constructive set theorypetera/mathlogaps-slides.pdf · 2008-09-10 · constructive set theory july,...

Constructive Set TheoryJuly, 2008, Mathlogaps workshop, Manchester

.

Peter Aczel

[email protected]

Manchester University

Constructive Set Theory – p.1/88

Plan of lectures

Lecture 1

1: Background to CST2: The axiom system CZF

Lecture 2

3: The number systems in CZF4: The constructive notion of set

Lectures 3,4 ?

5: Inductive definitions6: Locales and/or7: Coinductive definitions


1: Background to CST


Some brands of constructive mathematics

B1: Intuitionism (Brouwer, Heyting, ..., Veldman)B2: ‘Russian’ constructivism (Markov,...)B3: ‘American’ constructivism (Bishop, Bridges,...)B4: ‘European’ constructivism (Martin-Löf, Sambin,...)

B1,B2 contradict classical mathematics; e.g.B1 : All functions R → R are continuous,B2 : All functions N → N are recursive (i.e. CT).

B3 is compatible with each of classical maths, B1,B2and forms their common core.

B4 is a more philosophical foundational approach to B3.All B1-B4 accept RDC and so DC and CC.


Some liberal brands of mathematics using intuitionistic logic

B5: Topos mathematics (Lawvere, Johnstone,...)

B6: Liberal Intuitionism (Mayberry,...)

B5 does not use any choice principles.

B6 accepts Restricted EM.

B7: A minimalist, non-ideological approach: The aim is to do asmuch mainstream constructive mathematics aspossible in a weak framework that is common to allbrands, and explore the variety of possible extensions.


Some settings for constructive mathematics

type theoretical

category theoretical

set theoretical


Some contrastsclassical logic versus intuitionistic logic

impredicative versus predicative

some choice versus no choice

intensional versus extensional

consistent with EM versus inconsistent with EM


Mathematical TaboosA mathematical taboo is a statement that we may not wantto assume false, but we definately do not want to be able toprove.For example Brouwer’s weak counterexamples providetaboos for most brands of constructive mathematics; e.g. if

DPow(A) = b ∈ Pow(A) | (∀x ∈ A)[(x ∈ b) ∨ (x 6∈ b)]

then

(∀b ∈ DPow(N))[ (∃n)[n ∈ b] ∨ ¬(∃n)[n ∈ b] ]

is the Limited Excluded Middle (LEM) taboo.


Warning!There are two meanings of the word theory in mathematicsthat can be confused.

mathematical topic: e.g. (classical) set theory

formal system: e.g. ZF set theory

I will use constructive set theory (CST) as the name of a

mathematical topic and constructive ZF (CZF) as a specific

first order axiom system for CST.


Introducing CSTIt was initiated (using a formal system called CST) byJohn Myhill in his 1975 JSL paper.In 1976 I introduced CZF and gave an interpretation ofCZF+RDC in Martin-Löf’s dependent type theory. In myview the interpretation makes explicit a constructivelyacceptable foundational understanding of a constructiveiterative notion of set.By not assuming any choice principles, CZF allowsreinterpretations in sheaf models so that mathematicsdeveloped in CZF will apply to such models.CST allows the development of constructivemathematics in a purely extensional way exploiting thestandard set theoretical representation of mathematicalobjects.


2: The axiom system CZF


The axiom systemsZF and IZF

These axiom systems are formulated in predicate logicwith equality and the binary predicate symbol ∈.

ZF uses classical logic and IZF uses Intuitionistic logicfor the logical operations ∧,∨,→,⊥,∀,∃.

ZF = IZF + EM

ZF has a ¬¬-translation into IZF (H. Friedman).


The non-logical axioms and schemes ofZF and IZF

Extensionality

Pairing

Union

Separation

Infinity

Powerset

Collection (classically equivalent to Replacement)

Set Induction (classically equivalent to Foundation)

Collection (∀x ∈ a)∃yφ(x, y) → ∃b(∀x ∈ a)(∃y ∈ b)φ(x, y)

Set Induction ∀a[(∀x ∈ a)θ(x) → θ(a)] → ∀aθ(a)


The axiom systemCZF

This is the axiom system that is like IZF except that

the Separation scheme is restricted,

the Collection scheme is strengthened,

and the Powerset axiom is weakened to the SubsetCollection scheme.

CZF ⊆ IZF and CZF + EM = ZF .

CZF has the same proof theoretic strength asKripke-Platek set theory (KP ) or the system ID1 (i.e.Peano Arithmetic with axioms for an inductive definitionof Kleene’s second number class O).


The Restricted Separation SchemeRestricted Quantifiers We write

(∀x ∈ a)θ(x) ≡ ∀x[x ∈ a → θ(x)]

(∃x ∈ a)θ(x) ≡ ∃x[x ∈ a ∧ θ(x)]

A formula is restricted (bounded,∆0) if every quantifier in itis restricted.

The Scheme: ∃b∀x[x ∈ b ↔ (x ∈ a ∧ θ(x, . . .))]for each restricted formula θ(x, . . .).

• We write x ∈ a | θ(x, . . .) for the set b.


The Collection Principles ofCZF, 1

We write (∀∃ x∈ay∈b )θ for

(∀x ∈ a)(∃y ∈ b)θ ∧ (∀y ∈ b)(∃x ∈ a)θ.

Strong Collection

(∀x ∈ a)∃yφ(x, y) → ∃b(∀∃ x∈ay∈b )φ(x, y).

