Constructive Set Theory July, 2008, Mathlogaps Workshop, Manchester

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Constructive Set Theory July, 2008, Mathlogaps workshop, Manchester .

Manchester University

Constructive Set Theory – p.1/88 Plan of lectures

Lecture 1 1: Background to CST 2: The axiom system CZF Lecture 2 3: The number systems in CZF 4: The constructive notion of set Lectures 3,4 ? 5: Inductive deﬁnitions 6: Locales and/or 7: Coinductive deﬁnitions

Constructive Set Theory – p.2/88 1: Background to CST

Constructive Set Theory – p.3/88 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,B2 and forms their common core. B4 is a more philosophical foundational approach to B3. All B1-B4 accept RDC and so DC and CC.

Constructive Set Theory – p.4/88 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 as much mainstream constructive mathematics as possible in a weak framework that is common to all brands, and explore the variety of possible extensions.

Constructive Set Theory – p.5/88 Some settings for constructive mathematics type theoretical category theoretical set theoretical

Constructive Set Theory – p.6/88 Some contrasts classical logic versus intuitionistic logic impredicative versus predicative some choice versus no choice intensional versus extensional consistent with EM versus inconsistent with EM

Constructive Set Theory – p.7/88 Mathematical Taboos

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

DPow(A) = {b ∈ P ow(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.

Constructive Set Theory – p.8/88 Warning!

There are two meanings of the word theory in mathematics that 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.

Constructive Set Theory – p.9/88 Introducing CST

It was initiated (using a formal system called CST) by John Myhill in his 1975 JSL paper. In 1976 I introduced CZF and gave an interpretation of CZF+RDC in Martin-Löf’s dependent type theory. In my view the interpretation makes explicit a constructively acceptable foundational understanding of a constructive iterative notion of set. By not assuming any choice principles, CZF allows reinterpretations in sheaf models so that mathematics developed in CZF will apply to such models. CST allows the development of constructive mathematics in a purely extensional way exploiting the standard set theoretical representation of mathematical objects.

Constructive Set Theory – p.10/88 2: The axiom system CZF

Constructive Set Theory – p.11/88 The axiom systems ZF and IZF

These axiom systems are formulated in predicate logic with equality and the binary predicate symbol ∈. ZF uses classical logic and IZF uses Intuitionistic logic for the logical operations ∧, ∨, →, ⊥, ∀, ∃. ZF = IZF + EM ZF has a ¬¬-translation into IZF (H. Friedman).

Constructive Set Theory – p.12/88 The non-logical axioms and schemes of ZF 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)

Constructive Set Theory – p.13/88 The axiom system CZF

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 Subset Collection scheme.

CZF ⊆ IZF and CZF + EM = ZF . CZF has the same proof theoretic strength as Kripke-Platek set theory (KP ) or the system ID1 (i.e. Peano Arithmetic with axioms for an inductive deﬁnition of Kleene’s second number class O).

Constructive Set Theory – p.14/88 The Restricted Separation Scheme

Restricted 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 it is 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.

Constructive Set Theory – p.15/88 The Collection Principles of CZF, 1

x∈a We write (∀∃ y∈b )θ for

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

Strong Collection

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

Subset Collection

∃c∀z[(∀x ∈ a)(∃y ∈ b)φ(x,y,z)

′ x∈a → (∃b ∈ c)(∀∃ y∈b′ )φ(x,y,z)].

Constructive Set Theory – p.16/88 The Collection Principles of CZF, 2

Strong Collection can be proved in IZF using Collection and Separation. For if b is the set given by Collection then we get the set

{y ∈ b |∃x ∈ aφ(x,y)}

by Separation, which gives Strong Collection if used instead of b. Replacement can be proved in CZF using Strong Collection. 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 ∈ aφ(x,y)}.

Constructive Set Theory – p.17/88 Classes

Class terms: {x | φ(x,...)} a ∈ {x | φ(x,...)} ↔ φ(x,...) Identify each set a with the class {x | x ∈ a}. [A = B] ≡∀x[x ∈ A ↔ x ∈ B] Some Examples

V = {x | x = x} S A = {x |∃y ∈ Ax ∈ y} T A = {x |∀y ∈ Ax ∈ y} P ow(A) = {x | x ⊆ A} A × B = {x | (∃a ∈ A)(∃y ∈ B)x =(a, b)} where (a, b) = {{a}, {a, b}}.

