϶¯^{i}=∅ and y.ϵ¯^{i}= x.ϵ▫^{1+i}
for every i < 0.
The singleton completion theorem
Proposition:
Let S be a primary singleton completion
of a basic structure S_{0}.
Then
S is a basic structure
that is a faithful extension
of S_{0}.
Proof:
We consider an ϵ5-structure
S= (O, ϵ, …)
created from a basic structure S_{0}= (O_{0}, ϵ▫, …) in
the following steps:
Extend (O_{0}, .ɛɕ_{0}) to (O, .ɛɕ)
so that
(.ɛɕ) ∖ (.ɛɕ_{0})
is a bijection between
O_{0}.ɛɕ_{0}(0) ∖O_{0}.ɕɛ_{0}
and
O∖O_{0}.
(That is, define explicitly
a new primary singleton for every object from O_{0}.ɛɕ_{0}(0)
for which x.ɛɕ_{0} is undefined.)
There is obviously exactly one such extension, up to isomorphism.
Extend ϵ▫ and ϵ▫¯^{⁽*⁾}
to
ϵ and ϵ¯^{⁽*⁾}
according to the definition of primary singleton completion.
Now the proof consists in verifying that
S
is a basic structure that is a faithful extension of S_{0}
and is singleton complete.
Claim A:
For every old objects x, y
and every natural i,
(a)
x ϵ^{i} y ↔ x ϵ▫^{i} y,
(b)
x ϵ¯^{i} y ↔ x ϵ▫¯^{i} y.
For every old object y,
y.϶⊆ y.▫϶.↧.
Proof:
Let x, y be old objects and i a natural number.
We can assume that i > 1.
Then
x ϵ^{2} y ↔ x ϵ▫^{2} y
since if x ϵ a ϵ y for some new a then
x.ɛɕ = a,
thus y ∈ a.ϵ= x.ϵ▫.ϵ▫¯⊆ x.ϵ▫^{2},
x ϵ^{i} y ↔ x ϵ▫^{i} y
since new objects are unrelated by ϵ so that induction
argument can be used together with the previous case i = 2,
x ϵ¯^{i} y ↔ x ϵ▫¯^{i} y
since z.϶¯=∅ for every new object z.
Assume that y is an old object
and let a and b be objects such that
a.ɛɕ = b ϵ y and b is new.
By definition of b.ϵ, there is an old object u
such that a ϵ▫ u ϵ▫¯ y.
By definition of b.↥, it follows that b < u.
Claim B:
(b~2) is satisfied.
For every object y, and every integer i, j,
(ⅰ)
y.ϵ¯^{i}.ϵ¯^{j}⊆ y.ϵ¯^{i+j}
and
(ⅱ)
y.϶¯^{i}.϶¯^{j}⊆ y.϶¯^{i+j}.
(b~3) is satisfied:
For every objects x, y, z and every natural i,
if x ϵ y ϵ^{-i} z then x ϵ^{1-i} z.
(b~4) is satisfied:
For every objects x, y and every integer i,
x ϵ¯^{i} y ϵ¯^{-i} x↔x.ec(i) = y.
(b~5) is satisfied: O.ϵ≤r.
(b~6) is satisfied: T⊆O.϶.
(b~7)(a) is satisfied:
y.϶^{-i}⊆ {y}.ec(i)
for every y ∈T∪O.ɛɕ and every natural i.
(b~7)(b) is satisfied:
x.ϵ= x.ϵ¯ for every
x ∈ (T∪O.ɛɕ).ec^{∗}.
(b~8) is satisfied:
If x.ɛɕ = y
then:
(a)
{x} = y.϶,
(b)
x.ϵ^{i}= y.ϵ^{i-1} for every i ≤ 1,
(c)
(x,y) ∉ (ϵ¯).
For every new object y = x.ɛɕ,
y.mli = x.mli + 1.
Corollary: (b~10) is satisfied.
(Every object has a finite metalevel index.)
Proof:
For every new object u and every integer i
it follows by definition that
u.϶¯^{i}≠∅
↔
u.϶¯^{i}= {u} and i = 0.
This shows that
(ⅰ) is satisfied for all old objects y
and
(ⅱ) is satisfied for all objects y.
It remains to show that
(ⅰ) is satisfied if y is new.
Assume therefore that x.ɛɕ = y and y is new.
Denote Y = y.ϵ¯^{i}.ϵ¯^{j}.
For i > 1 it follows by definition of ϵ¯^{i} that
Y = y.ϵ¯^{i-1}.ϵ¯.ϵ¯^{j}
so that it is sufficient to show
y.ϵ¯.ϵ¯^{j}⊆ y.ϵ¯^{1+j}
and apply the induction argument.
For i = 1 we obtain
Y = x.ϵ▫.ϵ▫¯.ϵ▫¯^{j}⊆ x.ϵ▫.ϵ▫¯^{1+j}.
Denote Z = x.ϵ▫.ϵ▫¯^{1+j}.
It is sufficient to show that y.ϵ¯^{1+j}⊇ Z.
If j ≥ 0 then
y.ϵ¯^{1+j}= y.ϵ¯.ϵ¯^{j}= x.ϵ▫.ϵ▫¯^{1+j}= Z.
If j = -1 then
y.ϵ¯^{1+j}= y.↥= x.ϵ▫∪ {y}
⊇ Z.
If j < -1 then
y.ϵ¯^{1+j}= y.ϵ¯^{1+j}= x.ϵ▫^{2+j}⊇ Z.
For i = 0 we obtain
Y = y.↥.ϵ¯^{j}= y.ϵ¯^{j}∪ x.ϵ▫.ϵ¯^{j}.
Denote Z = x.ϵ▫.ϵ¯^{j}.
We should prove that y.ϵ¯^{j}⊇ Z.
If j > 0 then y.ϵ¯^{j}= x.ϵ▫.ϵ¯^{j}= Z.
If j = 0 then y.ϵ¯^{j}= y.↥∪ x.ϵ▫⊇ Z.
If j < 0 then y.ϵ¯^{j}= x.ϵ▫¯^{1+j}⊇ Z.
For i < 0 we obtain
Y = x.ϵ▫¯^{1+i}.ϵ▫¯^{j}⊆ x.ϵ▫¯^{1+i+j}.
Denote Z = x.ϵ▫¯^{1+i+j}.
It is sufficient to show that y.ϵ¯^{i+j}⊇ Z.
If i+j > 0 then
y.ϵ¯^{i+j}= x.ϵ▫.ϵ▫¯^{i+j}⊇ x.ϵ▫¯.ϵ▫¯^{i+j}= x.ϵ▫¯^{1+i+j}= Z.
If i+j = 0 then y.ϵ¯^{i+j}= {y} ∪ x.ϵ▫⊇ Z.
If i+j < 0 then y.ϵ¯^{i+j}= x.ϵ▫¯^{1+i+j}= Z.
We first show that for every object y and every natural i,
(ⅰ)
y.ϵ.ϵ^{-i}⊆ y.ϵ^{1-i}
and
(ⅱ)
y.϶^{-i}.϶⊆ y.϶^{1-i}.
Assume that y is new and let x be such that
x.ɛɕ = y.
Then
(ⅱ) is satisfied since
y.϶^{-i}.϶≠∅
↔
y.϶^{-i}.϶= {x} and i = 0.
To show (ⅰ), write
y.ϵ.ϵ^{-i}= x.ϵ▫.ϵ▫¯.ϵ▫¯^{-i}⊆ x.ϵ▫.ϵ▫¯^{1-i}.
Denote Z = x.ϵ▫.ϵ▫¯^{1-i}.
It is sufficient to show that y.ϵ^{1-i}⊇ Z.
If i = 0 then
y.ϵ^{1-i}= y.ϵ= x.ϵ▫.ϵ▫¯= Z.
If i = 1 then
y.ϵ^{1-i}= y.↥⊇ x.ϵ▫= Z.
If i > 1 then
y.ϵ^{1-i}= y.ϵ¯^{1-i}= x.ϵ▫¯^{2-i}⊇ Z.
Thus,
(ⅰ) and (ⅱ)
hold for every new y.
This shows that
(∗)
x ϵ y ϵ^{-i} z
→
x ϵ^{1-i} z
(i ∈ℕ)
is satisfied for every objects x, y, z such that
at least one of x or z are new.
(Note that this time the y variable is used in the middle place.)
It remains to show that (∗) holds if y is new.
Assume therefore
that
x, y, z are objects such that
x ϵ y ϵ^{-i} z and y is new.
Then necessary x.ɛɕ = y and
it is sufficient to show that
x.ϵ.ϵ^{-i}⊆ x.ϵ^{1-i}.
If i = 0 then
x.ϵ.ϵ^{-i}= y.↥= x.ϵ.
if i < 0 then
x.ϵ.ϵ^{-i}= x.ϵ▫.ϵ▫^{-i}⊆ x.ϵ▫^{1-i}= x.ϵ^{1-i}.
If x and y are old then
(b~4) in S_{0} applies.
If x or y is new then
x ϵ¯^{i} y ϵ¯^{-i} x↔i = 0 and x ≤ y ≤ x↔x.ec(i) = y.
If x.ɛɕ = y and y is new then
y.↥⊇ x.ϵ▫∋r.
This is a consequence of new objects being non-terminal.
Let y be from T∪O.ɛɕ
and let u be from y.϶^{-i}
for a natural i.
We should prove that y.ec(i) = u.
For u and y both being old this is asserted by
(b~7) in S_{0}.
If y is new then by the prescription for primary singleton completion,
i = 0 and u = y so that the requested equality is satisfied.
Let x be from (T∪O.ɛɕ).ec^{∗}.
If x is old then
x.ϵ= x.ϵ▫
(since x.ɛɕ is undefined)
and thus x.ϵ= x.ϵ¯.
If x is new then the same equality holds by the prescription for
primary singleton completion.
Assume that x.ɛɕ = y.
If y is old then so is x and
(b~8) in S_{0} applies:
(a)
{x} = y.϶= y.▫϶,
(b)
x.ϵ^{i}= x.ϵ▫^{i}= y.ϵ▫^{i-1}= x.ϵ^{i-1}
for every i ≤ 1,
(c)
(x,y) ∉ (ϵ¯).
Let x.ɛɕ = y and y be new.
By definition, y.ϵ^{-i}= x.ϵ▫^{1-i} for every i > 0,
so that
y ϵ^{1-(i+1)}r
↔
x ϵ^{1-i}r.
Since x is non-terminal it follows that x ϵ^{1-i}r
is satisfied for i > 0 (put i = x.mli).
As a consequence,
y.mli = x.mli+1.
Claim C:
S preserves the rank of old objects,
i.e.
for every old object x, x.d = x.d_{0}.
Axiom (b~11) is satisfied:
x.ϵ= x.ϵ¯
for every
x from O∖O.∍.
Proof:
Consider the ranking products(Ƣ_{0}^{}, ϵ▫, …)
and
(Ƣ, ϵ, …)
of
S_{0} and S, respectively.
Let y be an old object.
To prove that y.d = y.d_{0}
it is sufficient to show that
(∗)
for every (x,-i) from (y,0).϶
there is
an (u,-k) from (y,0).▫϶
such that
(x,-i).d ≤ (u,-k).d.
Assume (x,-i) ϵ (y,0)
and (x,-i) is not from Ƣ_{0}^{}
so that x is a new object.
Let a be such that a.ɛɕ = x.
The requested pair (u,-k) in (∗) is then found as follows.
If i = 0 then x ϵ y
so that, by claim A2, x ≤ u ϵ▫ y
for some u and thus (u,0) is the requested pair.
Assume further that i > 0. Then
x ϵ¯^{1-i} y (by definition of (Ƣ, ϵ)),
a ϵ▫^{2-i} y
(since, by definition, a ϵ▫^{1-j} y ↔ x ϵ^{-j} y
for every natural j),
(a,1-i) ϵ▫ (y,0)
(a consequence of a ϵ▫^{2-i} y).
Since a.mli = x.mli-1, it follows
that (x,-i).d ≤ (a,1-i).d
and thus (a,1-i) is the requested pair.
Finally, assume that y is new
and
x.ɛɕ = y.
Since y.϶= {x} and x is bounded
it follows that
(y,0).d equals the maximum of (x,0).d + 1 and
(y,-1).d + 1.
Because
(y,-1).d = y.mli-1 =
x.mli ≤ x.d = (x,0).d
the equality y.d = x.d+1 follows.
By claim C1, S and S_{0} have the same unbounded objects.
If x is an unbounded (and therefore old) object then,
x.ϵ= x.ϵ▫= x.ϵ▫¯= x.ϵ¯.
Extensional pre-completion
The diagram on the right shows an extension of
a powerclass complete basic structure S_{0}= (O_{0}, …)
to a
powerclass complete basic structure S= (O, …)
by an attachment of a free leafu.
New objects are those from u.ec^{∗}.
The difference between (O, ≤) and (O_{0}, ≤)
is indicated by dashed brown arrows (again in the reflexive transitive reduction).
The extension causes the following change for the m object
(of which u is a direct free leaf in S, i.e. u.ԏ= m):
m.▫∍= {a, b}⇢m.∍= {a, b, u.ec(2)}.
As a result, m becomes
extensionally consistent
(which was not the case in S_{0} since
m.▫∍= n.▫∍ and m ≰ n).
Moreover, all new objects are extensionally consistent
as well as all old objects that were extensionally consistent in S_{0}.
The above method can be used to achieve extensional consistency and
powerclass consistency for all objects.
Extensional pre-completion
Let
S= (O, ϵ, …)
be an ϵ5-structure
and
S_{0}= (O_{0}, ϵ▫, …)
a basic structure that is metaobject complete.
We say that
S is an extensional pre-completion of S_{0} if
S is an extension of S_{0}
such that the following conditions are satisfied:
.ec is total, well-founded and injective.
(So is therefore
(.ec) ∖ (.ec_{0}) in the restriction to new objects.)
.ɛɕ is the same as .ɛɕ_{0}.
For every new primary object x
and every i ≥ 0, j ≤ 1,
x.ec(i).϶= {x}.ec(i-1),
x.ec(i).϶¯^{j}= {x}.ec(i-j).
In particular, the set O_{0} of old objects is closed
w.r.t. .ϵ^{∗-∗}.
For every new primary object x,
there is a unique old primary object y
such that, denoting k = y.mli,
for every i ≥ 0, j ≤ 1,
x.ϵ¯^{i}= y.ϵ¯^{i-k}⊎ {x}.ec(i),
that is,
x.ԏ= y, i.e.
x is a direct free leaf of y,
x.ec(i).ϵ= x.ϵ¯^{i+1},
x.ec(i).ϵ¯^{j}= x.ϵ¯^{i+j}.
For every old object y, the set y.ԏ(-1) ∖O_{0}
of new direct free leaves of y has
exactly
0 elements if y is pre-consistent in S_{0},
i.e.
(◉)
all objects from y.ec^{∗} are
e+p consistent (extensionally consistent and powerclass consistent),
2 elements otherwise.
(There can be a finer prescription which adds exactly 1 new
direct free leaf of y in relevant cases.)
Observation:
Every metaobject complete basic structure S_{0}
has an extensional pre-completion S.
Proof:
Let
S_{0}= (O_{0}, ϵ▫, ϵ▫¯^{⁽*⁾}, r, .ec_{0}, ɛɕ_{0})
be
a metaobject complete basic structure and construct S as follows.
Let (N, O_{0}, .ec_{1}, .ԏ_{1})
be a two-sorted structure that is isomorphic to
(Ṅ, O_{0}, .ec_{1}, .ԏ_{1})
where
Ṅ = Y × {0,1} ×ℕ where
Y is the set of (old) objects that do not satisfy (◉),
.ec_{1} is the map Ṅ → Ṅ
such that (y,i,j).ec_{1}= (y,i,j+1),
.ԏ_{1} is the map Ṅ →O_{0}
such that (y,i,j).ԏ_{1}= y.
Let S= (O, ϵ, ϵ¯^{⁽*⁾}, r, .ec, .ɛɕ)
be an ϵ5-structure that is
an extension of
S_{0} defined by:
O=O_{0}⊎ N,
(.ec) = (.ec_{1}) ⊎ (.ec_{0}),
.ɛɕ = .ɛɕ_{0},
ϵ and ϵ¯^{⁽*⁾}
are extensions of
ϵ▫ and ϵ▫¯^{⁽*⁾}
according to Ⅲ and Ⅳ.
The y object in Ⅳ equals x.ԏ_{1}.
The extensional pre-completion theorem
Proposition:
Let S be an extensional pre-completion
of a metaobject complete basic structure S_{0}.
Then
S is metaobject complete,
extensionally consistent and powerclass consistent.
Proof:
Let S= (O, ϵ, …)
be an extensional pre-completion of
a metaobject complete basic structure S_{0}= (O_{0}, ϵ▫, …).
Claim A: (Observations)
For every old objects x, y
and every natural i,
(a)
x ϵ^{i} y ↔ x ϵ▫^{i} y,
(b)
x ϵ¯^{i} y ↔ x ϵ▫¯^{i} y.
For every new object x and every integer i,
x.϶^{i}= x.϶¯^{i}= {x}.ec(-i)
(i.e.
x ϵ^{i} y
↔ x ϵ¯^{i} y
↔ x.ec(i) = y
for every new objects x, y),
x.ϵ¯= x.ϵ.
Claim B:
(b~2) is satisfied.
For every objects x, y, z and every integer
k, ℓ
if x ϵ¯^{k} y ϵ¯^{ℓ} z then x ϵ¯^{k+ℓ} z.