Subset Collection

∃c∀z[(∀x ∈ a)(∃y ∈ b)φ(x, y, z)

→ (∃b′ ∈ c)(∀∃ x∈ay∈b′ )φ(x, y, z)].


The Collection Principles ofCZF, 2

Strong Collection can be proved in IZF using Collectionand Separation.

For if b is the set given by Collection then we get the set

y ∈ b | ∃x ∈ a φ(x, y)

by Separation, which gives Strong Collection if usedinstead of b.

Replacement can be proved in CZF using StrongCollection.

For if ∀x ∈ a ∃!y φ(x, y) and b is a set such that(∀∃ x∈a

y∈b )φ(x, y) then

b = y | ∃x ∈ a φ(x, y).


Classes -more examples

x ∈ A | φ(x, . . .) = x | x ∈ A ∧ φ(x, . . .)

. . . x . . . | x ∈ A = y | ∃x ∈ A y = . . . x . . .

Class functions For classes F,A,B let F : A → B ifF ⊆ A × B such that

(∀x ∈ A)(∃!y ∈ B)[(x, y) ∈ F ].

Also, if a ∈ A then let F (a) be the unique b ∈ B such that(a, b) ∈ F . By Replacement, if A is a set then so is

F (x) | x ∈ A.


The Fullness axiomFor classes A,B,C let C : A >− B if C ⊆ A × B suchthat

(∀x ∈ A)(∃y)[(x, y) ∈ C].

For sets a, b let

mv(a, b) = r ∈ Pow(a × b) | r : a >− b.

The Axiom

(∃c ∈ Pow(mv(a, b)))(∀r ∈ mv(a, b))(∃s ∈ c)[s ⊆ r]

Theorem: Given the other axioms and schemes of CZF, the

Subset Collection scheme is equivalent to the Fullness ax-

iom.


Myhill’s Exponentiation AxiomIf a is a set and B is a class let aB ≡ f | f : a → B.

If F : a → B then F (x) | x ∈ a is a set, and so is F , as

F = (x, F (x)) | x ∈ a.

So F ∈ aB.

The axiom: ab is a set for all sets a, b.

This is an immediate consequence of the Fullnessaxiom and so a theorem of CZF.

For if c ⊆ mv(a, b) is given by Fullness thenab = f ∈ c | f : a → b is a set by Restricted Separation.


‘Truth Values’Let 0 = ∅, 1 = 0 and Ω = Pow(1).

For each formula θ we may associate the class< θ >= x ∈ 1 | θ, where x is not free in θ. Then

θ ↔ < θ >= 1

and if θ is a restricted formula then < θ > is a set in Ω.

It is natural to call < θ > the truth value of θ.

the Powerset axiom is equivalent to“The class Ω is a set”,

the full Separation scheme is equivalent to“Each subclass of 1 is a set and so in Ω”.

With classical logic each subclass of 1 is either 0 or 1,so that the powerset axiom and the full separationscheme hold; i.e. we have ZF.


Set Terms, 1We can conservatively extend CZF to a theory CZFst byadding set terms, t, given by the syntax equation:

t ::= x | ∅ | t, t | ∪t | t ∩ t | t | x ∈ t,

where free occurrences of x in t1 are bound in t1 | x ∈ t2,and adding the following axioms.

y ∈ ∅ ↔ ⊥

y ∈ t1, t2 ↔ [y = t1 ∨ y = t2]

y ∈ ∪t ↔ (∃x ∈ t) y ∈ x

y ∈ t1 ∩ t2 ↔ [y ∈ t1 ∧ y ∈ t2]

y ∈ t1 | x ∈ t2 ↔ (∃x ∈ t2) y = t1


Set Terms, 2Theorem: For each restricted formula θ(x) and set term athere is a set term t such that

CZFst ⊢ t = x ∈ a | θ(x).

Corollary: Given the other axioms and schemes of CZF,the Restricted Separation Scheme is equivalent to theconjunction of the axioms

Emptyset: the empty class ∅ is a set,

Binary Intersection: the intersection class a ∩ b of sets a, b is aset.


The Infinity Axiom

Call a class A inductive if ∅ ∈ A and (∀x ∈ A)[x+ ∈ A],where x+ = x ∪ x.

Infinity Axiom: There is an inductive set.

Strong Infinity Axiom: There is a smallest inductiveset, ω = ∩x | x is an inductive set.

Full Infinity Scheme: There is a smallest inductive setthat is a subset of each inductive class.

In CZF, by making essential use of the Set Induction

Scheme, each instance of the full infinity scheme can be

derived.


3: The number systems inCZF

N 7→ Z 7→ Q 7→ R


Peano structures,1Call (N, 0, S) a Peano structure if the Dedekind-Peanoaxioms hold; i.e. N is a set, 0 ∈ N, S : N → N is injectivesuch that (∀n ∈ N)[S(n) 6= 0] and, for all sets A ⊆ N

[0 ∈ A] ∧ (∀n ∈ A)[S(n) ∈ A] → (∀n ∈ N)[n ∈ A].

It is a full Peano structure if this holds for all classes A.

In CZF, (ω, ∅, s) is a Peano structure, where s : ω → ω isgiven by s(n) = n+ for n ∈ ω.


Peano structures,2Theorem: In CZF, any Peano structure (N, 0, S) is fulland functions can be defined on N by iteration and,more generally by primitive recursion.