Constructive Set Theory – p.18/88 Classes -more examples

{x ∈ A | φ(x,...)} = {x | x ∈ A ∧ φ(x,...)} {...x... | x ∈ A} = {y |∃x ∈ Ay = ...x...}

Class functions For classes F,A,B let F : A → B if F ⊆ 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}.

Constructive Set Theory – p.19/88 The Fullness axiom

For classes A,B,C let C : A >− B if C ⊆ A × B such that

(∀x ∈ A)(∃y)[(x,y) ∈ C]. For sets a, b let

mv(a, b) = {r ∈ P ow(a × b) | r : a >− b}. The Axiom (∃c ∈ P ow(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 axiom.

Constructive Set Theory – p.20/88 Myhill’s Exponentiation Axiom

If 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 Fullness axiom and so a theorem of CZF. For if c ⊆ mv(a, b) is given by Fullness then ab = {f ∈ c | f : a → b} is a set by Restricted Separation.

Constructive Set Theory – p.21/88 ‘Truth Values’

Let 0 = ∅, 1 = {0} and Ω = P ow(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 separation scheme hold; i.e. we have ZF.

Constructive Set Theory – p.22/88 Set Terms, 1

We can conservatively extend CZF to a theory CZFst by adding 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

Constructive Set Theory – p.23/88 Set Terms, 2

Theorem: For each restricted formula θ(x) and set term a there 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 the conjunction of the axioms Emptyset: the empty class ∅ is a set, Binary Intersection: the intersection class a ∩ b of sets a, b is a set.

Constructive Set Theory – p.24/88 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 inductive set, ω = ∩{x | x is an inductive set}. Full Infinity Scheme: There is a smallest inductive set that 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.

Constructive Set Theory – p.25/88 3: The number systems in CZF

N 7→ Z 7→ Q 7→ R

Constructive Set Theory – p.26/88 Peano structures,1

Call (N, 0, S) a Peano structure if the Dedekind-Peano axioms hold; i.e. N is a set, 0 ∈ N, S : N → N is injective such 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 is given by s(n) = n+ for n ∈ ω.

Constructive Set Theory – p.27/88 Peano structures,2

Theorem: In CZF, any Peano structure (N, 0, S) is full and functions can be deﬁned on N by iteration and, more generally by primitive recursion. Iteration Scheme: For classes A and F : A → A, if a0 ∈ A then there is a unique H : N → A such that H(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. So Heyting Arithmetic can be interpreted in CZF. Corollary 2: In CZF, any two Peano structures are isomorphic. So the axioms for a Peano structure form a categorical axiom system.

Constructive Set Theory – p.28/88 The number sets Z and Q in CST, 1

Starting with the Peano structure (N, 0, S), the successive construction of ﬁrst the ordered ring (Z,...) of integers and then the ordered ﬁeld (Q,...) of rationals can be carried out in weak systems of CST much as in classical set theory. Both the constructions N 7→ Z and Z 7→ Q can be obtained using a quotient (A × B)/R, where A,B are suitably chosen sets and R is a set equivalence relation on the set A × B.

Constructive Set Theory – p.29/88 The number sets Z and Q in CST, 2

A × B is the set X = ∪{∪{(a, b) | a ∈ A} | b ∈ B} and the quotient X/R is the set {[x] | x ∈ X} where [x] = {x′ ∈ X | (x,x′) ∈ R}. Only the Union and Pairing axioms and the Replacement and Restricted Separation schemes are needed to get these sets.

Constructive Set Theory – p.30/88 The Cauchy continuum in CST, 1

We assume the axiom CC of countable choice: For all sets a, if r ∈ mv(N, a) then there is f : N → a such that f ⊆ 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. Using Exponentiation and Restricted Separation this class is a set. For f,g ∈ C let f ∼ g if (∀ǫ ∈ Q>0)(∃n ∈ N)(∀m ≥ n) |f(m) − g(m)| < ǫ.