(b~3) is satisfied:
For every objects x, y, z and every natural k,
if x ϵ y ϵ^{-k} z then x ϵ^{1-k} z.
(b~4) is satisfied:
(ϵ¯^{i}) ∩ (϶¯^{-i}) = .ec(i)
for every integer i.
(b~5) is satisfied: O.ϵ≤r.
(b~6) is satisfied: O=O.϶.
(b~7)(a) is satisfied:
z.϶^{-i}⊆ {z}.ec(i)
for every z ∈T∪O.ɛɕ and every natural i.
(b~7)(b) is satisfied:
x.ϵ= x.ϵ¯ for every
x ∈ (T∪O.ɛɕ).ec^{∗}.
(b~8) is satisfied:
If x.ɛɕ = y
then:
(a)
{x} = y.϶,
(b)
x.ϵ^{i}= y.ϵ^{i-1} for every i ≤ 1,
(c)
(x,y) ∉ (ϵ¯).
(b~10) is satisfied:
O.mli < ω.
(b~11) is satisfied:
x.ϵ¯= x.ϵ
for every
unbounded object x.
Proof:
Let x ϵ¯^{k} y ϵ¯^{ℓ} z, k, ℓ ∈ℤ.
Moreover, let (a,i) = (x.pr,x.eci)
so that x = a.ec(i).
Since the set O_{0} of old objects is
closed w.r.t. .ϵ^{∗-∗} there are 3 cases to check:
If all of x, y, z are new then
x.ec(k) = y and
y.ec(ℓ) = z so that
x.ec(k+ℓ) = z and thus x ϵ¯^{k+ℓ} z.
Assume that x and y are new and z is old.
Then
a.ec(i+k) = x.ec(k) = y ϵ¯^{ℓ} z→a ϵ¯^{i+k+ℓ} z→x = a.ec(i) ϵ¯^{k+ℓ} z.
Assume that x is new and y and z is old,
and denote m = a.ԏ.mli.
Then
a.ec(i) ϵ¯^{k} y ϵ▫¯^{ℓ} z
↔
a ϵ¯^{i+k} y ϵ▫¯^{ℓ} z
↔
a.ԏϵ▫¯^{i+k-m} y ϵ▫¯^{ℓ} z
→
a.ԏϵ▫¯^{i+k+ℓ-m} z
↔
a.ec(i) ϵ¯^{k+ℓ} z.
Let x ϵ y ϵ^{-k} z, k ∈ℕ.
If x is new then x ϵ¯ y and thus
(b~2) applies.
Otherwise, all of
x, y and z are old and thus
(b~3) in S_{0} applies.
(b~4)
follows by
O_{0}.ϵ^{∗-∗}=O_{0}
and claim A2.
Let x be a new object and let a,i,y be
such that a.ec(i) = x and a.ԏ= y.
Denote k = y.mli.
Then
x.ϵ=
a.ϵ¯^{i+1}= y.ϵ¯^{1+i-k}⊎ {a}.ec(i+1).
It follows that x.ϵ cannot contain
an old object z that is terminal in S_{0}.
(It would follow from z ∈ y.ϵ¯^{1+i-k}∩T
that z = y and k = 0 and thus z ϵ¯^{1+i} z
for i ≥ 0 which is not possible.)
Since by definition a.ec(i+1).↥= a.ϵ¯^{i+1} it follows that
a.ec(i+1) < y.ϵ¯^{1+i-k}≤r
and thus a.ec(i+1) < r.
This shows that x.ϵ≤r and, consequently,
O.ϵ≤r.
The O=O.϶ equality follows by:
x ϵ x.ec for every new object x.
Assume that z ∈T∪O.ɛɕ and i ∈ℕ.
If z is new then
z.϶^{-i}= {z}.ec(i) by definition of extensional pre-completion.
Assume further that z is old
and let a be a new primary object, j ∈ℕ
and denote k = a.ԏ.mli.
Then the following equivalences are satisfied:
a.ec(j) ϵ^{-i} z↔a ϵ¯^{j-i} z↔a.ԏϵ¯^{j-i-k} z↔a.ԏ= z and k = 0 and j = i.
Since z satisfies the
(◉) condition
the a.ԏ= z is disallowed.
This
shows that all objects from z.϶^{-i} are old and thus
z.϶^{-i}= z.▫϶^{-i}= {z}.ec(i)
by (b~7) in S_{0}.
(b~7)(b)
follows by
O_{0}.ϵ^{∗-∗}=O_{0}
and the
x.ϵ= x.ϵ¯ equality which holds for every new object x.
Assume that x.ɛɕ = y
so that both
x and y are old objects.
To show that y.϶ cannot contain new objects and thus
(b~8)(a) is satisfied,
proceed similarly as in the proof of (b~7).
Subsequently,
(b~8)(b)
follows by
O_{0}.ϵ^{∗-∗}=O_{0}
and
(b~8)(c)
follows by embedding of ϵ¯.
By embedding of ϵ¯^{i}, the metalevel index of old objects is preserved.
Since new objects are from T.ec^{∗}
and S is already asserted to be pre-basic,
it follows that
for every new object x,
x.mli = x.eci which is finite by definition.
(Apply (a) x.mli = 0 for x ∈T
and (b) x.ec(i).mli = x.mli + i.)
Since new objects are from T.ec^{∗},
every unbounded object x is old
and is subject to the x.ϵ¯= x.ϵ equality
as a consequence of O_{0}.ϵ=O_{0}.
Claim C:
S preserves the rank of old objects,
i.e.
for every old object x, x.d = x.d_{0}.
S is metaobject complete
and e+p consistent
(extensionally consistent and powerclass consistent).
Proof:
Observe first that since new objects are from T.ec^{∗}
and S is pre-basic it follows that
x.d = x.mli = x.eci for every new object x.
Consider the ranking products(Ƣ_{0}^{}, ϵ▫, …)
and
(Ƣ, ϵ, …)
of
S_{0} and S, respectively.
Observe that since new objects of S
are from T.ec^{∗}
it follows that new objects of
(Ƣ, ϵ, …) have zero index, that is,
they are of the form (x,0) where x is a new object of S.
Let z be an old object that is well-founded in (O, ϵ)
and thus in (Ƣ,ϵ).
It is sufficient to show that
for every new object ẋ from (z,0).϶ there is an
old object ḃ from (z,0).϶ such that
ẋ.d ≤ ḃ.d.
Assume therefore that ẋ = (x,0) is a new object from (z,0).϶
so that x ϵ z
and let a, i, k be such that
a.ec(i) = x and k = a.ԏ.mli.
Then
the requested ḃ object is found according to:
x ϵ z↔a.ԏϵ¯^{1+i-k} z
↔
(a)
i ≤ k and (a.ԏ, i-k) ϵ (z,0)
(by definition of (Ƣ,ϵ)) or
(b)
i ≥ k and a.ԏ.ec(i-k) ϵ z
(by properties of .ec).
In the (a) case, put ḃ = (a.ԏ, i-k),
in the (b) case, put ḃ = (a.ԏ.ec(i-k),0).
In both cases, ḃ.d = k + (i-k) = ẋ.d.
Since S_{0} is metaobject complete, and,
in the restriction to new objects, .ec is total
and identical to .ɛϲ, it follows that S is metaobject complete.
To show that S is e+p consistent, let x be an object.
If x is new then it is a terminal or a singleton
and is therefore e+p consistent by observations about
extensional consistency and powerclass consistency.
If x is an old object that is not e+p consistent in S_{0},
then x.pr is subject to the
(◉) condition
in the definition of extensional pre-completion.
As a consequence, x.pr has at least 2 direct free leaves.
Since their powerclass chains are infinite,
proposition B3 applies.
If x is an old object that is already e+p consistent in S_{0}
then
x is extensionally consistent in S since for every
object y,
∅≠ x.∍⊆ y.∍→y is old and
∅≠O_{0}∩ x.∍⊆O_{0}∩ y.∍→x ≤ y,
x is
powerclass consistent
in S since for every
non-empty set Y of old objects, all objects from Y.△ are old.
Rank pre-completion
The diagram on the right shows a basic structure S that is
powerclass complete,
extensionally consistent
(for every natural i,
r.ec(i).∍= b.ec(i).ec^{∗}
and
b.ec(i+1).∍= {b}.ec(i)
and thus
x.∍⊆ y.∍
→
x ≤ y for every non-terminal x,y),
singleton complete (since T.ec^{∗}=O.∍),
and
powerclass consistent
(r is not powerclass-like since {b,b.ec} has no
upper bound).
If ϖ= ω
(recall that ϖ is a
fixed limit ordinal
and ω is the least limit ordinal)
then S is also
and thus
pre-complete.
If ϖ > ω then S is not ∊-ranked
and needs to
be equipped with additional bounded objects to become pre-complete.
Note:
The S
structure shown by the diagram is a minimal basic structure
such that (a) and (b) are satisfied
and thus minimal such that (a)–(d) are satisfied.
It follows that if ϖ= ω then
S is a minimal pre-complete structure.
The omissible case ϖ= ω
Proposition:
Let S be a basic structure
such that
(a)
(∊) = (∊) ○ (∊^{0})
and
(b)
T.ϵ^{∗}=O
(i.e. S is ϵ-grounded
and therefore ϵ-ranked).
Then the following are satisfied:
For every object x,
(recall that r_{∊}(x) denotes the
∊-rank of x)
r_{∊}(x) = x.d
if x is well-founded in ϵ,
r_{∊}(x) ≥ ω
otherwise.
Corollary:
If ϖ= ω then S is ∊-ranked.
Proof:
Let W be the set of objects that are well-founded in ϵ
and denote r() the rank function in (W,ϵ).
Let x be an object from W.
If r(x) ≤ϖ then x.϶= x.∍
and thus r(x) = r_{∊}(x) = r_{ϵ}(x) = x.d.
Assume that r(x) > ϖ.
Then
ϖ≤ r(u) for some u from x.϶ and thus
ϖ= u.d
= sup(u.∍.d)
(by the induction assumption)
≤ sup(u.∍^{0}.d)
(since (∊) = (∊) ○ (∊^{0}))
≤ sup(x.∍.d)
(since u ϵ¯ x so that u.∍^{0}⊆ x.∍)
= r_{∊}(x).
It follows that r_{∊}(x) = r_{ϵ}(x) = x.d =ϖ.
Finally, if x ∈O∖ W then
x.϶^{i}≠∅ for every natural i
(a consequence of non-well-foundedness of x),)
↔
x ∈O.ϵ^{i} for every natural i
→
x ∈T.ϵ^{i} for infinitely many natural i
(since T.ϵ^{∗}=O)
↔
x ∈T.∊^{i} for infinitely many natural i
(by proposition A2)
→
r_{∊}(x) ≥ ω
(by definition of ∊-rank).
Rank pre-complete structure
We say that a basic structure S is rank pre-complete
if every primary object that is non-well-founded in ϵ
is ∊-ranked.
Observation:
Let S be a basic structure that is rank pre-complete.
If (∊) = (∊) ○ (∊^{0}) then
every non-well-founded object is ∊-ranked.
Corollary:
If, in addition, S is extensionally consistent and metaobject complete
then S is ∊-ranked.
As a consequence, for every object x,
r_{∊}(x) =ϖ
↔
r_{∊}(x.pr) =ϖ.
Let S be a basic structure that is
(a) is extensionally consistent,
(b) metaobject complete
(i.e.
(b_{1}) powerclass complete and
(b_{2}) singleton complete)
and
(c) rank pre-complete.
By the ∊-levelling proposition,
it follows from
(a) and (b_{1})
that
S is ∊-levelled, in particular
T.ϵ^{∗}=O.
Moreover, it follows from (b_{2}) that
(∊) = (∊) ○ (∊^{0})
so that, denoting W the set of objects that are well-founded in ϵ,
for every object x,
if x ∉ W then x is ∊-ranked according to
the previous proposition,
if x ∈ W then x is ∊-ranked according to
proposition 1 in the previous subsection.
Rank pre-completion
Let
S= (O, ϵ, ϵ¯^{⁽*⁾}, r, .ec, .ɛɕ)
be an ϵ5-structure
and
S_{0}= (O_{0}, ϵ▫, …)
a basic structure.
We say that
S is a rank pre-completion of S_{0} if
S is an extension of S_{0}
such that
(1)
.ec = .ec_{0} and .ɛɕ = .ɛɕ_{0},
and there is a (necessarily unique) map .τ
from O∖O_{0}
to O_{0}
such that the following are satisfied:
(2)
O.τ is the set all non-well-founded
primary objects of S_{0}
that are not ∊-ranked in S_{0}.
(3)
For every object y from O.τ,
if X = {y}.τ(-1)
(i.e. X is the fiber of y under .τ)
then the following holds:
(a)
X is closed w.r.t. .϶^{∗-∗}.
(b)
For every natural k > 0,
both y.϶^{-k}∩ X and X.϶^{-k} are empty.
(c)
X ⊆ y.϶∩ y.϶¯∩ y.↧.
(d)
(X, ϵ¯) = (X, ϵ) = (X, <) ≃ (ϖ, <).
In (d),
(X, <) ≃ (ϖ, <) means that
(X, <) is isomorphic to the strict order between
ordinals less than ϖ.
(4)
For every new object x,
every old object z
such that z ≠ x.τ
and every integer k ≤ 1,
x ϵ^{k} z ↔
x ϵ¯^{k} z ↔
x.τ ϵ▫¯^{i} z for some integer i ≤ k.
Note that ϵ▫^{k} and ϵ▫¯^{k} can be
used interchangeably here since
x.τ is unbounded.
Observation:
Every basic structure S_{0} has a unique rank pre-completion S,
up to isomorphism.
Proof:
Given a basic structure S_{0}= (O_{0}, ϵ▫, …),
proceed in the following steps.
Let Ṅ be the set of all pairs (x,i)
where x
an old primary objects that is not ∊-ranked in S_{0}
and i is an ordinal number less than ϖ.
Define relations ϵ, ≤, ϵ¯ and
ϵ^{-k}, k > 0, on Ṅ by
(x,i) ϵ (y,j)↔x = y and i < j,
(x,i) ≤ (y,j)↔x = y and i ≤ j,
(Ṅ, ϵ¯) = (Ṅ, ϵ)
and
(Ṅ, ϵ^{-k}) =∅.
Let (N, ϵ, ϵ¯^{⁽*⁾})
be isomorphic to
(Ṅ, ϵ, ϵ¯^{⁽*⁾}) via .ν : Ṅ → N
and let .τ be the unique map N →O_{0}
such that
(x,i).ν.τ = x.
Let S= (O, ϵ, ϵ¯^{⁽*⁾}, r, .ec, .ɛɕ)
be an extension of S_{0} such that
.ec = .ec_{0} and
.ɛɕ = .ɛɕ_{0},
O=O_{0}⊎ N,
(N, ϵ, ϵ¯^{⁽*⁾}) is a restriction of a reduct of S,
the intersection of ϵ and ϵ^{k}, k ≤ 1,
with N ×O_{0}
is given by (3bc) and (4).
The rank pre-completion theorem
Proposition:
Let S be a rank pre-completion
of a basic structure S_{0}.
Then
Proof:
The proof is accomplished in a series of claims below.
Assume that S= (O, ϵ, …)
is a rank pre-completion of
a basic structure S_{0}= (O_{0}, ϵ▫, …).
Claim A: (Observations)
The set O_{0} of old objects is closed w.r.t.
.ϵ^{∗-∗}.
Corollary:
For every old objects x, y
and every natural i,
(a) x ϵ^{i} y ↔ x ϵ▫^{i} y,
(b) x ϵ¯^{i} y ↔ x ϵ▫¯^{i} y.
For every new object x, x.mli = x.τ.mli.
For every new object x and every integer k,
x.϶¯^{k}= x.϶^{k} and x.ϵ¯^{k}= x.ϵ^{k}.
(In particular, x.ϵ= x.ϵ¯.)
For every new object x
and every integer k,
x ϵ^{k} x.τ
↔ x ϵ¯^{k} x.τ
↔ k ≥ 0.
The set O_{0}.▫∍ of old bounded objects is closed
w.r.t. .϶^{∗-∗}.
The set of old objects that are well-founded in (O_{0}, ϵ▫) is closed
w.r.t. .϶^{∗-∗}.
Proof:
The O_{0}=O_{0}.ϵ^{∗-∗} equality
is a consequence of (3a).
Let x be a new object and k a natural number.
If x.τ =r then
x ϵ^{1-k}r
↔
k ≤ 1 by (3bc).
Otherwise (4) applies:
x ϵ^{1-k}r↔x.τ ϵ▫^{1-i}r for some natural i ≥ k↔x.τ ϵ▫^{1-k}r.
It follows in both cases that x.mli = x.τ.mli.
The equalities follow directly from (3) and (4).
Let x be a new object and k an integer.
For k ≤ 1 the requested equivalences are asserted by (3bc).
Assume that k > 1 and denote X = {x}.τ.τ(-1).
Then it follows from (3d)
that x.ϵ¯^{k-1}∩ X = x.ϵ^{k-1}∩ X ≠∅
(the non-emptiness is a consequence of ϖ being a limit ordinal)
so that x.τ ∈ x.ϵ¯^{k-1}.ϵ¯∩ x.ϵ^{k-1}.ϵ.
Assume that x is a new object,
and z is an old object such that
x ϵ^{k} z for some integer k ≤ 1.
We have to show that z is unbounded in S_{0}.
If x.τ = z then z
is unbounded by definition of O.τ.
For x.τ ≠ z condition (4) applies so that
x.τ ϵ▫^{i} z for some integer i ≤ k.
Since x.τ is unbounded in S_{0},
so must be z.
Claim B:
(b~2) is satisfied.