Iteration Scheme: For classes A and F : A → A, ifa0 ∈ A then there is a unique H : N → A such thatH(0) = a0 and (∀n ∈ N)[H(S(n)) = F (H(n))].

Corollary 1: In CZF, given a Peano structure (N, 0, S)all the primitive recursive functions on N exist. SoHeyting Arithmetic can be interpreted in CZF.

Corollary 2: In CZF, any two Peano structures areisomorphic. So the axioms for a Peano structure form acategorical axiom system.


The number setsZ and Q in CST, 1Starting with the Peano structure (N, 0, S), thesuccessive construction of first the ordered ring (Z, . . .)of integers and then the ordered field (Q, . . .) ofrationals can be carried out in weak systems of CSTmuch as in classical set theory.

Both the constructions N 7→ Z and Z 7→ Q can beobtained using a quotient (A × B)/R, where A,B aresuitably chosen sets and R is a set equivalence relationon the set A × B.


The number setsZ and Q in CST, 2A × B is the set X = ∪∪(a, b) | a ∈ A | b ∈ B and thequotient X/R is the set [x] | x ∈ X where[x] = x′ ∈ X | (x, x′) ∈ R.

Only the Union and Pairing axioms and theReplacement and Restricted Separation schemes areneeded to get these sets.


The Cauchy continuum in CST, 1We assume the axiom CC of countable choice: For allsets a, if r ∈ mv(N, a) then there is f : N → a such thatf ⊆ r; i.e. (∀n ∈ N)[(n, f(n)) ∈ r].

f : N → Q is a Cauchy sequence if

(∀ǫ ∈ Q>0)(∃n ∈ N)(∀m ≥ n) |f(m) − f(n)| < ǫ.

Using CC, n can be given as a function of ǫ.

Let C be the class of all Cauchy sequences. UsingExponentiation and Restricted Separation this class is aset.

For f, g ∈ C let f ∼ g if

(∀ǫ ∈ Q>0)(∃n ∈ N)(∀m ≥ n) |f(m) − g(m)| < ǫ.


The Cauchy continuum in CST, 2The quotient set C/∼ is the set Rc of Cauchy reals.Addition and multiplication of Cauchy reals can bedefined in the usual way so that Rc forms a ring.

Each rational r can be identified with the Cauchy real[r#] where r#(n) = r for all n ∈ N. This gives a ringembedding of the ring Q in the ring Rc.

We can define a relation < on Rc such that, for f, g ∈ C,[f ] < [g] iff

(∃ǫ ∈ Q>0)(∃n ∈ N)(∀m ≥ n)[f(m) + ǫ < g(m)].

This makes Rc into an Archimedean pseudo-orderedring.


Archimedean pseudo-ordered ringsA relation < on a set R is a pseudo-ordering if, for allx, y, z ∈ R,1. ¬[x < y ∧ y < x],

2. [x < y] → [x < z ∨ z < y],

3. ¬[x < y ∨ y < x] → [x = y].A pseudo-ordered ring is a ring R with a pseudo-orderingcompatible with the ring structure; i.e. for all x, y, z ∈ R,1. [x < y] → [x + z < y + z],

2. [x < y ∧ 0 < z] → [xz < yz].It is Archimedean if, for all a ∈ R there is n ∈ N such that

a <

n︷︸︸︷

1 + · · · + 1 .


More on pseudo-orderingsLet < be a pseudo-ordering of a set R. Define ≤ on R:

x ≤ y ↔ ¬[y < x].

Then ≤ is a partial ordering of R; i.e. it is reflexive,transitive and antisymmetric.


Cauchy CompletenessTheorem (CZF+CC): (Rc, . . .) is the unique, up toisomorphism Archimedean pseudo-ordered field that isCauchy complete.

A pseudo-ordered ring, R, is Cauchy complete if everyCauchy sequence of elements of R converges to anelement of R.

f : N → R is a Cauchy sequence if

(∀ǫ > 0)(∃n)(∀m ≥ n) [f(n) − ǫ < f(m) < f(n) + ǫ],

and converges to a ∈ R if

(∀ǫ > 0)(∃n)(∀m ≥ n) [a − ǫ < f(m) < a + ǫ].


Dedekind CompletenessAn Archimedean pseudo-ordered field, R, is Dedekindcomplete if every upper-located subset has asupremum.

A subset X of R is upper-located if

(∀ǫ > 0)(∃x ∈ X))(∀y ∈ X)[y < x + ǫ].

and a ∈ R is a supremum of X if

(∀x ∈ X)[x ≤ a] ∧ (∀ǫ > 0)(∃x ∈ X)[a < x + ǫ].

Note: If a is a supremum of X then it is the lub of X;i.e.

(∀x ∈ X)[x ≤ a] ∧ (∀b ∈ R)[(∀x ∈ X)[x ≤ b] → [a ≤ b].


The continuum without choiceProposition (CZF): Let R be an Archimedeanpseudo-ordered field. Then1. If R is Dedekind complete then it is Cauchy

complete.2. Assuming CC, if R is Cauchy complete then R is

Dedekind complete.

Theorem (CZF): There is a unique, up to isomorphismDedekind complete, Archimedean, pseudo-orderedfield.

An upper-located X ⊆ Q is a Dedekind cut if X = X<,where X< = y ∈ Q | (∃x ∈ X)[y < x].

Theorem (CZF): The class Rd of all Dedekind cutsforms a set that can be made into a Dedekind complete,Archimedean, pseudo-ordered field.