Constructive Set Theory – p.31/88 The Cauchy continuum in CST, 2

The quotient set C/∼ is the set Rc of Cauchy reals. Addition and multiplication of Cauchy reals can be deﬁned 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 ring embedding 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

This makes Rc into an Archimedean pseudo-ordered ring.

Constructive Set Theory – p.32/88 Archimedean pseudo-ordered rings

A relation < on a set R is a pseudo-ordering if, for all x,y,z ∈ R, 1. ¬[x

Constructive Set Theory – p.33/88 More on pseudo-orderings

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

x ≤ y ↔ ¬[y

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

Constructive Set Theory – p.34/88 Cauchy Completeness

Theorem (CZF+CC): (Rc,...) is the unique, up to isomorphism Archimedean pseudo-ordered ﬁeld that is Cauchy complete. A pseudo-ordered ring, R, is Cauchy complete if every Cauchy sequence of elements of R converges to an element of R. f : N → R is a Cauchy sequence if

(∀ǫ > 0)(∃n)(∀m ≥ n) [f(n) − ǫ

and converges to a ∈ R if

(∀ǫ > 0)(∃n)(∀m ≥ n) [a − ǫ

Constructive Set Theory – p.35/88 Dedekind Completeness

An Archimedean pseudo-ordered ﬁeld, R, is Dedekind complete if every upper-located subset has a supremum. A subset X of R is upper-located if

(∀ǫ > 0)(∃x ∈ X))(∀y ∈ X)[y

and a ∈ R is a supremum of X if

(∀x ∈ X)[x ≤ a] ∧ (∀ǫ > 0)(∃x ∈ X)[a

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].

Constructive Set Theory – p.36/88 The continuum without choice

Proposition (CZF): Let R be an Archimedean pseudo-ordered ﬁeld. Then 1. 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 isomorphism Dedekind complete, Archimedean, pseudo-ordered ﬁeld. An upper-located X ⊆ Q is a Dedekind cut if X = X<, where X< = {y ∈ Q | (∃x ∈ X)[y

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

Constructive Set Theory – p.37/88 Binary Reﬁnement Axiom

The axiom: For every set A there is a set D of subsets of A such that if G0 ∪ G1 = A then there are U0, U1 ∈ D such 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 by Fullness 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 let r =(G0 × {0}) ∪ (G1 × {1}) ∈ mv(A, {0, 1}). Choose s ∈ C such that s ⊆ r.

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

Constructive Set Theory – p.38/88 Proof that Rd is a set, 1

Let C be the class of subsets X of Q such that X is open; i.e. ∀x ∈ X ∃y ∈ Xx

Proposition: Rd ⊆ C. So, by Restricted Separation, it sufﬁces to show that C is a set.

Constructive Set Theory – p.39/88 Proof that Rd is a set, 2

To show that C is a set: Let A = {(x,y) ∈ Q × Q | x

Constructive Set Theory – p.40/88 Proof that Rd is a set, 3

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

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 ′.

Constructive Set Theory – p.41/88 4: The constructive notion of set

Constructive Set Theory – p.42/88 Some terms for set-like notions set, class, type, sort collection, domain, manifold, universe category (in the philosophical, not algebraic, sense)

Constructive Set Theory – p.43/88 Some history,1

Boole: algebra of classes Dedekind: algebra of ideals Dedekind/Cantor: pointsets, sets of objects of our thought, Cantor: countable and uncountable sets, transﬁnite numbers Frege: naive set theory Cantor: inconsistent sets Zermelo/Hilbert/Burali-Forti/Russell: paradoxes of set theory Zermelo: axiomatic set theory

Constructive Set Theory – p.44/88 Some history,2

Russell/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

Constructive Set Theory – p.45/88 The cumulative hierarchy in ZF , V = Sα∈On Vα where for . Vα = Sβ<α P ow(Vβ) α ∈ On

V0 ⊆ V1 ⊆··· Vα ⊆ Vα+1 ⊆···

V0 = ∅,V1 = {∅},Vα+1 = P ow(Vα)

Vα = the set of sets formed by stage α

{x ∈ Vβ |··· x · · ·} ∈ Vα if β < α

Constructive Set Theory – p.46/88 The iterative, combinatorial notion of set

Zermelo, Scott, Schoenﬁeld, Boolos, Parsons,... Sets are extensional. Sets are formed in stages out of elements formed at earlier 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 {si}i∈I is a family of stages indexed by a set I then there is a stage later than all the si.