For every objects x, y, z and every integer
k, ℓ
if x ϵ¯^{k} y ϵ¯^{ℓ} z then x ϵ¯^{k+ℓ} z.
(b~3) is satisfied:
For every objects x, y, z and every integer ℓ,
if x ϵ y ϵ^{ℓ} z then x ϵ^{1+ℓ} z.
(b~4) is satisfied:
For every objects x, y and every integer k,
x ϵ¯^{k} y ϵ¯^{-k} x
↔
x.ec(k) = y.
(b~5) is satisfied: O.ϵ≤r.
(b~6) is satisfied: T⊆O.϶.
(b~7)(a) is satisfied:
x.϶^{-i}⊆ {x}.ec(i)
for every x ∈T∪O.ɛɕ and every natural i.
(b~7)(b) is satisfied:
x.ϵ= x.ϵ¯ for every
x ∈ (T∪O.ɛɕ).ec^{∗}.
(b~8) is satisfied:
If x.ɛɕ = y
then:
(a)
{x} = y.϶,
(b)
x.ϵ^{i}= y.ϵ^{i-1} for every i ≤ 1,
(c)
(x,y) ∉ (ϵ¯).
(b~10) is satisfied:
O.mli < ω.
Proof:
Assume x ϵ¯^{k} y ϵ¯^{ℓ} z, k,ℓ ≤ 1,
k+ℓ ≤ 1.
If all of x, y and z are new
then k, ℓ ≥ 0
and x ϵ¯^{k+ℓ} z follows by ordering of ordinals.
If x, y are new and z is old such that y.τ = z
then
A4 applies.
If x, y are new and z is old such that y.τ ≠ z
then k ≥ 0
and there is a j ≤ ℓ such that y.τ ϵ▫¯^{j} z.
Consequently,
j ≤ k+ℓ and x.τ ϵ▫¯^{j} z, and therefore
x ϵ¯^{k+ℓ} y.
If x is new and y and z old and x.τ = y = z
then
k, ℓ ≥ 0 and A3 applies.
If x is new and y and z old and x.τ = y ≠ z
then
k ≥ 0
so that ℓ ≤ k+ℓ and thus x ϵ^{k+ℓ} z
(using ℓ for i in (4)).
If x is new and y and z old and x.τ ≠ y
then
there is an i ≤ k such that x.τ ϵ▫¯^{i} y ϵ▫¯^{ℓ} z.
Consequently,
i+ℓ ≤ k+ℓ and x.τ ϵ▫¯^{i+ℓ} z,
and therefore
x ϵ¯^{k+ℓ} y.
If all of x, y and z are old
then x ϵ¯^{k+ℓ} y
follows by (b~2) in S_{0}.
Apply claims A3 and A5.
Let x, y be objects and k an integer
such that x ϵ¯^{k} y ϵ¯^{-k} x or x.ec(k) = y.
Since O_{0}.ϵ^{∗-∗}=O_{0} and
.ec = .ec_{0},
it follows that
either both x and y are old or both are new.
If both x and y are old then
(b~4) in S_{0} applies.
If both x and y are new then necessarily k = 0
and the
reflexivity and antisymmetry of (ϖ, ≤) applies.
O.ϵ≤r
since x ≤r for every new object x
and T_{0}.϶=∅.
(b~6) is a consequence of:
Every terminal object is old.
If x ∈T∪O.ɛɕ
then x is necessarily an old object that is bounded in S_{0}
so that claim A5 applies.
All objects from
(T∪O.ɛɕ).ec^{∗}
are old so that
O_{0}.ϵ^{∗-∗}=O_{0} applies.
If x.ɛɕ = y then both x and y are old
and bounded in S_{0}.
Consequently,
apply
the closedness of O_{0}.▫∍ w.r.t. .϶^{∗-∗},
the closedness of O_{0} w.r.t. .ϵ^{∗-∗},
and
ϵ▫¯^{i} being embedded into ϵ¯^{i} for every integer i.
This follows from x.mli = x.τ.mli
for every new object x (claim A2).
Claim C:
For every old object x, x.d = x.d_{0}.
(I.e. S preserves the rank of old objects.)
For every new object x,
x.d = x.mli + r_{ϵ}(x) < ϖ
(and thus r_{∊}(x) = r_{ϵ}(x)).
Axiom (b~11) is satisfied:
x.ϵ= x.ϵ¯
for every
x from O∖O.∍.
Every non-well-founded primary object x is ∊-ranked.
(That is, S is rank pre-complete.)
Proof:
Let X be the set of all old objects that are well-founded in ϵ▫.
Since X is closed w.r.t.
w.r.t. .϶^{∗-∗}
it follows that
x.d = x.d_{0}
for every x from X.
Objects
from O_{0}∖ X stay non-well-founded in ϵ
and have therefore rank ϖ both in S_{0} and S.
Denote N =O.τ(-1) the set of new objects and
proceed by well-founded
induction on (N, ϵ).
Let x be from N.
Since N.϶^{-k} is empty for every k > 0
and x.϶^{∗}.mli = {x}.mli
it follows by definition of .d
that
x.d =ϖ∧
(sup {a.d + 1 | a ϵ x}
∨
(x.mli + sup {i |
a ϵ^{i} x, i ∈ℕ})).
If x.϶=∅
(i.e. r_{ϵ}(x) = 0)
then a ϵ^{i} x ↔ i = 0
so that x.d =ϖ∧ (x.mli + 0) = x.mli.
Assume further that r_{ϵ}(x) > 0.
By induction assumption,
Since
r_{ϵ}(x) ≥ sup {i | a ϵ^{i} x, i ∈ℕ}
it follows that
x.d =ϖ∧ (x.mli + r_{ϵ}(x)).
Finally,
r_{ϵ}(x) < ϖ
since by definition of rank pre-completion,
(X, ϵ) ≃ (ϖ, <)
where X = {x}.τ.τ(-1).
This follows by
the closedness of O_{0} w.r.t. .ϵ^{∗}.
Let x be a non-well-founded primary object
and denote X = x.τ(-1).
It follows that
(X, ∊) = (X, ϵ) ≃ (ϖ, <)
and thus
r_{∊}(X) =ϖ.
Since X ⊆ x.∍ the statement follows.
ϵ-structure
In this section we develop a formal language
for families of structures based on a single relation of membership on a set.
ϵ-structure
By an ϵ-structure
we mean a structure (O, ϵ)
where
O is a set of objects, and
ϵ is the membership relation between objects.
There are no additional constraints for (O, ϵ).
Let r_{ϵ}() be the
ϖ-limited rank function
O→ϖ+1, i.e.
r_{ϵ}(x) =ϖ for every object x that is non-well-founded
in ϵ,
and
r_{ϵ}(x) =ϖ∧
sup {r_{ϵ}(a) + 1 | a ϵ x }
if x is well-founded in ϵ.
An object x is bounded if r_{ϵ}(x) < ϖ,
otherwise x is unbounded.
Similarly,
a set Y ⊆O is
bounded (resp. unbounded)
if sup { r_{ϵ}(x) | x ∈ Y } is less (resp. equal to) ϖ.
The following derived relations
≤, ∊, ∊^{0} and ϵ¯
between objects are distinguished.
Like in ϵ5-structures,
we use the symbols .ϵ/.϶ and .↥/.↧
for
the respective image/preimage operators of ϵ and ≤.
Similarly for ∊, ∊^{0} and ϵ¯.
≤ (the inheritance relation)
is a pre-order on O defined by
x ≤ y
↔
x = y or ∅≠ x.϶⊆ y.϶.
∊ (the bounded membership)
is the domain-restriction of ϵ to bounded objects,
i.e.
x ∊ y ↔
x ϵ y and r_{ϵ}(x) < ϖ.
∊^{0} is a transitive relation defined as
the domain-restriction of ≤ to bounded objects,
i.e.
x ∊^{0} y ↔
x ≤ y and r_{ϵ}(x) < ϖ.
ϵ¯ (the power membership)
is a subrelation of ϵ defined by
x ϵ¯ y ↔
x.↧⊆ y.϶.
Observations:
∊ is a well-founded relation of rank at most ϖ+1.
For every objects x and y the following are satisfied:
r_{ϵ}(x) =ϖ →
r_{∊}(x) = r_{ϵ}(x) (by the above equivalence).
We conclude that r_{ϵ}() and r_{∊}() coincide.
∊-structure
As a counterpart to ϵ-structures
we develop a language of ∊-structures.
We start with the ∊ symbol
which stands for an (intentionally) well-founded membership.
∊-structure
x ≤ y
↔
x = y or ∅≠ x.∍⊆ y.∍
x ∊^{0} y
↔
x ≤ y and x ∈O.∍
x ϵ¯ y
↔
x.∍^{0}⊆ y.∍
x ϵ y
↔
x ∊ y or x ϵ¯ y
By an ∊-structure
we mean a structure
(O, ∊)
where
O is the set of objects, and
∊ is the bounded membership between objects.
The derived relations ≤, ∊^{0}, ϵ¯ and ϵ
between objects are defined according to the table to the right.
The terminology for these relations is the same as in ϵ-structures.
Moreover,
the i-th power of ϵ, ϵ¯ and ∊
for a positive natural i is
defined in the usual sense of relational composition.
The 0-th power of ϵ or ϵ¯ equals ≤.
Observations:
Like in ϵ-structures,
(a) ≤ is a pre-order,
(b) ∊^{0} is transitive,
(c) (∊) ○ (≤) = (∊),
(d) (ϵ) ○ (≤) = (ϵ).
∊ equals the domain restriction of ϵ to O.∍.
O.∊=O.ϵ.
Proof:
We should prove that for every objects x, y,
x ∊ y
↔
x ϵ y and x ∈O.∍.
The →
direction follows by (∊) ⊆ (ϵ).
To show ←,
assume
x ϵ y and x ∈O.∍.
If x ϵ¯ y then, by definition of ϵ¯ and ∊^{0},
x ∈ x.∍^{0}⊆ y.∍ and thus x ∊ y.
Otherwise, if (x,y) ∉ (ϵ¯),
then x ∊ y follows directly by definition of ϵ.
Proposition:
For every objects x, y the following holds:
x.∍⊆ y.∍
→
x.϶¯⊆ y.϶¯.
x.∍⊆ y.∍
↔
x.϶⊆ y.϶.
x ≤ y
↔
x = y or ∅≠ x.϶⊆ y.϶.
Assume
(a) (∊) = (∊) ○ (∊^{0})
and
(b) O=O.∊∪O.∍.
Then for every objects x, y the following holds:
x.∍=∅
→
x.↧= x.∍^{0}= {x}.
{x} = y.∍
→
y.↧= y.∍^{0}⊇ {y}.
(If antisymmetry of ≤ is asserted then ⊇
can be replaced by =.)
x.∍^{0} is non-empty.
x ≤ y
↔
x.∍^{0}⊆ y.∍^{0}.
x.↧⊆ y.϶
↔
x.∍^{0}⊆ y.∍
(↔ x ϵ¯ y by definition of ϵ¯).
x.↧= y.϶
↔
x.∍^{0}= y.∍.
x.↧= y.϶
→
x.ϵ¯= y.↥.
x.↧= y.϶
↔
x.ϵ¯= y.↥
assuming x.↧= z.϶ for some object z.
Proof:
Assume
x.∍⊆ y.∍.
Then for every object u,
u ϵ¯ x ↔ u.∍^{0}⊆ x.∍
→ u.∍^{0}⊆ y.∍
↔ u ϵ¯ y.
The ←
direction follows by ∊
being a domain-restriction of ϵ,
→ follows from A1.
Apply the previous statement and the
O.ϵ=O.∊ equality.
Let x be such that x.∍=∅.
Then
x.↧= {x} by definition of ≤.
By definition of ∊^{0},
x.∍^{0}= x.↧∩O.∍.
Since x ∈O.∍ by (b),
the equality x.∍^{0}= {x} follows.
Apply condition (a).
For x.∍=∅ apply B1.
Otherwise, apply condition (a):
a ∊ x →
a ∊ b ∊^{0} x for some b.
If x ≤ y then x.↧⊆ y.↧.
Since z.∍^{0}= z.↧∩O.∍ for every object z,
the inclusion x.∍^{0}⊆ y.∍^{0} follows.
Conversely, assume
x.∍^{0}⊆ y.∍^{0}.
If x.∍=∅ then {x} = x.∍^{0}
and thus x ∊^{0} y, in particular
x ≤ y.
If x.∍≠∅ then apply (a):
x.∍^{0}⊆ y.∍^{0}
→
x.∍^{0}.∍⊆ y.∍^{0}.∍
→
x.∍⊆ y.∍
→
x ≤ y.
The →
direction follows by ∊^{0} and ∊
being the domain-restriction of ≤ and ϵ,
respectively, to O.∍.
To show ←,
assume
x.∍^{0}⊆ y.∍, i.e.
x ϵ¯ y.
Then for every object u,
u ≤ x
→
u.∍^{0}⊆ x.∍^{0}⊆ y.∍
→
u ϵ¯ y
→
u ϵ y.
(The first implication uses (a).)
This shows x.↧⊆ y.϶.
The →
direction follows by the same argument as with the previous statement.
To show ←,
assume
x.∍^{0}= y.∍.
Let u be an object from y.϶. Then
u ϵ¯ y
↔
u.∍^{0}⊆ y.∍
→
u.∍^{0}⊆ x.∍^{0}
↔
u ≤ x,
(u,y) ∉ (ϵ¯)
→
u ∊ y
→
u ∈ x.∍^{0}
→
u ≤ x.
This shows x.↧⊇ y.϶.
The converse inclusion is shown like in the previous statement.
Assume x.↧= y.϶.
(In particular y.϶≠∅.)
Then for every object u,
x ϵ¯ u
↔
x.↧⊆ u.϶
↔
y.϶⊆ u.϶
↔
y ≤ u.
(The last equivalence is due A3.)
This shows x.ϵ¯= y.↥.
Assume x.↧= z.϶ and x.ϵ¯= y.↥.
Using the previous statement we obtain
y.↥= z.↥
→
y ≤ z ≤ y
→
y.϶= z.϶.
∊-ϵ-structure
By an ∊-ϵ-structure we mean an ∊-structure
(O, ∊) satisfying the following conditions:
(∊-ϵ~1)
∊ is well-founded.
(∊-ϵ~2)
O.∍
equals the set of objects
whose ∊-rank
is strictly less than ϖ.
(In particular,
O=O.∊∪O.∍.)
(∊-ϵ~3)
(∊) = (∊) ○ (∊^{0}).
(∊-ϵ~4)
(ϵ) = (∊) ∪ (ϵ¯).
(Bounding monotonicity)
(∊-ϵ~5)
For every objects x, y,
x.∍⊆ y.∍
→
x.϶⊆ y.϶.
(Extensional consistency)
(∊-ϵ~6)
For every objects x, y,
x.∍^{0}⊆ y.∍
→
x.↧⊆ y.϶.
(Power-extensional consistency)
The horizontal line indicates that conditions
(∊-ϵ~4)–(∊-ϵ~6) are
redundant.
They are used for further reference.
Correspondence of ϵ-structures and ∊-structures
Proposition:
Let S= (O, ϵ, ≤, ∊, ∊^{0}, ϵ¯) be
a structure of 5 relations on a set O of objects
satisfying
(∊-ϵ~1)–(∊-ϵ~6).
Then the following are equivalent:
S is a definitional extension of an ϵ-structure.
(In this case, (∊-ϵ~1) is redundant.)
S is a definitional extension of an ∊-structure.
(That is, S is an ∊-ϵ-structure.)
That is, assuming (∊-ϵ~1)–(∊-ϵ~6),
conditions
(ⅰ) and
(ⅱ) in the following table are equivalent:
Relation
(ⅰ)
Definition in ϵ-structure
(ⅱ)
Definition in ∊-structure
∊
x ϵ y and r_{ϵ}(x) < ϖ
↔
x ∊ y
≤
x = y or ∅≠ x.϶⊆ y.϶
↔
x = y or ∅≠ x.∍⊆ y.∍
∊^{0}
x ≤ y and r_{ϵ}(x) < ϖ
↔
x ≤ y and x ∈O.∍
ϵ¯
x.↧⊆ y.϶
↔
x.∍^{0}⊆ y.∍
ϵ
x ϵ y
↔
x ∊ y or x ϵ¯ y
Proof:
ⅰ→ⅱ.
Assume that (O, ϵ) is an ϵ-structure in
which
(∊-ϵ~1)–(∊-ϵ~6) are
satisfied.
Observe first that (∊-ϵ~2)
has already been considered for ϵ-structures as
B(•).
As a particular consequence of this condition,
for every object x,
(α)
∅≠ x.϶ ↔ ∅≠ x.∍,
(β)
r_{ϵ}(x) < ϖ ↔ x ∈O.∍.
Using this, we can check the equivalences for the 5 relations.
For ∊ there is nothing to check.
For ∊^{0}, apply (β).
For ϵ, use (∊-ϵ~4).
≤:
Using (α), the equivalence for ≤ can be expressed as:
x.϶⊆ y.϶
↔
x.∍⊆ y.∍.
The → direction
follows by
∊ being a domain-restriction of ϵ.
The opposite direction is asserted by (∊-ϵ~5).
ϵ¯:
In the equivalence for ϵ¯,
the
→ direction follows
by ∊^{0} and ∊
being domain-restrictions of ≤ and ϵ,
respectively.
The opposite direction is asserted by (∊-ϵ~6).
ⅰ←ⅱ.
Assume that (O, ∊) is an
∊-ϵ-structure.