Binary Refinement AxiomThe axiom: For every set A there is a set D of subsetsof A such that if G0 ∪ G1 = A then there are U0, U1 ∈ Dsuch that U0 ∪ U1 = A and Ui ⊆ Gi for i = 0, 1.

Theorem (CZF): Binary Refinement.Proof: Let the set C ⊆ mv(A, 0, 1) be given byFullness so that (∀r ∈ mv(A, 0, 1))(∃s ∈ C)[s ⊆ r].

Let D = si | (s, i) ∈ C × 0, 1, where

si = x ∈ A | (x, i) ∈ s.

If G0 ∪ G1 = A letr = (G0 × 0) ∪ (G1 × 1) ∈ mv(A, 0, 1).

Choose s ∈ C such that s ⊆ r.

Finally let Ui = si for i = 0, 1.


Proof that Rd is a set, 1Let C be the class of subsets X of Q such that

X is open; i.e. ∀x ∈ X ∃y ∈ X x < y,

X is weakly upper-located; i.e. for all x, y ∈ Q

x < y ⇒ [x ∈ X ∨ y 6∈ X].

Proposition: Rd ⊆ C. So, by Restricted Separation, it

suffices to show that C is a set.


Proof that Rd is a set, 2To show that C is a set: Let

A = (x, y) ∈ Q × Q | x < y.

Choose the set D of subsets of A by Binary Refinement.For each V ∈ D let LV = x | (x, y) ∈ V for some y.

Claim: If L ∈ C then L = LV ′ for some V ′ ∈ D.

By Restricted Separation, P = V ′ ∈ D | LV ′ ∈ C is aset so that, by Replacement,

C = LV ′ | V ′ ∈ P is a set.

Proof of claim: Let L ∈ C. IfV = (x, y) ∈ A | x ∈ L and W = (x, y) ∈ A | y 6∈ L

then V ∪ W = A, as L is weakly located, so that we maychoose V ′ ⊆ V and W ′ ⊆ W , both in D, such thatV ′ ∪ W ′ = A. We will show that L = LV ′.


Proof that Rd is a set, 3Trivially LV ′ ⊆ LV ⊆ L.

To show that L ⊆ LV ′, let x ∈ L. Then, as L is open,

x < y for some y ∈ L

so that, as (x, y) ∈ A, either

(x, y) ∈ V ′ or (x, y) ∈ W ′.

But, if (x, y) ∈ W ′ then (x, y) ∈ W so that y 6∈ L,contradicting y ∈ L.

So we must have (x, y) ∈ V ′, so that x ∈ LV ′.


4: The constructive notion ofset


Some terms for set-like notions

set, class, type, sort

collection, domain, manifold, universe

category (in the philosophical, not

algebraic, sense)


Some history,1Boole: algebra of classes

Dedekind: algebra of ideals

Dedekind/Cantor: pointsets, sets of objects of ourthought,

Cantor: countable and uncountable sets, transfinitenumbers

Frege: naive set theory

Cantor: inconsistent sets

Zermelo/Hilbert/Burali-Forti/Russell: paradoxes of settheory

Zermelo: axiomatic set theory


Some history,2Russell/Poincare: vicious circle principle, impredicativity

Russell: ramified type theory, axiom of reducibility

Weyl: predicative mathematics

Ramsey: simple type theory

Mirimanoff: well-founded (wf) and non-wf sets

Fraenkel: Replacement Scheme

von Neumann: Axiom of Restriction/Foundation


The cumulative hierarchy in ZF

V =⋃

α∈On Vα,

where Vα =⋃

β<α Pow(Vβ) for α ∈ On.

V0 ⊆ V1 ⊆ · · ·Vα ⊆ Vα+1 ⊆ · · ·

V0 = ∅, V1 = ∅, Vα+1 = Pow(Vα)

Vα = the set of sets formed by stage α

x ∈ Vβ | · · ·x · · · ∈ Vα if β < α


The iterative, combinatorial notion of set

Zermelo, Scott, Schoenfield, Boolos, Parsons,...

Sets are extensional.

Sets are formed in stages out of elements formed atearlier stages.

A set is formed by collecting together its elements.

There are lots of stages:1. There is a stage.2. For each stage there is a later stage.3. There is a stage reflecting 1,2.4. If sii∈I is a family of stages indexed by a set I then

there is a stage later than all the si.


Types and ClassesA mathematical object is always given as an object ofsome type.

We write a : A for the judgement that a is an object oftype A.

A class on a type is the extension of a propositionalfunction on the type.

If B is a propositional function on the type A then itsextension is the class C = x : A | B(x).

For a : A the proposition that a is in the class C is B(a).

If also C ′ = x : A | B′(x) then (C = C ′) is theproposition

(∀x : A)[B(x) ↔ B′(x)].


What is a set ofelements of a type?

It is a collection into a whole of objects chosen from thetype.

e.g. given the type N of natural numbers we have setsof natural numbers such as 0, 0, 1, 0, 3, 18,

and sets 0, 2, 4, 6, . . . , 92, 2i | i < n for n : N,

and infinite sets such as 0, 2, 4, 6, . . . = 2i | i : N.

In general we can form sets of natural numbersai | i ∈ I with ai : N for i : I, where I is an index-sort.


SortsI use the word sort for something like- Bishop’s constructive notion of set,which I think is also something like- Martin-Löf’s type-theoretic notion of set or data-typeand something like- the category theorists’ notion of setwhen they talk about a category of sets.

A sort is an object that is conceptually prior to itselements.