Constructive Set Theory – p.47/88 Types and Classes

A mathematical object is always given as an object of some type. We write a : A for the judgement that a is an object of type A. A class on a type is the extension of a propositional function on the type. If B is a propositional function on the type A then its extension 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 the proposition ′ (∀x : A)[B(x) ↔ B (x)].

Constructive Set Theory – p.48/88 What is a set of elements of a type?

It is a collection into a whole of objects chosen from the type. e.g. given the type N of natural numbers we have sets of natural numbers such as {0}, {0, 1}, {0, 3, 18}, {} and sets {0, 2, 4, 6,..., 92}, {2i | i

Constructive Set Theory – p.49/88 Sorts

I 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-type and something like - the category theorists’ notion of set when they talk about a category of sets. A sort is an object that is conceptually prior to its elements. I need a distinct word in order to avoid confusion with the combinatorial notion of set, which is what axiomatic set theory is about. A set is formed out of its elements.

Constructive Set Theory – p.50/88 Sets, of elements of a type, 1

Given 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, {ai}i: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).

Constructive Set Theory – p.51/88 Sets, of elements of a type, 2

An equality relation, =A, on a type A is an assignment of a proposition (b =A c) to b, c : A so that the laws for an equivalence 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 may deﬁne the membership relation ∈A and extensional equality relation =Sub(A) as follows:

If α : Sub(A) is {ai | i : I} then, for a : A, (a ∈A α) is the proposition (∃i : I)[a =A ai].

If also β : Sub(A) is {bj | j : J} then (α =Sub(A) β) is the proposition

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

Constructive Set Theory – p.52/88 The type of iterative sets

The type V of iterative sets is the inductive type obtained 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 type having the introduction rule

I : Sort f : I → V sup(I,f) : V

So we have Sub(V ) = V .

Constructive Set Theory – p.53/88 Equality and membership on V

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

(∀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}. Also α ∈V β =def (∃j : J)(α =V bj).

Constructive Set Theory – p.54/88 Martin-Löf’s Philosophy

Notes on Constructive Mathematics, 1970. An intuitionistic theory of types:predicative part, in Logic Colloquium ’73, published 1975. Intuitionistic Type Theory, 1980 lectures, notes by Giovanni Sambin, published as a Bibliopolis book in 1984. On the meanings of the logical constants and the justiﬁcations of the logical laws, 1983 lectures, published in Nordic Journal of Philosophical Logic in 1996

Constructive Set Theory – p.55/88 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-theoretic interpretation of CST, JSL, vol 71 (2006), pp. 67-103.

Constructive Set Theory – p.56/88 5: Inductive Deﬁnitions

Constructive Set Theory – p.57/88 Examples of inductive deﬁnitions

ω 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 such that, 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 , but not HC, can be proved to be sets in CZF.

Constructive Set Theory – p.58/88 What is an inductive deﬁnition?

An inductive definition is a class of pairs. A pair (X, a) in an inductive definition will usually be written X/a and called an (inference) step of the inductive definition, with conclusion a and set X of premisses. If Φ is an inductive definition, a class I is Φ-closed if X ⊆ 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 class B, (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).

Constructive Set Theory – p.59/88 More Examples

The Set Induction Scheme expresses that V is the smallest class I such that a ⊆ I ⇒ a ∈ I.

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

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

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

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

Constructive Set Theory – p.60/88 Proof of the theorem

Given a class Φ of steps X/a, for each class Y let ΓY be the class of a such that there is a step X/a of Φ with X ⊆ Y . So Y is Φ-closed iff ΓY ⊆ Y .