Let r_{∊}() be the ∊-rank function so that
(∊-ϵ~2) can be expressed as
r_{∊}(x) < ϖ ↔ x ∈O.∍.
Since ∊ is a domain-restriction of ϵ to O.∍
(see observation 2)
it follows that
(O.∍, ∊) coincides with (O.∍, ϵ).
Recall that the .d function is by definition
the ϖ-limited rank w.r.t. ϵ.
We therefore obtain
x ∈O.∍ →
r_{∊}(x) = r_{ϵ}(x) < ϖ
(since r_{∊}(x) < ϖ and
(O.∍, ∊) = (O.∍, ϵ)),
x ∉O.∍ →
r_{∊}(x) = r_{ϵ}(x) =ϖ
(since r_{∊}(x) =ϖ and r_{∊}(x) ≤ r_{ϵ}(x)).
As a consequence,
(•)
r_{ϵ}(x) < ϖ ↔ x ∈O.∍.
Using this, we can check the equivalences for the 5 relations.
(∊):
Apply (•) and the fact
that ∊ is a domain-restriction of ϵ to O.∍.
(≤):
Apply Proposition A3.
(∊^{0}):
Apply (•).
(ϵ¯):
Apply Proposition B5.
(ϵ):
There is nothing to check.
Disallowed structures
The following diagrams show examples of
ϵ-structures
that are disallowed by
(∊-ϵ~4)–(∊-ϵ~6).
Each example violates just the indicated condition.
As usual,
ϵ
equals the composition
(→) ○ (≤)
where
→
is the (exact) relation indicated by blue arrows,
and ≤ is indicated by green arrows
(with possibly implicit arrow heads on the higher ends)
in its reflexive transitive reduction.
(ⅰ)
¬(∊-ϵ~4):
(a,b) ∈ (ϵ) ∖ ((∊) ∪ (ϵ¯))
(ⅱ)
¬(∊-ϵ~5):
a.∍= b.∍ and a.϶≠ b.϶
(ⅲ)
¬(∊-ϵ~6):
a.∍^{0}⊆ b.∍ and (a,b) ∉ (ϵ)
It is assumed that ϖ= ω.
Bounded objects are shown in beige color, the unbounded objects are in blue.
Bounded ∊-ϵ-structures
In contrast to
basic structures,
∊-ϵ-structures
need not have unbounded objects.
(Even an empty structure is allowed.)
Moreover, the following observation can be made:
Observation:
For every ∊-ϵ-structure (O, ∊),
the restriction
(O.∍, ∊)
is an ∊-ϵ-structure
(in which every object is bounded).
Pre-complete structure of ϵ
In this section we introduce a subfamily of
basic structures
that is closed w.r.t. hitherto described completion constructions.
As a particular consequence,
the structures appear as a (definitional extension of a) subfamily
of
∊-ϵ-structures.
Pre-complete structure of ϵ
We say that an ϵ5-structureS is pre-complete
if it is a basic structure such that
the following are satisfied.
For every object x,
r_{∊}(x) (the ∊-rank of x) equals x.d.
Similarly, a metaobject structure is
pre-complete if its correspondent basic structure is pre-complete.
Note:
In case ϖ= ω the (D) condition is redundant.
Proposition:
A pre-complete ϵ5-structure
S=(O, ϵ, ϵ¯^{⁽*⁾}, r, .ec, .ɛɕ)
is uniquely given by
ϵ or by ∊
according to the following table.
Moreover, S is an
∊-ϵ-structure.
Relation (subset)
(ⅰ)
Derived from ϵ
(ⅱ)
Derived from ∊
∊
x ϵ y and r_{ϵ}(x) < ϖ
↔
x ∊ y
≤
x = y or ∅≠ x.϶⊆ y.϶
↔
x = y or ∅≠ x.∍⊆ y.∍
∊^{0}
x ≤ y and r_{ϵ}(x) < ϖ
↔
x ≤ y and x ∈O.∍
ϵ¯
x.↧⊆ y.϶
↔
x.∍^{0}⊆ y.∍
ϵ
x ϵ y
↔
x ∊ y or x ϵ¯ y
.ec
x.↧= y.϶
↔
x.∍^{0}= y.∍
.ɛϲ
{x} = y.϶
↔
{x} = y.∍
r.↧=
O.ϵ
=
O.∊
Proof:
Let S= (O, ϵ, …)
be a pre-complete structure.
Make the following observations:
O=O.∊∪O.∍
(since x ≤r for every non-terminal object x and (A)
applies).
(∊) = (∊) ○ (∊^{0})
(since S is singleton-complete so that
(∊) = (.ɛϲ) ○ (∊^{0})).
O.∊^{0}=O
(by the previous two equalities).
O.∍ equals the set of objects
with bounded ∊-rank
(since S is ∊-ranked and O.∍=r.∍).
For every objects x, y, the following holds:
x.϶⊆ y.϶
↔
x.∍⊆ y.∍
(since S is extensionally consistent).
x.↧= y.϶
↔
x.∍^{0}= y.∍.
The → direction is by definition.
For the opposite direction, use powerclass completeness.
Assume x.∍^{0}= y.∍.
Since x.ec exists we have y.∍= x.ec.∍.
Since x.∍^{0} is non-empty it follows that x.ec = y.
x.↧⊆ y.϶
↔
x.∍^{0}⊆ y.∍.
This follows from (ϵ¯) = (.ec) ○ (≤):
x.↧⊆ y.϶
↔
x.ec.϶⊆ y.϶
↔
x.ec.∍⊆ y.∍
↔
x.∍^{0}⊆ y.∍.
This shows
the determination of S via ϵ or ∊
as well as
that conditions
(∊-ϵ~1)–(∊-ϵ~6)
are satisfied.
We have proved that the family of pre-complete ϵ5-structures
is definitionally equivalent to a family of ∊-structures.
The next subsection provides the corresponding axiomatization via ∊
and the definitional extension of ∊-structures.
The last axiom refers to
powerclass consistency which
we let be defined in ∊-structures the same way as in
ϵ5-structures.
Pre-complete structure via ∊
We say that an
∊-structure(O, ∊)
is
pre-complete if the following are satisfied:
(ep~1)
∊ is well-founded.
(ep~2)
∊ is weakly extensional:
for every x, y from O.∊,
if x.∍= y.∍ then x = y.
(ep~3)
O.∍ is the set of all objects x with bounded ∊-rank:
x ∈O.∍ ↔ r_{∊}(x) < ϖ.
(ep~4)
O.∍=r.∍
for some (necessarily unique) object r.
(ep~5)
For every object x,
there is an object y (= x.ec)
such that x.∍^{0}= y.∍.
(ep~6)
For every object x ∈O.∍,
there is an object y (= x.ɛϲ)
such that {x} = y.∍.
(ep~7)
For every object x,
if x is powerclass-like then x is a powerclass.
Observations:
A pre-complete ∊-structure is an
∊-ϵ-structure.
(The (∊) = (∊) ○ (∊^{0}) equality is follows by
(ep~6).)
The induced ∊-structure of a pre-complete ϵ5-structure
is pre-complete.
Proposition:
x ≤ y
↔
x = y or ∅≠ x.∍⊆ y.∍
r.∍
=
O.∍
x.ec = y
↔
x.∍^{0}= y.∍
x.ɛϲ = y
↔
{x} = y.∍
x ∊^{0} y
↔
x ≤ y and x ∈O.∍
x ϵ¯ y
↔
x.∍^{0}⊆ y.∍
x ϵ y
↔
x ∊ y or x ϵ¯ y
Let S_{a}= (O, ∊) be a pre-complete ∊-structure
and let
S_{b}= (O, ≤, r, .ec, .ɛϲ) be a structure
definitionally derived from S_{a} according to the table on the right.
Then S_{b} is a
metaobject structure
that is pre-complete.
Proof:
Assume that
S_{a} and S_{b} are as in the antecedent of the proposition.
Note first that in S_{a} (due (ep~6)),
(∊) = (.ɛϲ) ○ (≤),
which is exactly how ∊ is derived in
S_{b} –
thus ∊ is disambiguated.
Observe also that since r.∍≠∅,
we have x ≤r ↔ x.∍≠∅
so that T=O∖O.∊.
Subsequently, we verify that
S_{b} satisfies the properties of a
grounded metaobject structure:
(mo~1):
Inheritance, ≤, is a partial order.
(mo~2):
x ≤ y ↔ x.ec ≤ y.ec
for every objects x, y.
(mo~3):
Objects from T.ec^{∗}
are minimal in ≤.
(mo~4):
For every object x,
x.ec ≤r.
(mo~5)':
T.∊^{∗}=O.
(mo~6):
The singleton map, .ɛϲ, is injective.
(mo~7):
Objects from O.ɛϲ.ec^{∗}
are minimal in ≤.
(mo~8):
x ≤ y ↔ x.ɛϲ ≤ y.ec
for every objects x, y
such that x.ɛϲ is defined.
(mo~9)':
For every object x,
x.ɛϲ is defined ↔
r_{ϵ}(x) < ϖ.
(1). ≤ is a preorder by definition.
The antisymmetry of ≤ is asserted by
(ep~2).
(2).
Apply proposition A3:
x ≤ y
↔
x.∍^{0}⊆ y.∍^{0}.
(3).
Apply propositions B1 and B2.
(4).
Apply x.∍^{0}≠∅
(proposition B3).
(5).
Apply well-foundedness of ∊.
(6).
Injectivity of .ɛϲ follows by definition.
(7).
Apply proposition B2.
(8).
By (ep~3) and (ep~6),
x.ɛϲ is defined iff x ∈O.∍.
Therefore, if x.ɛϲ is defined then
x.ɛϲ ≤ y.ec
↔
x.ɛϲ.∍⊆ y.ec.∍
↔
{x} ⊆ y.ec.∍
↔
x ∊ y.ec
↔
x ≤ y.
(9).
Apply x ∈O.∍ ↔
r_{ϵ}(x) < ϖ which holds in
∊-ϵ-structures.
Finally,
being a definitional extension of
an ∊-ϵ-structure,
S_{b} is ∊-ranked.
The powerclass consistency condition is explicitly asserted by
(ep~7).
Alternative formulation of (ep~7)
Proposition:
Assume
that S= (O, ∊) is an ∊-structure
satisfying
(ep~1)—(ep~6)
(that is, S is pre-complete
up to (ep~7)).
Let x be an object such that
x is a powerclass-like.
Then for every object u the following are equivalent:
u.↧= x.϶
(↔
x.϶ is a principal ideal
↔
u.ec = x
↔
u.∍^{0}= x.∍
↔
x is a powerclass).
u.∍= x.∍.∍≠∅ or {u} = x.∍.
Note:
By (ep~2), u in (ⅱ)
is unique whenever x.∍.∍
is non-empty, that is, whenever x ∉T.ec.
Proof:
(ⅰ)→(ⅱ).
If
u.↧= x.϶
then
u.∍= u.↧.∍= x.϶.∍= x.∍.∍.
If in addition u.∊=∅, that is, x ∈T,
then {u} = {x}.ϲɛ = {x}.ce = x.∍.
To prove (ⅰ)←(ⅱ),
assume that x is an object such that
x is powerclass-like, that is,
(∗)
x.϶= x.϶¯ and
for every X ⊆ x.∍ such that the ∊-rank
of X is bounded,
there is an object y such that X ≤ y ∈ x.϶.
Now let u be an object satisfying (ⅱ).
If {u} = x.∍ then
u.↧= {u} = x.∍= x.϶.
Assume that u.∍= x.∍.∍≠∅.
We prove that u.∍^{0}= x.∍.
The ⊇ inclusion is immediate.
To prove u.∍^{0}⊆ x.∍, let a be from u.∍^{0},
(that is, a ∈ u.↧∩O.∍)
and denote
X = a.∍.ɛϲ the singleton image of a.∍.
Make the following observations about X:
The ∊-rank of X equals a.d + 1
and is therefore bounded.
X ⊆ x.∍.
a =∨X, that is, a
is the least upper bound of X, in particular, a ≤ X.△.
Now apply condition (∗) to X:
there is an y from X.△∩ x.϶.
Since a ≤ y ∈ x.϶= x.϶¯, it follows that a ϵ x
and therefore a ∊ x.
This proves u.∍^{0}⊆ x.∍.
Corollary:
Axiom (ep~7)' can be equivalently replaced by
(ep~7)'
For every object x,
if x is powerclass-like
then
x.∍.∍= u.∍ for some object u.
The pre-completion theorem
Proposition:
Every basic structure S_{0} has
a faithful extension S that is pre-complete.
Such an extension can be obtained in the following steps:
Let S_{1} be the rank pre-completion of S_{0}.
(Put S_{1}=S_{0} if ϖ= ω.)
Let S_{2} be the metaobject completion of S_{1}.
Let S=S_{3} be the extensional pre-completion of S_{2}.
Proof:
In each step, S_{i+1}
is a basic structure that is a faithful extension of S_{i}
therefore S is a basic structure that is
a faithful extension of S_{0}.
By the extensional
pre-completion theorem,
S is metaobject complete,
extensionally consistent and powerclass consistent.
Since S_{1} is
rank pre-complete so is S.
By the observations about rank pre-complete structures,
S is ∊-ranked and thus pre-complete.
Pre-completion
If S is obtained from S_{0}
as in the pre-completion theorem above, we call it the pre-completion
of S_{0}.
(a)
S_{0}
The following diagrams shows a pre-completion of a basic structure S_{0}
that contains two objects,
O_{0}= {r, a}.
It is assumed that ϖ= ω so that rank-precompletion
is skipped.
Instead, metaobject completion is shown in 2 steps.
(b) S_{1} … powerclass completion of S_{0}
(c) S_{2} … singleton completion of S_{1}
(d) S … extensional pre-completion of S_{2}
(and thus a pre-completion of S_{0})
Complete structure of ϵ
By a complete structure of ∊
(or a complete∊-structure)
we mean a structure
(O, ∊)
where
O is the set of objects
and ∊ is a relation between objects
satisfying the following conditions:
(co~1)
∊ is well-founded.
(co~2)
∊ is weakly extensional:
for every x, y from O.∊,
if x.∍= y.∍ then x = y.
(co~3)
O.∍ is the set of all objects whose ∊-rank is
less than ϖ.
(co~4)
For every subset X of O.∍ there is an object x
such that x.∍= X.
Observation:(O, ∊) is complete ↔(O, ∊)
is pre-complete
and satisfies (co~4).
Proof:(ep~1)–(ep~3)
are the same as
(co~1)–(co~3).
Subsequently,
(co~4)
implies
(ep~4)–(ep~7)
(use (ep~7)').
x ≤ y
↔
x = y or ∅≠ x.∍⊆ y.∍
r.∍
=
O.∍
x.ec = y
↔
x.∍^{0}= y.∍
x.ɛϲ = y
↔
{x} = y.∍
x ∊^{0} y
↔
x ≤ y and x ∈O.∍
A metaobject structure(O, ≤, r, .ec, .ɛϲ)
is complete if
it is definitionally derived from a complete ∊-structure
according to the table on the right.
An ϵ5-structure
S= (O, ϵ, ϵ¯^{⁽*⁾}, r, .ec, .ɛɕ)
is complete if it is a metaobject complete basic structure
whose correspondent metaobject structure is complete.
That is, S is (further) derived by
(ϵ¯)
=
(.ec) ○ (≤),
(ϵ)
=
(∊) ∪ (ϵ¯),
(ϵ^{-k})
=
(≤) ○ .ec(-k)
for every natural k > 0,
(.ɛɕ)
=
(.ɛϲ) ∖ (.ec).
We might also say that S is a complete structure of ϵ.
By the correspondence between pre-complete ϵ5-structures
and pre-complete ∊-structures,
an ϵ5-structure S is complete
iff it is a basic structure such that
For every object x,
r_{∊}(x) (the ∊-rank of x) equals x.d.
Adding a bottom
By (co~2)+(co~4) there is a one-to-one
correspondence between non-empty subsets of O.∍
and elements of O.∊.
For the purpose of convenience in expressing meet and join operations
in ≤
we introduce a bottom, r_{0},
that represents the empty set.
If S= (O, ϵ, ϵ¯^{⁽*⁾}, r, .ec, .ɛɕ)
is a complete
ϵ5-structure
then a
structure
S_{⊥}= (Θ, ϵ, ϵ¯^{⁽*⁾}, r, .ec, .ɛɕ)
is called a
lifted version of S
if it is an extension
of S such that
Θ=O⊎ {r_{0}},
(that is, r_{0} is the only new object)
r_{0}.϶^{i}=r_{0}.϶¯^{i}= {r_{0}} and
r_{0}.ϵ^{i}=r_{0}.ϵ¯^{i}=Θ
for every integer i,
(in particular, r_{0} < O)
r_{0}.ec =r_{0},
{r_{0}}.ɛɕ =∅.
Obviously, S_{⊥} is uniquely given up to isomorphism.
We did not introduce S_{⊥} as an ϵ5-structure
in order to preserve the notions of S.
(Otherwise we would have to change the definition of .mli
since r_{0} is a lower bound of H.)
The metalevel index and rank functions are extended by:
r_{0}.mli = ω and
r_{0}.d =ϖ.
We let the ∊ relation be the same so that
r_{0} is regarded as unbounded
(i.e. ∅=r_{0}.∍=r_{0}.∊).
For a subset X of O,
we let the values
of X.↧, X.↥, X.▽ or X.△
be the same as in S,
so that e.g.
H.▽ is empty as before.
Similarly for images and pre-images under ϵ^{i} or ϵ¯^{i},
i ∈ℤ.
Note that this convention only needs to be introduced for pre-images,
since O is closed w.r.t. images of
ϵ^{i} and ϵ¯^{i}.