I need a distinct word in order to avoid confusion withthe combinatorial notion of set, which is what axiomaticset theory is about.

A set is formed out of its elements.


Sets, of elements of a type, 1Given a type A, a set of elements of A is given by:1. a sort I, the index-sort of the set,2. a function f : I → A, also thought of as a family of

elements of A, aii:I , where ai = f(i) : A for i : I.

We may write the set as sup(I, f) or ai | i : I.

The elements of the set are the ai for i : I.

The sets of elements of A form a type Sub(A).


Sets, of elements of a type, 2An equality relation, =A, on a type A is an assignmentof a proposition (b =A c) to b, c : A so that the laws for anequivalence relation hold; i.e.

(∀x : A)[x =A x], (∀x, y : A)[x =A y → y =A x],(∀x, y, z : A)[x =A y → (y =A z → x =A z)].

Given an equality relation =A on a type A we maydefine the membership relation ∈A and extensionalequality relation =Sub(A) as follows:

If α : Sub(A) is ai | i : I then, for a : A, (a ∈A α) isthe proposition (∃i : I)[a =A ai].If also β : Sub(A) is bj | j : J then (α =Sub(A) β) isthe proposition

(∀i : I)(∃j : J)[ai =A bj ] ∧ (∀j : J)(∃i : I)[ai =A bj ].


The type of iterative setsThe type V of iterative sets is the inductive typeobtained by iterating the set-of operation.

The iterative sets are generated using the following rule.

Any set-of objects in V is an object in V

In Constructive Type Theory V is the inductive typehaving the introduction rule

I : Sort f : I → V

sup(I, f) : V

So we have Sub(V ) = V .


Equality and membership onV

We can recursively define (α =V β) for α, β : V using therule

(∀i : I)(∃j : J)[ai =V bj ] ∧ (∀j : J)(∃i : I)[ai =V bj ]

α =V β

where α = ai | i : I and β = bj | j : J.

Alsoα ∈V β =def (∃j : J)(α =V bj).


Martin-Löf’s PhilosophyNotes on Constructive Mathematics, 1970.

An intuitionistic theory of types:predicative part, in LogicColloquium ’73, published 1975.

Intuitionistic Type Theory, 1980 lectures, notes byGiovanni Sambin, published as a Bibliopolis book in1984.

On the meanings of the logical constants and thejustifications of the logical laws, 1983 lectures,published in Nordic Journal of Philosophical Logic in1996


The type-theoretic interpretation of CST

Aczel, 1978,choice principles:1982,inductive definitions:1986

Aczel and Rathjen, Notes on CST, Mittag-Leffler report,2000/2001

Gambino and Aczel, The generalised type-theoreticinterpretation of CST, JSL, vol 71 (2006), pp. 67-103.


5: Inductive Definitions


Examples of inductive definitionsω is the smallest class I such that ∅ ∈ I and

(∀x ∈ I) x+ ∈ I, where x+ = x ∪ x.

HF is the smallest class I such that, for all n ∈ ω,(∀f ∈ nI) ran(f) ∈ I.

HC is the smallest class I such that, for all a ∈ ω+

(∀f ∈ aI) ran(f) ∈ I.

For each class A, H(A) is the smallest class I suchthat, for all a ∈ A, (∀f ∈ aI) ran(f) ∈ I.

ω = H(2), HF = H(ω), HC = H(ω+)

Recall 0 = ∅, 1 = 0+ and 2 = 1+. Note that ω and HF , butnot HC, can be proved to be sets in CZF.


What is an inductive definition?An inductive definition is a class of pairs. A pair (X, a) in aninductive definition will usually be written X/a and called an(inference) step of the inductive definition, with conclusion aand set X of premisses.

If Φ is an inductive definition, a class I is Φ-closed ifX ⊆ I implies a ∈ I for each step X/a of Φ.

Theorem: There is a smallest Φ-closed class;i.e. a class I such that (i) I is Φ-closed and, for each classB, (ii) if B is Φ-closed then I ⊆ B. class.

The smallest Φ-closed class is unique and is called the class

inductively defined by Φ and is written I(Φ).


More ExamplesThe Set Induction Scheme expresses that V is thesmallest class I such that a ⊆ I ⇒ a ∈ I.

If R is a subclass of A×A such that Ra = x | (x, a) ∈ Ris a set for each a ∈ A then Wf(A,R) is the smallestsubclass I of A such that ∀a ∈ A [Ra ⊆ I ⇒ a ∈ I].

Note that Wf(A,R) = I(Φ), where Φ is the class ofsteps Ra/a for a ∈ A.

If Ba is a set for each a ∈ A then Wx∈ABx is the smallestclass I such that a ∈ A & f : Ba → I ⇒ (a, f) ∈ I.

Note that Wx∈ABx = I(Φ), where Φ is the class of stepsran(f)/(a, f) for a ∈ A and f : Ba → V .


Proof of the theoremGiven a class Φ of steps X/a, for each class Y let ΓYbe the class of a such that there is a step X/a of Φ withX ⊆ Y . So Y is Φ-closed iff ΓY ⊆ Y .

Γ is monotone; i.e. Y1 ⊆ Y2 ⇒ ΓY1 ⊆ ΓY2 and what iswanted is a least pre-fixed point of Γ.

The idea for the proof is to iterate the operator Γ into thetransfinite so that it ultimately closes up.

Call a class J of pairs an iteration class for Φ if, for allsets a, Ja = ΓJ∈a where Ja = x | (a, x) ∈ J andJ∈a =

⋃

x∈a Jx.