Γ is monotone; i.e. Y1 ⊆ Y2 ⇒ ΓY1 ⊆ ΓY2 and what is wanted is a least pre-ﬁxed point of Γ. The idea for the proof is to iterate the operator Γ into the transﬁnite so that it ultimately closes up. Call a class J of pairs an iteration class for Φ if, for all sets a, Ja = ΓJ∈a where Ja = {x | (a, x) ∈ J} and ∈a x J = Sx∈a J .

Lemma: Every inductive deﬁnition has an iteration class.

Constructive Set Theory – p.61/88 Proof of the lemma

A 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 that X/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. a a Let G = {(a, y)}∪ S Z. Then G is good so that y ∈ G ⊆ J . Thus ΓJ∈a ⊆ Ja.

Constructive Set Theory – p.62/88 Deﬁnition of I(Φ)

∞ a We show that J = Sa∈V J is the smallest Φ-closed class. • To show that J∞ is Φ-closed let X/y be a step of Φ for some set X ⊆ J∞. We must show that y ∈ J∞. −− As ∀y′ ∈ X ∃xy′ ∈ Jx, by Collection, there is a set a such that ∀y′ ∈ X ∃x ∈ ay′ ∈ Jx; i.e. X ⊆ J∈a. Hence y ∈ ΓJ∈a = Ja ⊆ J∞. Thus J∞ is Φ-closed. • Let I be Φ-closed, to show that J∞ ⊆ I we show that Ja ⊆ I by set-induction on a. Sowemayassumethe induction hypothesis that Jx ⊆ I for all x ∈ a. It follows that J∈a ⊆ I so that Ja = ΓJ∈a ⊆ ΓI ⊆ I, the inclusions holding because Γ is monotone and I is Φ-closed. Thus J∞ ⊆ I So we deﬁne I(Φ) = J∞.

Constructive Set Theory – p.63/88 Local Inductive Deﬁnitions

An inductive deﬁnition Φ is deﬁned to be local if ΓY is a set for 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.

Constructive Set Theory – p.64/88 When is the class I(Φ) a set?

A class B is a bound for Φ if, whenever X/y is a step of b Φ then X = ran(f) for some f ∈ Sb∈B X. 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 the Regular 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.

Constructive Set Theory – p.65/88 Regular Extension Axiom (REA)

A set A is regular if (A, ∈∩(A × A) is a transitive model of the Strong Collection Scheme; i.e. it is an inhabited set such that A ⊆ P ow(A) and if a ∈ A and R : a >− A then 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 regular set.

Constructive Set Theory – p.66/88 Tree Proofs,1

We will give a characterisation of I(Φ) in terms of a suitable 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 is the set of proofs of the premisses of the ﬁnal inference step X/a of the proof. We will call these trees proto-proofs. We will associate with each proto-proof p the set Steps(p) of the inference steps that it uses. Then a proto-proof p =(a, Z) will be a proof that a ∈ I(Φ) provided that Steps(p) ⊆ Φ.

Constructive Set Theory – p.67/88 Tree Proofs,2

Definition: The class P of proto-proofs is inductively defined to be the smallest class such that, for all pairs p =(a, Z), if Z ⊆ P then p ∈ P; i.e. P = I(Ψ), where Ψ is the class of steps Z/p for pairs p =(a, Z). Definition: Let concl : V 2 → V , Concl : P ow(V 2) → V and endstep : V × P ow(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 function Steps : P → P ow(P ow(V ) × V ) such that, for p =(a, Z) ∈ P, (∗) Steps(p) = {endstep(p)}∪ [{Steps(q) | q ∈ Z}.

Constructive Set Theory – p.68/88 Tree Proofs, 3

Definition: For each inductive definition Φ we define the class 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.

Constructive Set Theory – p.69/88 Set Compactness

Theorem (CZF +): For each set S and each set P ⊆ P ow(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 P ow(S) × S, where S is a set, then there is a set B of subsets of S such that, for each class X,

a ∈ I(Φ,X) ⇐⇒ a ∈ I(Φ,X0) for some X0 ∈ B such that X0 ⊆ X.