When referring to objects we mean elements of O.
We denote ∨ (∨) and ∧ (∧)
the join and meet operations in (Θ, ≤), respectively.
That is, if {x, y} ∪ X ⊆Θ then
∧X= y
↔
y is the greatest lower bound of X in
(Θ, ≤),
(note that ∧X exists ↔ X ≠∅)
x ∧ y =∧{x,y},
∨X= y
↔
y is the least upper bound of X in
(Θ, ≤),
x ∨ y =∨{x,y}
(whenever ∨{x,y} exists).
Furthermore, we also introduce the binary operation of a difference
between elements of Θ.ϵ:
z = x − y↔z.∍= x.∍∖ y.∍.
Observations:
Θ.ϵ=O.∊∪ {r_{0}}.
(Recall that
O.∊=O.ϵ=O∖T,
therefore,
Θ.ϵ
is the set of non-terminal objects together with r_{0}.)
(Θ.ϵ, ≤)
is a complete atomic Boolean algebra,
isomorphic to
(ℙ(O.∍), ⊆).
The set of atoms equals the set
O.ɛϲ of singletons.
The join and meet operations are the restrictions of
∨ and ∧ to
subsets of Θ.ϵ.
For every X ⊆Θ, the
join ∨X is defined
iff
either X ∩T=∅ or
X ⊆ {x,r_{0}} for some terminal x.
For every non-terminal object x,
x =∨x.∍.ɛϲ.
For every non-empty set X of non-terminal objects,
(∨X).d =∨X.d.
For every objects x, y,
x.ec ∧ y.ec = (x ∧ y).ec.
Proof:
Let ∅≠ X ⊆O.∊ and u =∨X
so that
u.∍= X.∍.
u.d ≤∨X.d follows by:
for every a ∊ u there is an x ∈ X such that a ∊ x.
u.d ≥ ∨X.d follows by monotonicity of
.d w.r.t. ≤.
Let x and y be objects.
Then for every bounded object a the following holds:
a ∊ (x ∧ y).ec
↔
a ≤ (x ∧ y)
↔
a ≤ x and a ≤ y
↔
a ∊ x.ec and a ∊ y.ec
↔
a ∊ (x.ec ∧ y.ec).
The union map
In a complete structure (O, ∊), the
union map is denoted .υ
and is defined as a partial map between objects by
x = y.υ
↔
x =∨y.∍.
In the lifted version S_{⊥} we let r_{0}.υ=r_{0}.
We let the integer powers of
.υ
be denoted and defined
in a similar way to that of .ec or .ɛϲ.
The 0-th power of .υ is the identity map on O.϶^{-1}
(which is the domain of .υ, see observations below).
Observations:
Each of the following conditions is equivalent to x = y.υ:
x.∍= y.∍^{2} and x ϶^{-1} y.
x.↥= y.ϵ^{-1}.
The domain of .υ equals the set O.϶^{-1}
of anti-members.
Since
O.϶^{-1}=r.϶^{-1}⊎T.϶^{-1}
and
T.϶^{-1}=T.ec =T.ɛϲ,
y.υ is defined
↔
(a) {y} ∪ y.∍⊆O.∊
or
(b)
{x} = y.∍ for some x ∈T.
The following inclusion chain applies:
(ϵ^{-1}) ∩ (϶)
⊆
(.υ)
⊆
(ϵ^{-1}).
The .ec and .ɛϲ maps are
subrelations of the inverse of .υ.
(A consequence of
(.ec) ∪ (.ɛϲ) ⊆ (ϵ) ∩ (϶^{-1}).)
If x = y.υ then
x.d = sup { i | i < y.d }. That is,
x.d + 1
= y.d
if y.d is a successor ordinal,
x.d
= y.d
if y.d is a limit ordinal.
Stages
For every ordinal number i,
the i-th stage of S
is denoted V_{i} and defined by
V_{i}= { x ∈V| x.d < i }.
That is, V_{i} is the set of objects whose rank
is strictly less than i. In particular,
V_{0}=∅,
V_{1}=O∖O.∊=T
is the ground stage (consisting of terminal objects),
V_{ϖ}=O.∍, is the set of bounded objects,
V_{ϖ+1}=O is the last stage.
Accordingly,
for each 0 < i ≤ϖ,
the i-th stage object is denoted r_{i}
and defined as the unique object
such that
V_{i}=r_{i}.∍.
The additional bottom r_{0} is
regarded as the 0-th stage object.
Each of the following diagrams shows a powerclass complete structure
that is a restriction of a complete structure of ϵ
to powerclass chains of
terminal or stage objects.
In the (a) case there is exactly one terminal object, the (b) structure contains
at least two terminals.
(a)
(b)
Observations:
Each stage object belongs to the 1-st metalevel.
Each stage object is primary except for r_{1} in the (a) case.
Proposition A:
For every object x and every ordinal i ≤ϖ,
x ∊r_{i}
↔
x.d < i,
x ≤r_{i}
↔
x.d ≤ i and x ∉T.
r_{ϖ}=r.
For every non-zero ordinal i < ϖ,
r_{i+1}=r_{1}∨r_{i}.ec.
r_{i+1}=r_{i}∨r_{i}.ec.
Proof:
(a) follows by definition.
To show (b), assume that x is a non-terminal object. Then
x.d ≤ i
↔ x.∍.d < i
(by definition of ∊-rank)
↔
x.∍⊆r_{i}.∍
(by (a))
↔
x ≤r_{i}
(by definition of ≤).
r_{ϖ}.∍=∪{r_{i}.∍| i < ϖ}=O.∍=r.∍.
It follows that r_{ϖ}=r.
x ∊r_{i+1}
↔
x.d < i+1
↔
x.d ≤ i
↔
x ∈T∪r_{i}.↧
(by A1b)
↔
x ∈r_{1}.∍∪r_{i}.↧
(since r_{1}.∍=T).
As a result, r_{i+1}.∍=r_{1}.∍∪r_{i}.↧,
that is,
r_{i+1}=r_{1}∨r_{i}.ec.
This follows by r_{1}≤r_{i} < r_{i+1}.
Proposition B:
For every non-terminal object x and every ordinal i < ϖ,
the following are satisfied.
x.ec ∧r_{i+1}= (x ∧r_{i}).ec.
x.ec =∨{ (x ∧r_{i}).ec | i < ϖ }.
Proof:
Assume that x is a non-terminal object and 0 < i < ϖ.
x.ec ∧r_{i+1}
= x.ec ∧ (r_{1}∨r_{i}.ec)
(by A3)
= x.ec ∧r_{i}.ec
(since x.ec is disjoint with r_{1})
For every non-zero cardinal number κ
there is a complete structure whose ground stage has cardinality κ.
Two complete structures are isomorphic
iff
they have the same cardinality of the ground stage.
If .ν is an isomorphism between
(O, ∊) and (V, ∊)
then .ν is uniquely given by its restriction to the ground stage
of (O, ∊) by
x.ν.∍= x.∍.ν
for every object x from O.∊.
Proof:
Proceed by transfinite recursion.
For each ordinal i ≤ϖ+1, define a set V_{i} by
V_{0}
=∅,
V_{1}
= κ × {0},
V_{i+1}
= V_{i}∪
{ (X,i) | X ⊆ V_{i} and for every j < i,
X ⊈ V_{j} }
if i > 0,
V_{i}
=∪{V_{j}| j < i}
if i is a limit ordinal.
Subsequently, let (V, ∊) be an ∊-structure
such that V = V_{ϖ+1} and
for every
(X,i), (Y,j) from V,
(X,i) ∊ (Y,j)↔(X,i) ∈ Y.
We claim that (V, ∊) is a complete structure
whose i-th stage is V_{i} for every i ≤ϖ+1.
In particular, the ground stage is V_{1} and thus has cardinality
κ.
To prove this, make the following observations.
Assume that
(X,i) and (Y,j) are from V
and k ≤ϖ+1.
Then the following holds:
(X,i) ∈ V_{k}↔i < k.
(X,i) ∊ (Y,j)→i < j
and thus ∊ is well-founded.
(Y,j).∍= Y whenever j > 0
and thus
∊ is weakly extensional.
r(X,i) = i where r() is the ∊-rank function.
By weak extensionality and well-founded recursion,
.ν is injective.
As a consequence, .ν is an embedding of
(O,∊)
into
(V,∊).
The surjectivity of .ν is proved by
well-founded recursion over (V, ∊).
Powerclass cumulation
For an ordinal i,
the i-th (powerclass) cumulation,
.ⅇc(i),
is a map Θ.ϵ→Θ.ϵ defined by transfinite recursion:
x.ⅇc(0)
= x,
x.ⅇc(i)
= x ∨ x.ⅇc(i-1).ec
if i is a successor ordinal,
x.ⅇc(i)
=∨{x.ⅇc(j) | j < i}
if i is a limit ordinal.
There is a single recursive formula for every ordinal i:
x.ⅇc(i) = x ∨∨{ x.ⅇc(j).ec | j < i }.
We also introduce special notation for .ⅇc(1) and .ⅇc(ϖ):
x.ⅇc = x ∨ x.ec
(the first cumulation of x),
x.ⅇc(⋆) = x.ⅇc(ϖ)
(the full cumulation of x).
For convenience, the following table shows the definition
with x.ⅇc(i) abbreviated to x_{i}.
Expressed using ∨
Expressed via .∍
i = 0
x_{0}= x
x_{0}.∍= x.∍
i is a successor ordinal
x_{i}= x ∨ x_{i-1}.ec
x_{i}.∍= x.∍∪
{ u |∅≠ u.∍⊆ x_{i-1}.∍ }
i is a limit ordinal
x_{i}=∨{x_{j}| j < i}
x_{i}.∍=∪{ x_{j}.∍| j < i }
Observations:
The set O is closed w.r.t. each .ⅇc(i).
The bottom r_{0} is a fixpoint of .ⅇc(i)
for every ordinal i.
An object x is a fixpoint of .ⅇc↔x ϵ x.
Proof:x ϵ x
↔ x ϵ¯ x
↔ x.ec ≤ x
↔ x ∨ x.ec = x.
Stage objects arise by powerclass cumulation of the ground stage object
r_{1}:
r_{1+i}=r_{1}.ⅇc(i)
for every ordinal i ≤ϖ.
(A consequence of monotonicity of .ec.)
For every x, y from Θ.ϵ and every
ordinals i, j,
i ≤ j
→
x.ⅇc(i) ≤ x.ⅇc(j),
x ≤ y
→
x.ⅇc(i) ≤ y.ⅇc(i).
That is, for every non-zero ordinal i,
.ⅇc(i) is (a) increasing and (b) monotone.
Proposition:
Assume that x is a non-terminal object
and i, j ordinal numbers.
x.ⅇc(i).ⅇc(j) = x.ⅇc(i+j).
x.ⅇc(i).d = (x.d + i) ∧ϖ.
(That is, the i-th cumulation increases the
rank of x by i whenever x.d + i ≤ϖ.)
Proof:
Assume that x
is a non-terminal object, i, j ordinal numbers and
proceed by transfinite induction over j.
x.ⅇc(i).ⅇc(j)
= x.ⅇc(i) ∨∨{x.ⅇc(i).ⅇc(k).ec | k < j}
(by definition of .ⅇc(j))
= x.ⅇc(i)
∨∨{x.ⅇc(i+k).ec | k < j}
(by the induction assumption)
= x.ⅇc(i) ∨ x
∨∨{x.ⅇc(i+k).ec | k < j}
(since x ≤ x.ⅇc(i))
= x.ⅇc(i) ∨ x
∨∨{x.ⅇc(i+k).ec | i+k < i+j}
(since k < j ↔ i+k < i+j)
= x.ⅇc(i) ∨ x.ⅇc(i+j)
(by definition of x.ⅇc(i+j),
using x.ⅇc(n) ≤ x.ⅇc(i) for n ≤ i)
= x.ⅇc(i+j)
(since x.ⅇc(i) ≤ x.ⅇc(i+j)).
Assume that x is a non-terminal object and proceed by transfinite induction.
x.ⅇc(i).d
= (x ∨∨{x.ⅇc(k).ec | k < i}).d
(by definition of .ⅇc(i))
= x.d ∨∨{x.ⅇc(k).ec.d | k < i}
(by properties of .d)
= x.d ∨∨{(x.ⅇc(k).d + 1) ∧ϖ| k < i}
(by properties of .ec)
= x.d ∨∨{(x.d + k + 1) ∧ϖ| k < i}
(by the induction assumption)
= x.d + ∨{(k + 1) ∧ϖ| k < i}
= (x.d + i) ∧ϖ.
The .ⅇc(⋆) operator
The following proposition shows that
the ϖ-th cumulation map .ⅇc(⋆)
is a closure operator on the set
Θ.ϵ=O.∊∪ {r_{0}}.
The fully cumulated objects are exactly the circular objects,
i.e. for every object x,
x ϵ x↔x ∈O.ⅇc(⋆).
That is,
O.ⅇc(⋆) are the fixpoints of .ⅇc
– using the observation
that x ϵ x ↔ x.ⅇc = x.
Proposition:
For every x from Θ.ϵ
and every ordinals i, j,
the following are satisfied.
x.ⅇc(j) ∧r_{i}≤ x.ⅇc(i).
x.ⅇc(⋆).ec ≤ x.ⅇc(⋆).
x.ⅇc(ϖ+i) = x.ⅇc(⋆).
Corollary:
.ⅇc(⋆)
is a closure operator on Θ.ϵ.
That is, .ⅇc(⋆) is increasing, monotone and
idempotent:
.ⅇc(⋆).ⅇc(⋆) = .ⅇc(⋆).
Proof:
Let x be from Θ.ϵ.
Proceed by transfinite induction over pairs (i,j).
For i = 0 the inequality reduces to
r_{0}≤ x which is satisfied by definition of r_{0}.
For j = 0 the inequality reduces to
x ∧r_{i}≤ x.ⅇc(i)
which follows by x ≤ x.ⅇc(i).
If i and j are both successor ordinals then
x.ⅇc(j) ∧r_{i}
=(x ∨ x.ⅇc(j-1).ec) ∧r_{i}
(by definition of .ⅇc(j))
= (x ∧r_{i})
∨
(x.ⅇc(j-1).ec ∧r_{i})
(by distributivity of ∨ and ∧)
≤∨{ (x.ⅇc(i).ec | i < ϖ }
(by the previous proposition (a))
≤∨{ (x ∨ x.ⅇc(i).ec | i < ϖ }
= x.ⅇc(⋆)
(by definition of .ⅇc(⋆) = .ⅇc(ϖ)).
By the previous proposition,
.ec is decreasing on O.ⅇc(⋆),
therefore, for every x from O.ⅇc(⋆),
x.ⅇc = x ∨ x.ec = x,
which shows that .ⅇc(⋆).ⅇc = .ⅇc(⋆).
Completion
This section describes the final step of completion
of basic structures of ϵ:
pre-complete structure →
complete structure.
For a given pre-complete structure S= (O, ∊)
we construct a faithful embedding .ν
of S into a complete structure V= (V, ∊)
whose ground stage V_{1}=V∖V.∊
has the same cardinality
as the set T=O∖O.∊
of terminal objects of S.
As a consequence we obtain the following:
Completion theorem:
Every basic structure can be faithfully extended to (embedded into)
a complete structure.
Since every complete structure is pre-complete
and both the pre-completion
as well as the .ν embedding described in this section are idempotent,
we can speak about completion of basic structures.
The completion theorem can be shortly stated as:
Every basic structure has a completion.
Embedding sequence
Let S= (O, ∊)
and V= (V, ∊) be ∊-structures
such that
S is pre-complete and
V is complete.
We say that a transfinite sequence
.ν_{0}, .ν_{1}, …, .ν_{ϖ}= .ν
of maps from O to V
is an embedding sequence (w.r.t. S and V)
if the following are satisfied:
The restriction of .ν_{i} to terminals is for every i
identical and forms a bijection
between T and V_{1}.
(⁎)
The restriction of .ν_{i} to the set
O.∊
of non-terminal objects x is defined according to the following
table.
Expressed via .∍
Expressed using ∨
i = 0
x.ν_{0}.∍= x.∍¯.ν_{0}.ec.∍∪
x.∍.ν_{0}
x.ν_{0}=∨(x.∍¯.ν_{0}.ec ∪
x.∍.ν_{0}.ɛϲ)
i is a successor ordinal
x.ν_{i}.∍= x.϶¯.ν_{i-1}.ec.∍∪
x.∍.ν_{0}
x.ν_{i}=∨(x.϶¯.ν_{i-1}.ec ∪
x.∍.ν_{0}.ɛϲ)
i is a limit ordinal
x.ν_{i}.∍=∪{ x.ν_{k}.∍| k < i }
x.ν_{i}=∨{ x.ν_{k}| k < i }
Notes:
(⁎)
To avoid notational conflicts, we refer to the set of terminals of V
by V_{1} (the ground stage of V).
Similarly, the inheritance root of V is referred to by
r_{ϖ} (the ϖ-th stage object).
We can consider .ν_{i} to be defined for every ordinal i.
Since .ec and .ν commute (as shown below) it follows
that
.ν_{ϖ}= .ν_{ϖ+1}= … = .ν_{i}
for every ordinal i ≥ ϖ.
Observations A:
For a non-terminal x,
x.ν_{0} is defined by well-founded recursion,
using well-foundedness of ∊,
x.ν_{i} for 0 < i ≤ϖ
is defined by transfinite induction over i.
Note that ϶¯ is used,
not ∍¯.
By definition,
.ec.∍= .∍^{0}.