Lemma: Every inductive definition has an iteration

class.


Proof of the lemmaA set G of ordered pairs is defined to be good if

(∗) Ga ⊆ ΓG∈a for all sets a.Let J be the union of all good sets.• We must show that Ja = ΓJ∈a.• If y ∈ Ja then, for some good set G,

y ∈ Ga ⊆ ΓG∈a ⊆ ΓJ∈a.

Thus Ja ⊆ ΓJ∈a. For the converse let y ∈ ΓJ∈a so thatX/a is a step of Φ for some X ⊆ J∈a. So

∀y′ ∈ X ∃G [ G is good and y′ ∈ G∈a ].

By Strong Collection there is a set Z of good sets such that

∀y′ ∈ X ∃G ∈ Z y′ ∈ G∈a.

Let G = (a, y) ∪⋃

Z. Then G is good so that y ∈ Ga ⊆ Ja.Thus ΓJ∈a ⊆ Ja.


Definition of I(Φ)

We show that J∞ =⋃

a∈V Ja is the smallest Φ-closed class.• To show that J∞ is Φ-closed let X/y be a step of Φ forsome set X ⊆ J∞. We must show that y ∈ J∞.−− As ∀y′ ∈ X ∃x y′ ∈ Jx, by Collection, there is a set asuch that ∀y′ ∈ X ∃x ∈ a y′ ∈ Jx; i.e. X ⊆ J∈a. Hencey ∈ ΓJ∈a = Ja ⊆ J∞. Thus J∞ is Φ-closed.• Let I be Φ-closed, to show that J∞ ⊆ I we show thatJa ⊆ I by set-induction on a. So we may assume theinduction hypothesis that Jx ⊆ I for all x ∈ a. It follows thatJ∈a ⊆ I so that Ja = ΓJ∈a ⊆ ΓI ⊆ I, the inclusions holdingbecause Γ is monotone and I is Φ-closed. Thus J∞ ⊆ I

So we define I(Φ) = J∞.


Local Inductive DefinitionsAn inductive definition Φ is defined to be local if ΓY is a setfor each set Y .Proposition: If Φ is local then Ja and J∈a are sets for all a.

This has an easy proof by Set Induction.


When is the classI(Φ) a set?A class B is a bound for Φ if, whenever X/y is a step ofΦ then X = ran(f) for some f ∈

⋃

b∈BbX.

Φ is bounded if Φ has a set bound and, for each set X,the class of conclusions y of steps X/y in Φ is a set.

Note that if Φ is a set then it is bounded.

CZF+ = CZF + REA, where REA is theRegular Extension Axiom

Theorem (CZF+): If Φ is bounded then it is local and I(Φ)is a set. Examples: For each set A,

H(A) is a set,

Wf(A,R) is a set, if R is a set,

Wx∈ABx is a set, if Bx is a set for each x ∈ A.


Regular Extension Axiom (REA)A set A is regular if (A,∈ ∩(A × A) is a transitive modelof the Strong Collection Scheme; i.e. it is an inhabitedset such that A ⊆ Pow(A) and if a ∈ A and R : a >− Athen there is b ∈ A such that

∀x ∈ a ∃y ∈ b (x, y) ∈ R and ∀y ∈ b ∃x ∈ a (x, y) ∈ R

The axiom REA: Every set is a subset of a regular set.

Classically, if α is a regular ordinal then Vα is a regularset.


Tree Proofs,1We will give a characterisation of I(Φ) in terms of asuitable notion of tree proof.

These will be well-founded trees, each given as a pair(a, Z), where a is the conclusion of the proof and Z isthe set of proofs of the premisses of the final inferencestep X/a of the proof.

We will call these trees proto-proofs. We will associatewith each proto-proof p the set Steps(p) of the inferencesteps that it uses.

Then a proto-proof p = (a, Z) will be a proof thata ∈ I(Φ) provided that Steps(p) ⊆ Φ.


Tree Proofs,2Definition: The class P of proto-proofs is inductively definedto be the smallest class such that, for all pairs p = (a, Z), ifZ ⊆ P then p ∈ P; i.e. P = I(Ψ), where Ψ is the class ofsteps Z/p for pairs p = (a, Z).Definition: Let concl : V 2 → V , Concl : Pow(V 2) → V andendstep : V × Pow(V 2) → V be given by

concl(p) = a

Concl(Z) = concl(q) | q ∈ Z

endstep(p) = (Concl(Z), a)

for all pairs p = (a, Z).Lemma: There is a unique class functionSteps : P → Pow(Pow(V ) × V ) such that, for p = (a, Z) ∈ P,

(∗) Steps(p) = endstep(p) ∪⋃

Steps(q) | q ∈ Z.


Tree Proofs, 3Definition: For each inductive definition Φ we define theclass P(Φ) of Φ-proofs as follows.

P(Φ) = p ∈ P | Steps(p) ⊆ Φ.

Theorem (CZF): For each inductive definition Φ

I(Φ) = I ′,

where I ′ = concl(p) | p ∈ P(Φ).

• The theorem is a consequence of the following two claims.

Claim 1: concl(p) ∈ I(steps(p)) for all p ∈ P.

Claim 2: I ′ is Φ-closed.


Set Compactness

Theorem (CZF+): For each set S and each setP ⊆ Pow(S) there is a set B of subsets of P × S such that,for each class Φ ⊆ P × S,

a ∈ I(Φ) ⇐⇒ a ∈ I(Φ0) for some Φ0 ∈ B such that Φ0 ⊆ Φ.