Constructive Set Theory – p.70/88 6: Point-free Topology

Constructive Set Theory – p.71/88 Classical point-free topology

In classical point-set topology a topology on a set X of points is a subset τ of P ow(X) of the open sets. The set τ must have X ∈ τ and be closed under arbitrary unions and binary intersections. τ, when partially ordered by inclusion, forms a nice complete lattice, sometimes called a frame or locale. A complete lattice is a poset L in which every subset Y has both a sup W Y and an inf V Y . In fact only sups are needed, as

^ Y = _{a ∈ L |∀x ∈ Y a ≤ x}. A frame/locale is a complete lattice satisfying the distributive law a ∧ W Y = W{a ∧ x | x ∈ Y }.

Constructive Set Theory – p.72/88 A problem for CST!

According to the classical deﬁnition there are no interesting complete lattices! e.g. neither 2 = {0, 1} nor Ω = P ow(1) is a complete lattice. 2 complete implies Ω = 2, and Ω complete implies Ω is a set, which is a taboo. What to do? We must allow topologies of open sets, and so also complete lattices, to be classes.

New Deﬁnition: A W-semi-lattice is a poclass L such that every subset Y has a sup W Y . It is a frame/locale if it has a top ⊤ and binary meets such that the frame distributive law holds. Example: For each set A, P ow(A) is a frame.

Constructive Set Theory – p.73/88 Another problem for CST!

The deﬁnition V Y = W{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 W-semi-lattice L is a set of generators for L if, for every a ∈ L, Ga = {x ∈ S | x ≤ a} is a set such that a = W Gx. L is set-generated if it has a set of generators. Proposition: If L has a set S of generators then every subset Y has an inf given by ^ Y = _{a ∈ L |∀x ∈ S a ≤ x}. Example: A set basis for a topological space is a set of generators for its locale of opens.

Constructive Set Theory – p.74/88 Formal Topologies, 1

A formal topology (A, A) consists of a set A and an operator A : P ow(A) → P ow(A) such that for sets U, V ⊆ A, U ⊆AU U ⊆AV ⇒AU ⊆AV , AU ∩AV ⊆A(U ↓∩V ↓), where U ↓= Sx∈U A{x}. The notion of a formal topology was ﬁrst introduced by Giovanni Sambin as a constructive approach to point-free topology in the setting of dependent type theory. Instead of the operator A it is usual to use the cover relation ⊳ where a ⊳ U ⇐⇒ a ∈AU. Up to isomorphism there is a one-one correspondence between formal topologies and set-generated locales:

Constructive Set Theory – p.75/88 Formal Topologies, 2

Each formal topology (A, A) determines a set-generated locale

Sat(A, A) = {U ∈ P ow(A) |AU = U},

partially ordered by ⊆, with set {A{x} | x ∈ A} of generators. Every set-generated locale, with set G of generators, is isomorphic to Sat(A, A), where (A, A) is the formal topology with A = G and AU = GW U for U ∈ P ow(A).

Constructive Set Theory – p.76/88 Formal Points

Let (A, A) be a formal topology. A formal point is a set α ⊆ A such that ∃a a ∈ α, ∀a, b ∈ α ∃c ∈ α c ∈ A{a}∩A{b}, ∃a ∈ α a ∈AU ⇒ ∃a ∈ α a ∈ U, for all sets U ⊆ A. There may not be any formal points. The class P t(A, A) of formal points may not form a set. If it is a set then the set forms a topological space with the set {Ba | a ∈ A} of basic open sets, where Ba = {α ∈ P t(A, A) | a ∈ α}.

Constructive Set Theory – p.77/88 A Galois adjunction Classically there is a Galois adjunction between the category T op of topological spaces and the category Loc of locales: The maps of T op are the continuous functions, while the maps L → L′ of Loc are the frame maps L′ → L; i.e. the functions L′ → L that preserve the frame structure, top, sups and binary meets. To each topological space is associated its locale of open sets and to each locale is associated its topological space of ‘points’. There is a CZF version of the Galois adjunction in my paper " Aspects of General Topology in Constructive Set Theory", in the proceedings of the second workshop of Formal Topology in a special issue of the Annals of Pure and Applied Logic, 137 (2006) 3-29. There are complications because certain classes that

are sets classically cannot be shown to be setsConstructive in CZF. Set Theory – p.78/88 7: Coinductive Deﬁnitions

Constructive Set Theory – p.79/88 What is a co-inductive deﬁnition?