In case of i = 0, we can even use .↧ instead of .∍^{0}.
Observations B:
For a bounded object x,
x.ν_{0}= x.ν_{i}= x.ν
for every ordinal i.
For every object x,
x.ν_{i}≤ x.ν_{i+1}≤ x.ν
for every ordinal i.
Proof:
If x is a bounded non-terminal object then
x.϶¯= x.∍¯
so that the prescriptions for x.ν_{0} and x.ν_{1}
are coincident.
Proceed by transfinite induction over i.
The case i = 0 follows by definition of
.ν_{0} and .ν_{1}
(using the fact that ∊ is a restriction of ϵ).
For a successor ordinal i we apply the induction assumption
(in particular,
x.϶¯.ν_{i-1}.ec.∍⊆ x.϶¯.ν_{i}.ec.∍):
Theorem:
Let
.ν_{0}, .ν_{1}, …, .ν_{ϖ}= .ν
be an embedding sequence
w.r.t.
S and V as in the previous subsection.
Then
.ν is a faithful embedding of S into V.
That is,
.ν is an embedding of the basic structure
(O, ϵ, ϵ¯^{⁽*⁾}, r, .ec, .ɛɕ)
into the basic structure
(V, ϵ, ϵ¯^{⁽*⁾}, r_{ϖ}, .ec, .ɛɕ).
.ν is faithful w.r.t.
.pr, ϵ^{i}, ϵ¯^{i} and .d.
Proof:
The proof is accomplished in the series of claims A–E below.
Claim A
Claim A:
For every objects x, y from O the following holds.
x.ν_{0}.d = x.ν.d = x.d.
x ∊ y ↔ x.ν_{0}∊ y.ν_{0}.
(That is, y.∍.ν_{0}= y.ν_{0}.∍∩O.ν_{0}
whenever y is non-terminal.)
x ≤ y ↔ x.ν_{0}≤ y.ν_{0}.
(That is, y.↧.ν_{0}= y.ν_{0}.↧∩O.ν_{0}.
In particular, .ν_{0} is injective.)
Proof:
Proceed by well-founded induction on (O, ∊).
x.ν_{0}.d
= sup { b.d + 1 |
b ∈ x.∍¯.ν_{0}.↧∪ x.∍.ν_{0} }
= sup { b.d + 1 |
b ∈ x.∍.ν_{0} }
(since .d is monotone in V)
= sup { a.ν_{0}.d + 1 | a ∊ x }
= sup { a.d + 1 | a ∊ x }
(since a.ν_{0}.d = a.d by the induction assumption)
= x.d.
If x is bounded, then
x.ν_{0}= x.ν.
If x is unbounded, then
ϖ= x.d = x.ν_{0}.d ≤ x.ν.d ≤ϖ.
In both cases, we obtain x.ν_{0}.d = x.ν.d
as a consequence.
We first show the
→ direction.
By (1), if x is bounded then so is x.ν_{0}.
Therefore
The ← direction
is shown by well-founded induction on (V, ∊).
If y.ν_{0} has rank 0 (i.e. y is terminal),
then the implications are trivially satisfied.
Otherwise denote
d = y.d and assume that for every non-terminal object u
such that u.d < d,
(a) u.∍.ν_{0}= u.ν_{0}.∍∩O.ν_{0}
and
(b) u.↧.ν_{0}= u.ν_{0}.↧∩O.ν_{0}.
Let x.ν_{0}∊ y.ν_{0}.
By definition of y.ν_{0}, there exists an object b from O
such that
either
x.ν_{0}≤ b.ν_{0} and b ∊¯ y,
or
x.ν_{0}= b.ν_{0} and b ∊ y.
If b is such that (ⅱ)
is satisfied then x = b
(since by the induction assumption, .ν_{0} is injective on objects
of rank less than d), thus x ∊ y.
If b is such that (ⅰ)
is satisfied then we obtain
x ≤ b
(by induction assumption for x and b
– both x and b have rank less than d),
x ∊¯ y (since x ≤ b ∊¯ y).
For a non-terminal x we obtain
x.ν_{0}≤ y.ν_{0}
↔
x.ν_{0}.∍⊆ y.ν_{0}.∍
→
x.ν_{0}.∍∩O.ν_{0}⊆
y.ν_{0}.∍∩O.ν_{0}
↔
x.∍.ν_{0}⊆ y.∍.ν_{0}
(by the just proved (a)),
↔
x.∍⊆ y.∍
(by injectivity of .ν_{0} due to the induction assumption),
↔
x ≤ y.
Claim B
Claim B:
For every objects x, y from O
and every ordinal i, the following are satisfied.
x ∊ y ↔ x.ν_{0}∊ y.ν_{i}.
(That is, y.∍.ν_{0}= y.ν_{i}.∍∩O.ν_{0}
whenever y is non-terminal.)
x ≤ y ↔ x.ν_{i}≤ y.ν_{i}.
(That is, y.↧.ν_{i}= y.ν_{i}.↧∩O.ν_{i}.
In particular, .ν_{i} is injective.)
As a particular consequence for i =ϖ,
x ∊ y ↔ x.ν ∊ y.ν,
x ≤ y ↔ x.ν ≤ y.ν.
Proof:
Proceed by transfinite induction over i.
The case i = 0 has been proved in A2.
Assume that i > 0 and that
(a)
x ∊ y ↔ x.ν_{0}∊ y.ν_{k},
(b)
x ≤ y ↔ x.ν_{k}≤ y.ν_{k}.
for every k < i.
We proceed similarly as in A2 and show first the
→ direction.
↔
x.ν_{k}≤ y.ν_{k}
for every k < i
(by the induction assumption)
→
x.ν_{i}≤ y.ν_{i}
(by definition of .ν_{i} for a limit i).
←:
Let x.ν_{0}∊ y.ν_{i}
and assume that i is a successor ordinal.
By definition of y.ν_{i}, there exists an object b from O
such that
either
x.ν_{0}≤ b.ν_{i-1} and b ϵ¯ y,
or
x.ν_{0}= b.ν_{0} and b ∊ y.
If (ⅱ) is satisfied then
x = b and thus x ∊ y.
If (ⅰ)
is satisfied we obtain
x ≤ b
(by the induction assumption),
x ∊¯ y (since x ≤ b ϵ¯ y and x is bounded).
Assume now that i is a limit ordinal. Then
x.ν_{0}∊ y.ν_{i}
↔
x.ν_{0}∈∪{ y.ν_{k}.∍| k < i }
(by definition of y.ν_{i}).
↔
x.ν_{0}∊ y.ν_{k} for some ordinal k < i
↔
x ∊ y
(by the induction assumption).
For a non-terminal x we obtain
x.ν_{i}≤ y.ν_{i}
↔
x.ν_{i}.∍⊆ y.ν_{i}.∍
→
x.ν_{i}.∍∩O.ν_{0}⊆
y.ν_{i}.∍∩O.ν_{0}
↔
x.∍.ν_{0}⊆ y.∍.ν_{0}
(by the just proved (a))
↔
x.∍⊆ y.∍
(by injectivity of .ν_{0})
↔
x ≤ y.
Claim C
Claim C:
For every object x from O
and every ordinals i, j, the following is satisfied.
(x.ν_{j}∧r_{i}) ≤ x.ν_{i}.
Corollary:
If i ≤ j then
x.ν_{j}∧r_{i}= x.ν_{i}∧r_{i}.
x.ec.ν_{i+1}= x.ν_{i}.ec.
x.ec.ν = x.ν.ec.
x.pr.ν = x.ν.pr.
(That is, .ν preserves primary objects.)
Proof:
The inequality holds trivially for bounded objects x.
(Recall that if x ∈O.∍ then x.ν_{i}= x.ν_{j}.)
Assume further that x is unbounded and
proceed by transfinite induction over pairs (i,j).
For i = 0 the inequality holds by definition of r_{0}
as the artificial bottom object.
For j = 0 the inequality holds by the monotonicity of the
embedding sequence
(if j ≤ i then .ν_{j}≤ .ν_{i}).
If i is a limit ordinal then
x.ν_{j}∧r_{i}= x.ν_{j}∧∨{ r_{k}| k < i }
=∨{ x.ν_{j}∧r_{k}| k < i }
(by infinite distributivity)
≤∨{ x.ν_{k}| k < i }
(by the induction assumption)
= x.ν_{i}.
Similarly, if j is a limit ordinal then
x.ν_{j}∧r_{i}=∨{ x.ν_{k}| k < j }∧r_{i}
=∨{ x.ν_{k}∧r_{i}| k < j }
(by infinite distributivity)
≤∨{ x.ν_{i}| k < j }
(by the induction assumption)
= x.ν_{i}.
Assume finally that both i and j are successor ordinals.
By definition,
x.ν_{j}.∍= x.϶¯.ν_{j-1}.ec.∍∪ x.∍.ν_{0}.
By B1a, x.∍.ν_{0}⊆ x.ν_{i}.∍,
therefore,
it is sufficient to show that
for every object a from x.϶¯,
(a.ν_{j-1}.ec ∧r_{i}) ≤ x.ν_{i}.
If a is bounded then
a.ν_{j-1}.ec = a.ν_{0}.ec ≤ x.ν_{0}≤ x.ν_{i}.
Assume further that a is unbounded, in particular,
a.ν_{j-1} is non-terminal.
Then for every bounded object b from V,
b ∊ (a.ν_{j-1}.ec ∧r_{i})
↔
b ∊ (a.ν_{j-1}∧r_{i-1}).ec
(by proposition B1
(a.ν_{j-1} is non-terminal))
↔
b ≤ (a.ν_{j-1}∧r_{i-1})
(by definition of .ec)
→
b ≤ a.ν_{i-1}
(by the induction assumption)
→
b ∈ x.϶¯.ν_{i-1}.ec.∍
(since a is from x.϶¯)
→
b ∊ x.ν_{i}
(since x.϶¯.ν_{i-1}.ec.∍⊆ x.ν_{i}.∍).
The equality is shown by
x.ec.ν_{i+1}.∍
= x.ec.϶¯.ν_{i}.ec.∍∪
x.ec.∍.ν_{i}
(by definition of .ν_{i+1},
using .ν_{0}= .ν_{i} on O.∍)
If x is bounded then, using the previous statement,
x.ec.ν
= x.ec.ν_{0}= x.ec.ν_{1}= x.ν_{0}.ec = x.ν.ec.
Assume that x is unbounded, in particular, non-terminal.
Then
x.ec.ν
= x.ec.ν ∧r_{ϖ}
(since x.ec.ν ≤r_{ϖ})
= x.ec.ν ∧∨{ r_{i}| i < ϖ }
(by definition of r_{ϖ})
=∨{ x.ec.ν ∧r_{i}| i < ϖ }
(by infinite distributivity)
=∨{ x.ec.ν_{i}∧r_{i}| i < ϖ }
(since x.ec.ν ∧r_{i}=
x.ec.ν_{i}∧r_{i} by C1)
=∨{ x.ec.ν_{i+1}∧r_{i}| i < ϖ }
(since y.ν_{i}≤ y.ν_{i+1}≤ y.ν for y = x.ec)
=∨{ x.ν_{i}.ec ∧r_{i}| i < ϖ }
(since .ec.ν_{i+1}= .ν_{i}.ec by C2)
=∨{ (x.ν_{i}∧r_{i}).ec | i < ϖ }
(by proposition B1
(x.ν_{i} is non-terminal))
=∨{ (x.ν ∧r_{i}).ec | i < ϖ }
(since x.ν_{i}∧r_{i}= x.ν ∧r_{i} by C1)
Let x be an object from O such that
x.ν is a powerclass,
that is x.ν = u.ec for some u from V.
Recall that by definition of .ec,
u.↧= x.ν.϶.
We should prove that
x is a powerclass as well
(equivalently, that u is from O.ν).
If x.϶ contains a terminal object a then,
since a.ν is also terminal (in V),
{a.ν} = x.ν.϶, so that a.ν = u.
We can therefore further assume that x.϶∩T=∅.
By pre-completeness of S it is sufficient to show that x
is
powerclass-like,
that is
x.϶= x.϶¯, and
X.△∩ x.϶≠∅
for every non-empty bounded subset X of x.϶.
The (a) equality follows from the embedding properties of
.ν w.r.t. ϵ and ϵ¯.
To prove (b),
assume that X is a non-empty bounded subset of x.϶.
We obtain
c =∨X.ν exists
(since X ∩T=∅
and thus X.ν ∩V_{1}=∅)
c ∈ x.ν.϶,
(since X.ν ⊆ x.ν.϶= u.↧)
c ∈ x.ν.∍,
(since X is bounded and therefore so is c)
c ≤ a.ν for some a from x.϶,
(since x.ν.∍= x.϶.ν.ec.∍= x.϶.ν.↧∩V.∍)
X ≤ a,
(since X.ν ≤ c ≤ a.ν)
X.△∩ x.϶≠∅.
(since a ∈ X.△∩ x.϶)
Claim D
Claim D:
For every object x from O such that x ϵ x
and every ordinal i, the following is satisfied.
x.ν_{i}.ec ≤ x.ν_{i+1},
x.ν_{0}.ⅇc(i) ≤ x.ν_{i}.
r.ν =r_{ϖ}.
Proof:
Let x be such that x ϵ x.
(a) follows directly by definition of .ν_{i+1}.
(b) is proved by transfinite induction over i.
For i = 0, (b) reduces to x.ν_{0}≤ x.ν_{i}
which holds by
observation B2.
If i is a successor ordinal then
x.ν_{0}.ⅇc(i)
= x.ν_{0}∨ x.ν_{0}.ⅇc(i-1).ec
(by definition of .ⅇc(i))
≤ x.ν_{0}∨ x.ν_{i-1}.ec
(by the induction assumption)
≤ x.ν_{0}∨ x.ν_{i}
(since x.ν_{i-1}.ec ≤ x.ν_{i} by (a))
≤∨{ x.ν_{k}| k < i }
(by the induction assumption)
= x.ν_{i}.
The equality r.ν =r_{ϖ} is shown by
r_{ϖ}=r_{1}.ⅇc(ϖ)
≤r.ν_{0}.ⅇc(ϖ)
(since r_{1}≤r.ν_{0} and .ⅇc(ϖ)
is monotonic)
≤r.ν
(since rϵr so that (1b) applies for
i =ϖ)
≤r_{ϖ}
(since r_{ϖ} is the top of V.∊).
Claim E
Claim E:
For every bounded object x from O,
x.ɛɕ.ν = x.ν.ɛɕ.
For every objects x, y from O
and every integer n,
x ϵ¯^{n} y
↔
x.ν ϵ¯^{n} y.ν.
For every objects x, y from O
and every natural n,
x ϵ^{n} y
↔
x.ν ϵ^{n} y.ν.
Proof:
Let x, y be objects from O
such that x.ɛɕ = y,
i.e. {x} = y.∍ and y is primary.
We should prove that
x.ν.ɛɕ = y.ν, equivalently,
(a) {x}.ν = y.ν.∍ and (b) y.ν is primary.
The (b) condition follows by claim C4.
The (a) condition is shown by
y.ν.∍
= y.∍¯.ν.↧∪ y.∍.ν
(since y is bounded )
= y.∍.ν
(since y.∍¯=∅)
= {x}.ν
(since {x} = y.∍).
This follows by powerclass completeness of both S and V
(so that (ϵ¯^{i}) = (≤) ○ .ec(i) ○ (≤)
in both S and V).
Let x, y be objects from O and n
a positive natural number.
For n = 1 we have
x ϵ y↔x ∊ y or x ϵ¯ y↔x.ν ∊ y.ν or x.ν ϵ¯ y.ν↔x.ν ϵ y.ν.
Assume further that n > 1.
It is then sufficient to show that
x.ν ϵ^{n} y.ν
→
x ϵ^{n} y.
(The opposite implication follows by embedding of ϵ.)
Assume therefore that x.ν ϵ^{n} y.ν.
Since x.ν.ϵ¯^{n}= x.ν.ϵ^{n}
for unbounded x,
we can in addition assume that x is bounded so that
x.ν ∊^{n} y.ν.
x.ν ∊^{n} y.ν
↔
x.ν ∊^{n-1} b ∊ y.ν for some b from V
(by relational composition)
↔
x.ν ∊^{n-1} b ≤ u.ν ∊ y.ν
for some b from V and u from y.∍
(by definition of y.ν.∍)
→
x.ν ∊^{n-1} u.ν ∊ y.ν
for some u from y.∍
(since (∊^{n}) ○ (≤) ⊆ (∊^{n}))
→
x ∊^{n-1} u ∊ y for some u from O
(by the induction assumption)
↔
x ∊^{n} y.
Superstructures
Recall that the set O of objects of a complete
∊-structure
S= (O, ∊)
equals the (ϖ+1)-th stage of S.
In this section we provide an axiomatization for an arbitrary stage.
This introduces a generalization
in which the complete structures of ϵ
are viewed as (ϖ+1)-superstructures.
Superstructure
By an (abstract) superstructure we mean
an ∊-structure(V, ∊)
(so that
V is a set of objects and
∊ is a relation between objects)
such that
the following conditions hold.
(as~1)
∊ is well-founded.
(as~2)
∊ is weakly extensional:
for every x, y from V.∊,
if x.∍= y.∍ then x = y.
(as~3)
For every non-empty set X of objects
such that r(X) < r(V)(⁎)
there is an object x such that x.∍= X.
By an α-superstructure
we mean a superstructure
(V, ∊) whose ∊-rank equals α.
Note:(⁎)
We use r(X) to refer to
the ∊-rank
of a subset X of V.
Similarly, for an object x we let
r(x) be the rank of x in ∊.