Definition: For each class X let

I(Φ, X) = I(Φ ∪ (∅ × X)).

Theorem (CZF+): If Φ is a subset of Pow(S) × S, where Sis a set, then there is a set B of subsets of S such that, foreach class X,

a ∈ I(Φ, X) ⇐⇒ a ∈ I(Φ, X0) for some X0 ∈ B

such that X0 ⊆ X.


6: Point-free Topology


Classical point-free topologyIn classical point-set topology a topology on a set X ofpoints is a subset τ of Pow(X) of the open sets.

The set τ must have X ∈ τ and be closed underarbitrary unions and binary intersections.

τ , when partially ordered by inclusion, forms a nicecomplete lattice, sometimes called a frame or locale.

A complete lattice is a poset L in which every subset Yhas both a sup

∨Y and an inf

∧Y .

In fact only sups are needed, as∧

Y =∨

a ∈ L | ∀x ∈ Y a ≤ x.

A frame/locale is a complete lattice satisfying thedistributive law a ∧

∨Y =

∨a ∧ x | x ∈ Y .


A problem for CST!According to the classical definition there are nointeresting complete lattices!e.g. neither 2 = 0, 1 nor Ω = Pow(1) is a completelattice.

2 complete implies Ω = 2, and Ω complete implies Ω is aset, which is a taboo.

What to do?

We must allow topologies of open sets, and so alsocomplete lattices, to be classes.

New Definition: A∨

-semi-lattice is a poclass L suchthat every subset Y has a sup

∨Y .

It is a frame/locale if it has a top ⊤ and binary meetssuch that the frame distributive law holds.Example: For each set A, Pow(A) is a frame.


Another problem for CST!The definition

∧Y =

∨a ∈ L | ∀x ∈ Y a ≤ x. does not

work, as the sup is of a class that may not be a set.

What to do?

Definition: A subset G of a∨

-semi-lattice L is a set ofgenerators for L if, for every a ∈ L, Ga = x ∈ S | x ≤ ais a set such that a =

∨Gx.

L is set-generated if it has a set of generators.

Proposition: If L has a set S of generators then everysubset Y has an inf given by

∧

Y =∨

a ∈ L | ∀x ∈ S a ≤ x.

Example: A set basis for a topological space is a set ofgenerators for its locale of opens.


Formal Topologies, 1A formal topology (A,A) consists of a set A and anoperator A : Pow(A) → Pow(A) such that for setsU, V ⊆ A,

U ⊆ AU

U ⊆ AV ⇒ AU ⊆ AV ,AU ∩ AV ⊆ A(U ↓ ∩V ↓), where U ↓=

⋃

x∈U Ax.

The notion of a formal topology was first introduced byGiovanni Sambin as a constructive approach topoint-free topology in the setting of dependent typetheory. Instead of the operator A it is usual to use thecover relation ⊳ where a ⊳ U ⇐⇒ a ∈ AU .

Up to isomorphism there is a one-one correspondencebetween formal topologies and set-generated locales:


Formal Topologies, 2Each formal topology (A,A) determines aset-generated locale

Sat(A,A) = U ∈ Pow(A) | AU = U,

partially ordered by ⊆, with set Ax | x ∈ A ofgenerators.

Every set-generated locale, with set G of generators, isisomorphic to Sat(A,A), where (A,A) is the formaltopology with A = G and AU = GW

U for U ∈ Pow(A).


Formal PointsLet (A,A) be a formal topology. A formal point is a setα ⊆ A such that

∃a a ∈ α,∀a, b ∈ α ∃c ∈ α c ∈ Aa ∩ Ab,∃a ∈ α a ∈ AU ⇒ ∃a ∈ α a ∈ U , for all sets U ⊆ A.

There may not be any formal points.

The class Pt(A,A) of formal points may not form a set.If it is a set then the set forms a topological space withthe set Ba | a ∈ A of basic open sets, whereBa = α ∈ Pt(A,A) | a ∈ α.


A Galois adjunctionClassically there is a Galois adjunction between thecategory Top of topological spaces and the categoryLoc of locales:

The maps of Top are the continuous functions, while themaps L → L′ of Loc are the frame maps L′ → L; i.e. thefunctions L′ → L that preserve the frame structure, top,sups and binary meets.To each topological space is associated its locale ofopen sets and to each locale is associated itstopological space of ‘points’.There is a CZF version of the Galois adjunction in mypaper " Aspects of General Topology in ConstructiveSet Theory", in the proceedings of the secondworkshop of Formal Topology in a special issue of theAnnals of Pure and Applied Logic, 137 (2006) 3-29.There are complications because certain classes thatare sets classically cannot be shown to be sets in CZF.


7: Coinductive Definitions


What is a co-inductive definition?Let Φ be a class of steps. For each class Y let

ΓY =⋃

X∈Pow(Y )a | X/a ∈ Φ.

Y is Φ-closed if X ⊆ Y ⇒ a ∈ Y for all X/a ∈ Φ.Y is Γ-closed if ΓY ⊆ Y . These notions coincide.I(Φ) is the smallest Φ-closed class.

Y is Φ-progressive if a ∈ Y ⇒ X)(Y for all X/a ∈ Φ,where X)(Y if X ∩ Y has an element [Sambin].