Let Φ be a class of steps. For each class Y let ΓY = SX∈P ow(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 ) and X ⊆ 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.

Constructive Set Theory – p.80/88 The streams example

Let 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 ‘ﬁnal coalgebras’, but ‘Φ-progressive’ seems the right notion for my applications. I expect that the general theory applies to either notion.

Constructive Set Theory – p.81/88 Review of inductive deﬁnitions

Theorem [CZF]: There is a smallest Φ-closed class I(Φ), called the class inductively deﬁned by Φ. More generally, for each class U there is a smallest class AU = I(Φ, U) that is Φ-closed and includes U. Theorem [CZF+REA]: If Φ is a set then I(Φ) is a set. More generally AU is a set for each subset U of S. Each inductively generated formal topology (S, A) is obtained using a suitable set Φ ⊆ P ow(S) × S.

Constructive Set Theory – p.82/88 The problem of Coinduction in CZF

We would like to show that under suitable conditions on Φ there is a largest Φ-progressive class J (Φ) that, under further conditions is a set. This would be called the class coinductively defined by Φ. More generally we would like to be able to define a largest Φ-progressive subclass AU of U, for each subclass U of S. The hope is that this might give us the method to coinductively define a binary positivity predicate on a formal topology. But we need to add a new axiom to CZF.

Constructive Set Theory – p.83/88 Relation Reﬂection Scheme (RRS)

Relation Reﬂection Scheme (RRS): For classes R,X, if R : X >− X then for each a0 ∈ X there is a subset Y of X 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 classes such that R : X >− X then for each a0 ∈ X there is f : N → X such that f(0) = a0 and (f(n),f(n + 1)) ∈ R for 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 RDC except that X and R are required to be sets. RRS is provable in ZF, so seems not to be a choice principle - GOOD!.

Constructive Set Theory – p.84/88 Coinductive deﬁnitions of classes

Theorem [CZF+RRS]: Let Φ be a class of steps such that Φa = {X | X/a is a Φ-step} is a set for every a. Then there is a largest Φ-progressive class J . Moreover, for each class U there is a largest Φ-progressive subclass J U of U. Proof: Observe that the union of any class of Φ-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 J U = [{Y ∈ P ow(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 ⊆ J U is similar.

Constructive Set Theory – p.85/88 Proof that B ⊆ J ,1

Claim: Let A = P ow(B). Then ′ ′ (∀Y ∈ A)(∃Y ∈ A) ∀a ∈ Y ∀X ∈ Φa X)(Y . Proof of claim: Given Y ∈ A let ΦY = Φ ∩ (V × Y ). Y Then Φ = Sa∈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, Y ∀s ∈ Φ ∃z[z ∈ Xs ∩ B]. So, by Strong Collection, there is a set Y ′ such that Y ′ ′ Y ∀s ∈ Φ ∃z ∈ Y [z ∈ Xs ∩ B] & ∀z ∈ Y ∃s ∈ Φ [z ∈ Xs ∩ B]. ′ Y So Y ∈ A and, if a ∈ Y and X ∈ Φa then s = X/a ∈ Φ and hence X)(Y ′, completing the proof of the claim.

Constructive Set Theory – p.86/88 Proof that B ⊆ J ,2

To show that B is a subclass of J let b ∈ B. It sufﬁces to show that b ∈ U for some Φ-progressive set U, as then b ∈ U ⊆ J . By the claim and RRS there is a set Z ⊆ A = P ow(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 a Φ-step, then a ∈ Y for some Y ∈ Z so that X)(Y ′ for some Y ′ ∈ Z and hence X)(U. Thus b ∈ U for some Φ-progressive set U.

Constructive Set Theory – p.87/88 Coinductive deﬁnitions of sets

Theorem[CZF+*REA]: If Φ is a set then the largest Φ-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 is union-closed; i.e. (∀x ∈ A) ∪x ∈ A, and (A, ∈∩(A × A) is a transitive model of the 2nd order version of the scheme RRS.

Constructive Set Theory – p.88/88