Observations:
The only 0-superstructure is the empty structure.
Every 1-superstructure is of the form (V, ∅)
where V is non-empty.
V.∍=V
↔
α is a limit ordinal or zero.
Axiom (as~2) can be left out if
in (as~3),
an object x is replaced by
a unique object x.
If r(V) is a successor ordinal α+1
then
(as~3) can be replaced by
a conjuction of the following conditions:
(a)
V.∍
is the set of all objects x such that
r(x) < α.
(co~3) if α =ϖ
(b)
For every subset X of V.∍ there is an object x
such that x.∍= X.
(co~4)
The labels on the right indicate the correspondence to axioms of
complete ∊-structures.
Corollary: For a structure S= (O, ∊)
the following are equivalent:
S is an (ϖ+1)-superstructure.
S is a complete ∊-structure.
Definitional extension
The following table shows the definitional extension of an
α-superstructure (V, ∊).
Inheritance
x ≤ y
↔
x = y or ∅≠ x.∍⊆ y.∍
Bounded inheritance
x ∊^{0} y
↔
x ≤ y and x ∈V.∍
Power membership
x ϵ¯ y
↔
∅≠ x.∍^{0}⊆ y.∍
Object membership
x ϵ y
↔
x ∊ y or x ϵ¯ y
Anti-membership
x ϵ^{-1} y
↔
∅≠ x.∍⊆ y.∍^{0}
Powerclass map
x.ec = y
↔
∅≠ x.∍^{0}= y.∍
Singleton map
x.ɛϲ = y
↔
{x} = y.∍
Union map
x = y.υ
↔
x.∍= y.∍^{2} and
y ϵ^{-1} x
Metalevel index
x.mli
=
min {i | x ∈V_{1}.∊^{i}, i ∈ℕ }
Rank
x.d
=
sup {a.d + 1 | a ∊ x}
i-th cumulation map
x.ⅇc(i) = y
↔
x ∈V.∊ and
y = x ∨∨{x.ⅇc(j).ec | j < i }
i-th stage
V_{i}
=
{ x | x.d < i }
(1+i)-th stage object
r_{1+i}
=
r_{1}.ⅇc(i)
where
r_{1}.∍=V_{1}=V∖V.∊
For a natural i > 0,
let ϵ^{i}/ϵ¯^{i}/ϵ^{-i}/∊^{i}
be the i-th relational composition of
let ϵ/ϵ¯/ϵ^{-1}/∊ with itself.
The 1-st stage V_{1}=V∖V.∊
is the ground stage of terminal objects.
Objects from V.∍ are bounded,
the remaining are unbounded.
Lifting by an additional bottomr_{0}
is defined like for complete ∊-structures.
Observations:
There is no (common) definition of inheritance root r
since if α is a limit ordinal then
V=V.∍ and thus
there is no object x such that x.∍=V.∍.
The metalevel index function .mli
is defined using ∊-levelling.
Definitions of ϵ¯ and .ec are adjusted
by the additional condition ∅≠ x.∍^{0}.
This is because in (α+2)-superstructures,
this condition is not implicitly guaranteed.
(In particular, there are unbounded singletons.)
As a consequence, the .ec map is not total in general.
The i-th cumulation map .ⅇc(i) is total
on V.∊ for every
ordinal i iff
α is a successor of a limit ordinal.
If α is a limit ordinal and x ∈V.∊ then:
x.ⅇc(i) is defined ↔x.d + i < α.
In (α+2)-superstructures,
.ⅇc(i) cannot be total on V.∊
for i > 1 since
.ec is not total.
Embedding of superstructures
In this section we make the final preparation
for the embedding of object membership
into the von Neumann universe.
We aim at the identification
(O, ∊) ↔ (𝕍_{ϖ+1}, ∈)
where
𝕍_{ϖ+1} is the partial von Neumann universe
of all well-founded pure sets whose rank is at most ϖ.
We are then interested in
(α+1)-superstructures (Z,∊),
where α is a limit ordinal,
which are regularly embedded into S= (O, ∊):
Terminals of (Z,∊) are
singletons of S and have constant rank.
All objects of (Z,∊) are bounded in S,
i.e. Z ⊆O.∍.
In particular, α < ϖ.
Subsequently, we can change the choice of ϖ to α
and regard (Z,∊) as an embedded
complete structure of ϵ.
In the rest of the section we assume that
S= (O, ∊)
is a complete ∊-structure
(equivalently, (ϖ+1)-superstructure),
with the definional extension introduced before.
Strata and the bottom stratum operator
For objects s, x,
s is said to be a stratum ofx if
either s is terminal and s = x or
s is non-terminal and there is a (necessarily unique) ordinal
number i such that
s.∍= { u | u ∊ x, u.d = i }.
(That is,
s.∍ consists of those members of x that have rank i.)
An object is called a stratum if it is a stratum of itself.
For an object x,
the bottom stratum of x is
denoted x.ϱ and defined as the unique stratum
of x with the least rank,
the height of x is denoted x.ℎ
and equals 1+i where i is the unique ordinal such that
x.ϱ.d + i = x.d.
Observations:
For every object x,
x.ϱ exists and is bounded,
x.ℎ exists (since x.ϱ≤ x),
x.ϱ.ϱ= x.ϱ
(that is, .ϱ is idempotent),
x ∈O.ϱ
↔
x is a stratum
↔
all objects from x.∍ have the same rank
↔
x.ℎ= 1,
x.ϱ.d is either zero or a successor ordinal,
x.ϱ.∍= { a | a ∊ x
and
a.d ≤ b.d for all b ∊ x }.
For every ordinal i < ϖ,
r_{i+1}−r_{i} is a stratum or r.
Moreover, for every object s,
s is a stratum of rank i+1↔s ≤r_{i+1}−r_{i}↔∅≠ s.∍⊆V_{i+1}∖V_{i}.
The ground stage object r_{1} is the bottom stratum
of every stage object r_{i}, 0 < i ≤ϖ.
Proposition:
.ϱ and .ec commute.
In particular, if s is a stratum then so is s.ec.
Powerclass cumulation preserves the bottom stratum,
i.e.
x.ϱ= x.ⅇc(i).ϱ
for every non-terminal object x and every non-zero ordinal i.
Proof:
Assume first that
x is a terminal object.
Then {x} = x.ec.∍
so that x.ec is a stratum and thus
x.ec.ϱ= x.ec.
Since by definition x.ϱ= x
it follows that
x.ec.ϱ= x.ϱ.ec.
Assume further that x is non-terminal.
Then for every non-terminal a,
a ∊ x.ec.ϱ
↔ a ∊ x.ec
and a.d ≤ b.d for every b ∊ x.ec
(by definition of .ϱ)
↔
a.∍⊆ x.∍
and a.d ≤ b.d
for every b such that ∅≠ b.∍⊆ x.∍
(by definition of .ec)
↔
a.∍⊆ x.∍
and a.∍.d ≤ u.d
for every u ∊ x
(by definitin of .d)
↔
a ∊ x.ϱ.ec.
Assume that x is a non-terminal object and i is a non-zero ordinal.
Then for every non-terminal a,
a ∈ x.ⅇc(i+1).∍∖ x.∍
↔
a.∍⊆ x.ⅇc(i).∍
(by definition of .ⅇc(i+1))
→
u.d < a.d for some u from
x.ⅇc(i).∍
(by definition of .d)
→
u.d < a.d for every u from
x.ϱ.∍
(by the induction assumption).
This shows x.ϱ.∍.d ≤ x.ⅇc(i+1).∍.d.
Since x.ϱ≤ x ≤ x.ⅇc(i+1) the statement follows.
Regularly cumulated objects
We say that an object b is a (regular) cumulation base
if it is a non-terminal stratum such that
b.∍⊆T∪O.ɛϲ.
(That is, all members of b are either terminals
or singletons of equal rank.)
An object z is said to be regularly cumulated
if z = b.ⅇc(α)
for some cumulation base b and
some ordinal α.
Observations:
Every regularly cumulated object z has a unique cumulation base b
given by
b = z.ϱ.
(The cumulation base of z equals the bottom stratum of z.)
If z is a regularly cumulated object
then so is
z.ⅇc(i) for every ordinal i.
Stage objects r_{i}, 0 < i ≤ϖ,
are regularly cumulated.
If b.d > 1 then b.υ exists
and thus b.υ.d refers to the ordinal predecessor of b.d.
Proof:
This is a consequence of .ⅇc(α) preserving the bottom stratum
(proposition 2).
Proposition:
Let z be a regularly cumulated object
and denote b = z.ϱ its cumulation base and Z = z.∍.
For every object x,
x ≤ z and x.d < z.d→x ∊ z.
The following are equivalent:
(ⅰ)
z ϵ z
(i.e. z is fully cumulated).
(ⅱ)
z.d =ϖ.
(i.e. z is unbounded).
Let i be the unique ordinal such that b.d + i = z.d.
(That is, z.ℎ= 1+i is the
height of z.)
Then
z = b.ⅇc(i).
The following closure properties are satisfied:
Z.↧= Z.
Z.∊∩ Z = Z ∖ b.∍,
that is,
b.∍= Z ∖ Z.∊,
that is, b.∍ is the set of ∊-minimal objects
in (Z, ∊).
Z.∍∩ Z = Z.∍∖ b.∍^{2},
that is,
(Z ∖ b.∍).∍⊆ Z.
In particular,
(Z, ∊, ≤) is a substructure of (O, ∊, ≤).
Let α_{0} be such that b.d = α_{0} + 1
and
let r() be the rank function in the well-founded
relation (Z, ∊).
Then
α_{0} + r(x) = x.d
for every object x from z.∍= Z,
α_{0} + r(x.∍) = x.d
for every object x from z.↧.
(Z, ∊) is an α-superstructure
where
α = z.ℎ
is the height of z.
Proof:
Assume that b, z and Z
are as in the antecedent of the proposition.
Let i be an ordinal such that z = b.ⅇc(i).
Proceed by transfinite induction over i.
If i = 1
then z = b is a stratum
so that x.d = z.d for every object x from z.↧
and the requested implication follows trivially.
Assume now that i is a successor ordinal, i > 1,
and let y = b.ⅇc(i-1) so that z = b ∨ y.ec.
Let x be an object from z.↧.
Assume x ≤ y. Then x ϵ y.ec ≤ z
thus x ϵ z.
It follows that x ∊ z if, in addition, x.d < ϖ.
Assume x ≰ y
and let u be from x.∍∖ y.∍.
It follows from u ∊ x ≤ z that u ∊ z and thus
u ≤ y.
Consequently,
u.d ≮ y.d
(by the induction assumption),
u.d = y.d
(since u.d ≤ y.d as a consequence of u ≤ y),
x.d ≥ z.d
(since
x.d ≥ u.d + 1 = y.d + 1 ≥ z.d).
Therefore x.d < z.d cannot be satisfied
so that the requested implication follows trivially.
Assume finally that i is a limit ordinal and let again
x be an object from z.↧.
If x ≤ b.ⅇc(j) for some j < i
then x ϵ b.ⅇc(j+1) ≤ z
and thus x ϵ z.
Assume that
x ≰ b.ⅇc(j) for every j < i.
Then there is a sequence
{ u_{j}| j < i }
such that u_{j}∈ x.∍∖ b.ⅇc(j).∍.
Consequently,
(ⅰ)→(ⅱ)
follows by definition of .d.
For the reverse implication assume that z.d =ϖ.
Then for every object x,
x ∊^{0} z↔x ≤ z and x.d < ϖ↔x ≤ z and x.d < z.d→x ∊ z.
The last implication is by (1).
This shows that z.∍^{0}⊆ z.∍, i.e. z ϵ¯ z.
Let i be such that b.d + i = z.d
and
let j be such that b.ⅇc(j) = z.
Apply proposition 2
about the rank increment by cumulation.
If z.d < ϖ then
b.ⅇc(j).d = b.d + j and thus i = j.
Assume that z.d =ϖ.
Since b.d + j ≥ z.d it follows that i ≤ j.
Let k be such that i + k = j
and thus b.ⅇc(j) = b.ⅇc(i).ⅇc(k).
Since b.ⅇc(i) is fully cumulated (by the previous proposition)
it follows that
b.ⅇc(j) = b.ⅇc(i).
The statements follow by definition of powerclass cumulation.
(For (a) use b.∍= b.∍.↧ which follows by
minimality of objects from b.∍.)
Assume that α_{0} is such that b.d = α_{0} + 1.
To show (a), let x be an object from Z.
Assume first that x ∊ b.
Then x.∍∩ Z =∅
and thus r(x) = 0.
Since
x.d = α_{0} the (a) equality follows.
Assume further that x is from Z.∊. Then
x.d
= sup { a.d + 1 | a ∊ x }
(by definition of .d)
= sup { α_{0} + r(a) + 1 | a ∊ x }
(by the induction assumption)
= α_{0} + sup { r(a) + 1 | a ∊ x }
(by properties of ordinal numbers)
= α_{0} + r(x)
(by definition of r(x)).
To show (b) let x be from z.↧,
that is, ∅≠ x.∍⊆ Z. Then
x.d
= sup { a.d + 1 | a ∊ x }
= sup { α_{0} + r(a) + 1 | a ∊ x }
(by (a))
= α_{0} + r(x.∍)
(by definition of r(x.∍)).
Let α_{0} be such that b.d = α_{0} + 1.
Well-foundedness of (Z,∊) follows by well-foundedness of
∊ (and has already been used for the definition of r()).
Similarly, weak extensionality of (Z,∊)
follows by weak extensionality of ∊ and
(Z ∖ b.∍).∍⊆ Z
(by 4c).
To verify
that
(as~3) is satisfied in (Z,∊),
assume that
X is a non-empty subset of Z.∍∩ Z
such that r(X) < r(Z)
(where r() is again the rank function in (Z,∊)).
By extensional completeness in (O,∊),
there is an object x such that X = x.∍.
For every ordinal i < ϖ,
the cardinality of V_{i+1}∖V_{i} is
at least i.
Let (α, κ) be a pair of non-zero ordinals
such that κ is in addition a cardinal
and
κ + α < ϖ.
Then there is a regularly cumulated object z
whose superstructure (Z,∊)
(where Z = z.∍) has the following properties:
The rank of (Z,∊) equals α.
The cardinality of the ground stage Z ∖ Z.∊ equals κ.
Proof:
First note that the cardinality of V_{i} is at least i
since for every k < i
there is an object x_{k} from V_{i}
such that x_{k}.d = k
(we can choose x_{k}=r_{k} for k ≠ 0).
Consider now the difference D =V_{i+1}∖V_{i}.
For i = 0 the statement holds trivially so that we can further assume that
i > 0.
By axioms
(co~2) and
(co~4),
for every non-empty subset X of V_{i},
there is a unique x from V_{i+1}, such that x.∍= X.
That is,
there is a bijection between V_{i+1}∖V_{1}
and ℙ(V_{i}) ∖ {∅}
(where ℙ(V_{i}) is the powerset of V_{i}).
Since V_{1}⊆V_{i} we can make the following observations:
If V_{i} is finite then
the cardinality of V_{i+1} is at least twice as much as the
cardinality of V_{i}.
If V_{i} is infinite then
the cardinality of D equals the cardinality of V_{i+1}.
In both cases it follows that D is of cardinality at least i.
Let α be an ordinal and κ a cardinal such that
κ + α < ϖ.
By the previous proposition,
there is a subset X of V_{κ+1}∖V_{κ}
such that card(X) = κ.
Consequently,
X.ɛϲ is a subset of
V_{κ+2}∖V_{κ+1}
such that card(X.ɛϲ) = κ
(since .ɛϲ is injective and increases rank by 1).
Let b be the unique object such that b.∍= X.ɛϲ
and denote z = b.ⅇc(i)
where α = 1+i and Z = z.∍.
Then
b is a regular cumulation base
(since all members of b are singletons of rank κ + 1),
(Z,∊) is an α-superstructure
(since ϖ is a limit ordinal so that
b.d + i = z.d < ϖ),
b.∍ has cardinality κ
(by the choice of X).
Embedded (α+1)-superstructure
Proposition:
Let z be a regularly cumulated object of height α+1
where α is a limit ordinal.
Denote Z = z.∍.
Assume that z.υ exists.
Then the following table describes
a definitional extension of the (α+1)-superstructure
(Z,∊).
(The z subscript is used to distinguish between
definitional extensions of (Z,∊) and (O,∊).)
Inheritance root
r
=
z ∧ z.υ
r.∍=
Z ∩ Z.∍
Ground stage object
b
=
z − r.ec
b.∍=
Z ∖ r.↧
For every x, y from Z :
Inheritance
x ≤_{z} y
↔
x ≤ y
↔
x.∍⊆ y.∍
Union map
x = y.υ_{z}
↔
x = y.υ
↔
x.∍= y.∍^{2}
Singleton map
x.ɛϲ_{z}= y
↔
x.ɛϲ = y
↔
{x} = y.∍
Powerclass map
x.ec_{z}= y
↔
r ∧ x.ec = y
↔
r.∍∩ x.↧= y.∍
Rank
b.υ.d + x.d_{z}
=
x.d
Proof:
Let r be the inheritance root of (Z,∍),
i.e, the unique object such that r.∍_{z}= Z.∍_{z}.
Let b be the ground stage object of (Z,∍),
i.e, the unique object such that r.∍_{z}= Z ∖ Z.∊_{z}.
Let x be an arbitrary object from Z.
Then prescriptions for r, b and .ec_{z}
follow from the following equalities:
r.∍=
r.∍_{z}= Z.∍_{z}= Z ∩ Z.∍= z.∍∩ z.∍^{2}= z.∍∩ z.υ.∍,
The difference between .d and .d_{z} follows
by
proposition 5.