)( is dual to ⊆: X)(Y ⇐⇒ (∃a ∈ X)(a ∈ Y ) andX ⊆ Y ⇐⇒ (∀a ∈ X)(a ∈ Y ).

Y is Γ-progressive if Y ⊆ ΓY .

J (Γ) = largest Γ-progressive class.

J (Φ) = largest Φ-progressive class.J (Φ, B) = largest Φ-progressive subclass of B.


The streams exampleLet Φ be the class of steps x/(n, x) for n ∈ N.Then ΓY = N × Y .

In the universe of hypersets J (Γ) is the set of streams,but J (Φ) is the largest class Y such that(n, x) ∈ Y ⇒ x ∈ Y ; i.e. the class V of all sets.

J (Φ, N × V ) is the set of streams.

‘Γ-progressive’ seems the right notion for ‘finalcoalgebras’, but ‘Φ-progressive’ seems the right notionfor my applications.

I expect that the general theory applies to either notion.


Review of inductive definitionsTheorem [CZF]: There is a smallest Φ-closed class I(Φ),called the class inductively defined by Φ. More generally, for eachclass U there is a smallest class AU = I(Φ, U) that is Φ-closedand includes U .

Theorem [CZF+REA]: If Φ is a set then I(Φ) is a set. Moregenerally AU is a set for each subset U of S.

Each inductively generated formal topology (S,A) isobtained using a suitable set Φ ⊆ Pow(S) × S.


The problem of Coinduction in CZFWe would like to show that under suitable conditions onΦ there is a largest Φ-progressive class J (Φ) that,under further conditions is a set. This would be calledthe class coinductively defined by Φ. More generally wewould like to be able to define a largest Φ-progressivesubclass AU of U , for each subclass U of S.

The hope is that this might give us the method tocoinductively define a binary positivity predicate on aformal topology.

But we need to add a new axiom to CZF.


Relation Reflection Scheme (RRS)Relation Reflection Scheme (RRS): For classes R,X, ifR : X >− X then for each a0 ∈ X there is a subset Y ofX such that a0 ∈ Y and R ∩ (Y × Y ) : Y >− Y .

This is an easy consequence of the scheme:Relative Dependent Choices (RDC): If X,R are classessuch that R : X >− X then for each a0 ∈ X there isf : N → X such that f(0) = a0 and (f(n), f(n + 1)) ∈ Rfor all n ∈ N.

For, given f we can let Y = ran(f). In fact:

Theorem (CZF): RDC is equivalent to RRS+DC,where the dependent choices axiom, DC, is like RDCexcept that X and R are required to be sets.

RRS is provable in ZF, so seems not to be a choiceprinciple - GOOD!.


Coinductive definitions of classesTheorem [CZF+RRS]: Let Φ be a class of steps such thatΦa = X | X/a is a Φ-step is a set for every a. Then there is alargest Φ-progressive class J . Moreover, for each class U there isa largest Φ-progressive subclass JU of U .

Proof: Observe that the union of any class ofΦ-progressive sets is a Φ-progressive class.

In particular we obtain the Φ-progressive class

J =⋃

Y | Y is a Φ-progressive set

and for each class U the Φ-progressive class

JU =⋃

Y ∈ Pow(U) | Y is a Φ-progressive set

Let B be a Φ-progressive class. We show that B ⊆ J .The proof that if B ⊆ U then B ⊆ JU is similar.


Proof that B ⊆ J ,1Claim: Let A = Pow(B). Then

(∀Y ∈ A)(∃Y ′ ∈ A) ∀a ∈ Y ∀X ∈ Φa X)(Y ′.

Proof of claim: Given Y ∈ A let ΦY = Φ ∩ (V × Y ).

Then ΦY =⋃

a∈Y (Φa × a) and so is a set.

For any step s = X/a let Xs be the set X.

As B is Φ-progressive and Y ⊆ B,

∀s ∈ ΦY ∃z[z ∈ Xs ∩ B].

So, by Strong Collection, there is a set Y ′ such that

∀s ∈ ΦY ∃z ∈ Y ′[z ∈ Xs ∩ B] & ∀z ∈ Y ′∃s ∈ ΦY [z ∈ Xs ∩ B].

So Y ′ ∈ A and, if a ∈ Y and X ∈ Φa then s = X/a ∈ ΦY

and hence X)(Y ′, completing the proof of the claim.


Proof that B ⊆ J , 2To show that B is a subclass of J let b ∈ B. It suffices toshow that b ∈ U for some Φ-progressive set U , as thenb ∈ U ⊆ J .

By the claim and RRS there is a set Z ⊆ A = Pow(B)such that b ∈ Z and

(∀Y ∈ Z)(∃Y ′ ∈ Z) ∀a ∈ Y ∀X ∈ Φa X)(Y ′.

Let U = ∪Z. Then b ∈ U . Also if a ∈ U , with X/a aΦ-step, then a ∈ Y for some Y ∈ Z so that X)(Y ′ forsome Y ′ ∈ Z and hence X)(U .

Thus b ∈ U for some Φ-progressive set U .


Coinductive definitions of setsTheorem[CZF+*REA]: If Φ is a set then the largestΦ-progresive class J is a set.

The axiom *REA states that every set is a subset of a*-regular set where a regular set A is *-regular if it isunion-closed; i.e. (∀x ∈ A) ∪x ∈ A, and (A,∈ ∩(A × A)is a transitive model of the 2nd order version of thescheme RRS.


constructive set theorypetera/mathlogaps-slides.pdf · 2008-09-10 · constructive set theory july,...

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