Embedded basic structure
Proposition:
Let again z be a regularly cumulated object of height α+1
where α is a limit ordinal,
z.υ exists
and Z = z.∍.
Assume in addition that
S_{o}= (O, …)
is a basic structure (of rank α+1) such that
(Z,∊) is a faithful extension of S_{o}.
Let T be the set of terminals of S_{o} and
r the inheritance root.
Then
T
=
(O.∍∖ O).ɛϲ ∩ O
(the set of terminal objects of S_{o}),
r.∍
=
O.∍∖ T.∍
(the inheritance root of both S_{o} and (Z,∊)),
Z
=
r.∍∪ r.↧
(i.e. z = r ∨ r.ec = r.ⅇc).
In particular,
Z is given by O
(in the ambient (ϖ+1)-superstructure (O, ∊)).
The definitional extension of (Z,∊) described in the previous subsection
is applicable to S_{o}.
Proof:
Let r be the inheritance root of (Z,∍),
i.e, the unique object such that r.∍_{z}= Z.∍_{z}.
Let b be the ground stage object of (Z,∍),
i.e, the unique object such that r.∍_{z}= Z ∖ Z.∊_{z}.
Let T be the set of terminal objects of S_{o}.
Then
O = (O ∖ T) ⊎ T,
O.∍= (O ∖ T).∍⊎ T.∍
(since T ⊆ b.∍ and O ∖ T ⊆ Z.∊
and Z ∩ b.∍^{2}=∅),
O.∍= r.∍⊎ T.∍
(since {r} ⊆ O ∖ T ⊆ Z and
r.∍_{z}= Z.∍_{z}).
By embedding of S_{o} into (Z,∊),
T ⊎ O.ɛϲ_{o}=O.ɛϲ ∩ O,
(since T ⊆ b.∍ and b.∍⊆O.ɛϲ),
T ⊎ O.ɛϲ_{o}= O.∍.ɛϲ ∩ O,
(since (.ɛϲ) ⊆ (∊)),
T = (O.∍.ɛϲ ∖ O.ɛϲ) ∩ O
(since O.ɛϲ_{o}= O.ɛϲ ∩ O),
T = (O.∍∖ O).ɛϲ ∩ O
(since .ɛϲ is injective).
This shows the equalities for T and r.
The z = r.ⅇc equality follows by
r = b.ⅇc(α).
Embedding into the von Neumann universe of sets
In this section we provide the final embedding of bounded object membership
∊ into set membership ∈ between well-founded sets.
If not stated otherwise, we will use the term class in
the sense of set theory as a collection of sets.
Let 𝕍 denote
the universal class and assume the ZFC axiom of foundation (a.k.a. axiom of regularity):
for every non-empty set x, there exists a ∈ x such that
a ∩ x =∅.
As a consequence, 𝕍 coincides with
the von Neumann universe of well-founded sets.
Note that the axiom says that (𝕍, ∈)
is a well-founded relation
provided that we extend the definition of well-foundedness to proper classes.
In addition to the standard powerset operator
𝕍→𝕍
which is denoted by ℙ
we define two sub-operators, ℙ_{+} and ℙ_{₁}.
For every set x, let
ℙ_{+}(x)
=
ℙ(x) ∖ {∅}
(ℙ_{+}(x) is the set of non-empty subsets of x),
ℙ_{₁}(x)
=
{ {u} | u ∈ x }
(ℙ_{₁}(x) is the set of singleton subsets of x).
For a natural number i we let ℙ^{i}
denote the i-th power of ℙ in the usual sense
(i.e.
ℙ^{0}(x) = x and
ℙ^{i+1}(x) =ℙ(ℙ^{i}(x))).
Similarly with ℙ_{+} and ℙ_{₁}.
Powerset cumulation
For every ordinal α,
the
α-th (powerset) cumulation
is denoted ℙ_{⋆}^{α}
and defined recursively as an operator 𝕍→𝕍 by
ℙ_{⋆}^{0}(x)
= x,
ℙ_{⋆}^{α}(x)
= x ∪ℙ_{+}(ℙ_{⋆}^{α-1}(x))
if α is a succesor ordinal,
ℙ_{⋆}^{α}(x)
=∪{ ℙ_{⋆}^{β}(x) | β < α }
if α is a limit ordinal.
By a single
recursive formula,
ℙ_{⋆}^{α}(x) =
x ∪∪{ℙ_{+}(ℙ_{⋆}^{β}(x))
| β < α }
(the α-th cumulation of x).
Note:
We deviate from standard definitions
[][]
by
using ℙ_{+} instead of ℙ in the above formulae.
A correspondence is then obtained by a substitution
x ↦ x ∪ {∅}.
In
[],
ℙ_{⋆}^{ω}(x ∪ {∅})
is called the superstructure over x.
In
[],
ℙ_{⋆}^{α}(x ∪ {∅}) is called an
α-superstructure (for infinite α).
The von Neumann hierarchy
The von Neumann universe 𝕍 is the class of all well-founded sets.
If the axiom of foundation is not assumed
and 𝕌 denotes the universal class of all sets,
then
𝕍 is obtained as a subclass of 𝕌 by transfinite recursion
via the cumulative hierarchy of sets 𝕍_{α}:
𝕍_{α}=∪{ ℙ(𝕍_{β}) | β < α },
that is,
𝕍_{0}=∅,
𝕍_{α}=ℙ(𝕍_{α-1})
if α is a successor ordinal,
𝕍_{α}=∪{ 𝕍_{β}| β < α }
if α is a limit ordinal,
𝕍=∪{ 𝕍_{α}| α ∈On }.
The axiom of foundation can be expressed as 𝕍=𝕌.
The rank function r() on 𝕍 is defined by
r(x) = α
↔
x ∈𝕍_{α+1}∖𝕍_{α}.
𝕍_{α} then contains exactly the sets
x such that r(x) < α.
Proposition:
For every ordinal α, the following are satisfied:
𝕍_{1+α}=ℙ_{⋆}^{α}({∅}).
(𝕍_{α}, ∈) is an α-superstructure
whose ground stage equals
{∅}
(whenever α > 0).
For every non-terminal bounded object x of (𝕍_{α}, ∈),
the powerclass of x
equals ℙ_{+}(x).
The set-representation theorem
Theorem:
Every basic structureS_{0}= (O_{0}, …)
can be represented as a well-founded set O
according to the following table.
In particular, there is a set V such that
(V, ∈)
is a complete ∊-structure
such that all elements of the ground stage
V_{1} are singleton sets of equal rank,
S_{0} is faithfully embedded into (V, ∈).
Terminal objects
T
=
O∩ℙ_{₁}(∪O∖O)
Inheritance root
r
=
∪O∖∪T
For every x, y from O :
Bounded membership
x ∊ y
↔
x ∈ y
Inheritance
x ≤ y
↔
x ⊆ y
Singleton map
x.ɛϲ = y
↔
{x} = y
Powerclass map
x.ec = y
↔
r∩ℙ(x) = y
Power membership
x ϵ¯ y
↔
r∩ℙ(x) ⊆ y
Object membership
x ϵ y
↔
r∩ℙ(x) ⊆ y or x ∈ y
Anti-membership
x ϵ^{-1} y
↔
r∩ℙ(y) ⊇ x
Complete extension (V,∈)
Last stage
V
=
r∪ℙ_{+}(r)
Ground stage
V_{1}
=
V∖ℙ(r)
Ground rank
α_{0}
=
r(∪V_{1})
The i-th stage
V_{i}
=
V∩𝕍_{α0+i}
The i-th metalevel (i < ω)
=
ℙ^{i}(r) ∖ℙ^{i+1}(r)
Union map
x.υ
=
∪x
Proof:
Proceed in the following steps:
Let S_{0}= (O_{0}, …) be a basic structure of ϵ.
Let S_{1}= (O_{1}, …)
be a completion of S_{0}.
Express S_{1} as an ∊-structure (O_{1}, ∊).
Let i be the cardinality of the ground stage of S_{1}.
Let α = i + ϖ + ω.
Let V be a regularly cumulated object
in the (α+1)-superstructure (𝕍_{α+1}, ∈)
such that
the rank of (V, ∈) equals ϖ+1,
the cardinality the ground stage of (V, ∈) equals i,
∅∉V
(this is implicitly satisfied for i ≠ 1).
Let .ν be an isomorphism between
(O_{1}, ∊) and (V, ∈).
Let O be the set O_{0}.ν.
The existence of S_{1} follows by the
completion theorem.
The existence of V follows by the
cardinality assertion.
(The roles of α and ϖ are exchanged.)
The equivalences and equalities stated in the table follow from the equalities
x.ec_{v}
=
ℙ_{+}(x),
x.υ_{v}
=
∪x,
x.ɛϲ_{v}
=
{x},
X.ɛϲ_{v}
=
ℙ_{₁}(X)
which hold in
the (α+1)-superstructure (𝕍_{α+1}, ∈)
for every sets x and X
that are both elements and subsets of
𝕍_{α}∖ {∅}.
(The equalities show the correspondence between set theoretic notation
and the abstract notation of superstructures so that the
last two
propositions of the previous
section can be applied.
In addition, we have used ℙ instead of ℙ_{+}
in the table where applicable.)
Union in basic structures
Consider the definitional extension of a
complete structureS= (O, ∊).
The following table suggests that
the union map .υ
is another candidate to be introduced into basic structures
by abstraction,
just like .ec and .ɛϲ.
Power membership
(ϵ¯) = (.ec) ○ (≤)
(x.ec is the least power container of x)
Bounded membership
(∊) = (.ɛϲ) ○ (≤)
(x.ɛϲ is the least bounded container of x)
Anti-membership
(ϵ^{-1}) = (.υ) ○ (≤)
(x.υ is the least anti-container of x)
We have already observed that in complete structures,
(ϵ^{-1}) ∩ (϶) is a subrelation (thus a submap) of .υ
(which is in turn a subrelation of ϵ^{-1}).
This suggests to regard
(ϵ^{-1}) ∩ (϶) as the implicit part of the abstraction
of .υ in basic structures.
We let the complementary, explicit part of .υ be denoted by .ⱷ
and call it the non-member union map.
The situation is then summarized by the following table.
(We can already consider the table to apply to the general case of basic
structures, with Domain replaced by Potential domain.)
Map between objects
Domain
Map parts
Implicit
Explicit
Powerclass map
.ec
O
∅
.ec
= (ϵ¯) ∩ (϶^{-1})
Singleton map
.ɛϲ
O.∍
(.ɛϲ) ∩ (.ec)
.ɛɕ
= (.ɛϲ) ∖ (ϵ¯)
(primary singleton map)
Union map
.υ
O.϶^{-1}
(ϵ^{-1}) ∩ (϶)
.ⱷ
= (.υ) ∖ (϶)
(non-member union map)
The (b~9) axiom
The corresponding generalization of basic structures
would be based on ϵ6-structures
(instead of ϵ5-structures)
which would contain .ⱷ as an additional constituent of the signature.
The presumed axiomatization of .ⱷ
is established by the reserved(b~9)
axiom which is shown below together with
the similar axiom (b~8).
(b~8)
If x.ɛɕ = y then:
(a) {x} = y.϶,
(b) x.ϵ^{i}= y.ϵ^{i-1} for every i ≤ 1,
(c) (x,y) ∉ (ϵ¯).
(b~9)
If x = y.ⱷ then:
(a_{1})
x.϶¯= y.϶.϶¯,
(a_{2})
x.϶= y.϶^{2},
(b) x.ϵ^{i}= y.ϵ^{i-1} for every i ≤ 0,
(c) (x,y) ∉ (ϵ).
Subsequently, the union map, .υ,
is defined as a partial map between objects by
x = y.υ
↔
{x} = (y.ϵ^{-1}∩ y.϶) ∪ {y}.ⱷ.
We say that S is
union complete
if x.υ is defined for every object x from
O.϶^{-1}.
The domains of ϵ^{-1} and of (ϵ^{-1}) ∖ (϶)
are the potential domains of .υ and .ⱷ,
respectively.
We let the integer powers, inverses and transitive closures of
.υ and .ⱷ
be denoted and defined
in a similar way to that of .ec.
The 0-th powers are identities on the respective potential domain.
Every pair (y,x) from (ϵ^{-1}) ∩ (϶) satisfies conditions
(b~9)(a_{1})(a_{2})(b).
If, in addition,
(b~9) is assumed
then for every objects x, y,
x = y.υ
→
1 + x.mli = y.mli.
Proof:
Assume y ϵ^{-1} x ϵ y and y ϵ^{-1} x' ϵ y.
Then
(a) x ϵ y ϵ^{-1} x' and
(b) x' ϵ y ϵ^{-1} x
so that (a) x ≤ x' and (b) x' ≤ x.
Assume y ϵ^{-1} x ϵ y. Then:
(a_{1})
x.϶¯= y.϶.϶¯:
a ϵ¯ b ϵ y
→
a ϵ¯ b ϵ y ϵ^{-1} x
→
a ϵ¯ b ≤ x
→
a ϵ¯ x.
(a_{2})
x.϶= y.϶.϶:
(The same proof with ϵ¯ replaced by ϵ.)
(b)
x.ϵ^{i}= y.ϵ^{i-1} for every i ≤ 0:
x ϵ^{i} a
→
y ϵ^{-1} x ϵ^{i} a
→
y ϵ^{i-1} a
→
x ϵ y ϵ^{i-1} a
→
x ϵ^{i} a.
This is a consequence of
(b~9)(b).
Rank adjustment
The diagram on the right shows that the (old) definition of the
rank function .d
is no longer acceptable
since the
.ⱷ map adds new constraints to it.
The purple arrow indicates that
a = b.ⱷ.
We consider the diagram to show two structures:
S_{0} and its extension S with the dashed part being the difference.
Since a.d is finite (and thus a non-limit ordinal)
a faithful interpretation of .d should assert that
a.d + 1 = b.d.
However, this is only satisfied in the S structure.
In S_{0},
a.d_{0}= b.d_{0}= 2
(according to the not-yet-adjusted definition of the rank function).
The following is an unverified proposal for the definition of the rank function
.d
in an ϵ6-structure S= (O, ϵ, …, .ⱷ).
Define path membership to be a system
E = {ℇ^{i}| i ∈ℤ}
of relations between objects
generated by the following rules.
(ℇ^{i}) ○ (ℇ^{j}) ⊆ (ℇ^{i+j}) for every
integers i, j.
(ϵ^{i}) ⊆ (ℇ^{i}) for every integer i.
(.ⱷ) ⊆ (ℇ^{-1}).
Define path-ranking product to be a structure (Ƣ, ϵ)
where
Ƣ equals (O⊎ {ȶ}) ×ℕ
(with ȶ standing for a fictitious terminal object)
and
ϵ is a relation on Ƣ such that
for every objects x,y and every natural i, j,
(x,i) ϵ (y,j)↔x ℇ^{1+i-j} y,
(ȶ,i) ϵ (y,j)↔1+i-j = y.mli,
(ȶ,i) ϵ (ȶ,j)↔1+i-j = 0,
(x,i) ϵ (ȶ,j)↔
false.
For every object x, let x.d be the
ϖ-limited rank of (x,0)
in (Ƣ, ϵ).
Union 1-completion
The below is a prescription for
equipping a basic structure with missing union objects.
Since new objects have .ⱷ undefined
we speak about 1-completion.
Let
S= (O, ϵ, …)
be an ϵ6-structure
and
S_{0}= (O_{0}, ϵ▫, …, .ⱷ_{0})
a basic ϵ6-structure.
We say that
S is a union 1-completion of S_{0} if
S is an extension of S_{0}
such that the following are satisfied:
(.ec) = (.ec_{0}),
(.ɛɕ) = (.ɛɕ_{0}), and
O.ⱷ(-1) =O_{0}.ⱷ_{0}(0).
(.ⱷ) ∖ (.ⱷ_{0})
is injective.
(That is, for every new object x there is exactly one old
object y such that x = y.ⱷ.)
(.ⱷ) ∖ (.ⱷ_{0}) is defined according to the
following table.
We assume that a, b are old objects.
In (ⅰ) we assume that
a.ⱷ and b.ⱷ are defined and new,
in (ⅱ) we assume that
a.ⱷ is defined and new
in (ⅲ) we assume that
b.ⱷ is defined and new.
(α)
(β)
(γ) (i ∈ℕ)
ⅰ
a.ⱷϵ b.ⱷ
↔
b ∈ a.ϵ▫^{-1}.ϵ▫¯.ϵ▫
↔
a.ⱷϵ¯ b.ⱷ
a.ⱷϵ¯^{-i} b.ⱷ
↔
b ∈ a.ϵ▫¯^{-i}∪a.ϵ▫¯^{-i-1}.ϵ▫
ⅱ
a.ⱷϵ b
↔
b ∈ a.ϵ▫^{-1}.ϵ▫¯
↔
a.ⱷϵ¯ b
a.ⱷϵ¯^{-i} b
↔
b ∈ a.ϵ▫¯^{-i-1}
ⅲ
a ϵ b.ⱷ
↔
b ∈ a.ϵ▫^{2}
a ϵ¯ b.ⱷ
↔
b ∈ a.ϵ▫¯.ϵ▫
a ϵ¯^{-i} b.ⱷ
↔
b ∈ a.ϵ▫¯^{-i+1}∪a.ϵ▫¯^{-i}.ϵ▫
Completion conjecture
Conjecture:
The completion theorem holds also for basic structures with the union map
(i.e. ϵ6-structures satisfying
(b~1)–(b~11)
with an adjusted definition of .d).
More specifically,