Constructive Set Theory July, 2008, Mathlogaps workshop, Manchester - - PowerPoint PPT Presentation
Constructive Set Theory July, 2008, Mathlogaps workshop, Manchester . Peter Aczel petera@cs.man.ac.uk Manchester University Constructive Set Theory p.1/88 Plan of lectures Lecture 1 1: Background to CST 2: The axiom system CZF Lecture 2
July, 2008, Mathlogaps workshop, Manchester . Peter Aczel
petera@cs.man.ac.uk
Manchester University
Constructive Set Theory – p.1/88
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
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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.
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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.
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type theoretical category theoretical set theoretical
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classical logic versus intuitionistic logic impredicative versus predicative some choice versus no choice intensional versus extensional consistent with EM versus inconsistent with EM
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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 ∈ Pow(A) | (∀x ∈ A)[(x ∈ b) ∨ (x ∈ b)]}
then
(∀b ∈ DPow(N))[ (∃n)[n ∈ b] ∨ ¬(∃n)[n ∈ b] ]
is the Limited Excluded Middle (LEM) taboo.
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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.
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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
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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).
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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)
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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
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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, . . .).
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We write (∀∃ x∈a
y∈b )θ for
(∀x ∈ a)(∃y ∈ b)θ ∧ (∀y ∈ b)(∃x ∈ a)θ.
Strong Collection
(∀x ∈ a)∃yφ(x, y) → ∃b(∀∃ x∈a
y∈b )φ(x, y).
Subset Collection
∃c∀z[(∀x ∈ a)(∃y ∈ b)φ(x, y, z) → (∃b′ ∈ c)(∀∃ x∈a
y∈b′)φ(x, y, z)].
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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 ∈ a φ(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 ∈ a φ(x, y)}.
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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} A = {x | ∃y ∈ A x ∈ y} A = {x | ∀y ∈ A x ∈ y} Pow(A) = {x | x ⊆ A} A × B = {x | (∃a ∈ A)(∃y ∈ B)x = (a, b)}
where (a, b) = {{a}, {a, b}}.
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{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 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}.
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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 ∈ 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.
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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.
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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 separation scheme hold; i.e. we have ZF .
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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
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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.
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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.
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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) = 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.
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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
categorical axiom system.
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Starting with the Peano structure (N, 0, S), the successive construction of first the ordered ring (Z, . . .)
rationals can be carried out in weak systems of CST much as in classical set theory. Both the constructions N → Z and Z → Q can be
suitably chosen sets and R is a set equivalence relation
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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.
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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.
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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 < g(m)].
This makes Rc into an Archimedean pseudo-ordered ring.
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A relation < on a set R is a pseudo-ordering if, for all
x, y, z ∈ R,
A pseudo-ordered ring is a ring R with a pseudo-ordering compatible with the ring structure; i.e. for all x, y, z ∈ R,
It is Archimedean if, for all a ∈ R there is n ∈ N such that
a <
n
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Let < 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.
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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 < f(m) < f(n) + ǫ],
and converges to a ∈ R if
(∀ǫ > 0)(∃n)(∀m ≥ n) [a − ǫ < f(m) < a + ǫ].
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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 < 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].
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Proposition (CZF): Let R be an Archimedean pseudo-ordered field. Then
complete.
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 < x]}. Theorem (CZF): The class Rd of all Dedekind cuts forms a set that can be made into a Dedekind complete, Archimedean, pseudo-ordered field.
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The axiom: For every set A there is a set D of subsets
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.
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Let 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 ∈ X].
Proposition:
Rd ⊆ C.
So, by Restricted Separation, it suffices to show that C is a set.
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To 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 a set so that, by Replacement,
C = {LV ′ | V ′ ∈ P} is a set.
Proof of claim: Let L ∈ C. If
V = {(x, y) ∈ A | x ∈ L} and W = {(x, y) ∈ A | y ∈ L}
then V ∪ W = A, as L is weakly located, so that we may choose V ′ ⊆ V and W ′ ⊆ W, both in D, such that
V ′ ∪ W ′ = A. We will show that L = LV ′.
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Trivially 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 ∈ L, contradicting y ∈ L. So we must have (x, y) ∈ V ′, so that x ∈ LV ′.
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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
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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
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α∈On Vα,
β<α Pow(Vβ) for α ∈ On.
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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:
there is a stage later than all the si.
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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)].
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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
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 numbers
{ai | i ∈ I} with ai : N for i : I, where I is an index-sort.
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I use the word sort for something like
which I think is also something like
and something like
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.
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Given a type A, a set of elements of A is given by:
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).
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An equality relation, =A, on a type A is an assignment
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].
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The type V of iterative sets is the inductive type
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 .
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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).
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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
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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.
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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 .
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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).
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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 .
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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
J∈a =
x∈a Jx.
Lemma: Every inductive definition has an iteration
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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.
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.
Let G = {(a, y)} ∪ Z. Then G is good so that y ∈ Ga ⊆ Ja. Thus ΓJ∈a ⊆ Ja.
Constructive Set Theory – p.62/88
We show that J∞ =
a∈V Ja is the smallest Φ-closed class.
some set X ⊆ J∞. We must show that y ∈ J∞.
−− As ∀y′ ∈ X ∃x y′ ∈ Jx, by Collection, there is a set a
such that ∀y′ ∈ X ∃x ∈ a y′ ∈ Jx; i.e. X ⊆ J∈a. Hence
y ∈ ΓJ∈a = Ja ⊆ J∞. Thus J∞ is Φ-closed.
Ja ⊆ I by set-induction on a.
So we may assume the 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∞.
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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
A class B is a bound for Φ if, whenever X/y is a step of
Φ then X = ran(f) for some f ∈
b∈B bX.
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.
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A set A is regular if (A, ∈ ∩(A × A) is a transitive model
set such that A ⊆ Pow(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.
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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.
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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 : Pow(V 2) → V and
endstep : 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 function
Steps : P → Pow(Pow(V ) × V ) such that, for p = (a, Z) ∈ P, (∗) Steps(p) = {endstep(p)} ∪
Constructive Set Theory – p.68/88
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)}.
Claim 1: concl(p) ∈ I(steps(p)) for all p ∈ P. Claim 2: I′ is Φ-closed.
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Theorem (CZF +): For each set S and each set
P ⊆ 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 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.
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Constructive Set Theory – p.71/88
In classical point-set topology a topology on a set X of points is a subset τ of Pow(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 Y and an inf Y . In fact only sups are needed, as
A frame/locale is a complete lattice satisfying the distributive law
a ∧ Y = {a ∧ x | x ∈ Y }.
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According to the classical definition there are no interesting complete lattices! e.g. neither 2 = {0, 1} nor Ω = Pow(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 -semi-lattice is a poclass L such that every subset Y has a sup 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, Pow(A) is a frame.
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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 of generators for L if, for every a ∈ L, Ga = {x ∈ S | x ≤ a} is 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 every subset Y has an inf given by
Example: A set basis for a topological space is a set of generators for its locale of opens.
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A formal topology (A, A) consists of a set A and an
U, V ⊆ A, U ⊆ AU U ⊆ AV ⇒ AU ⊆ AV , AU ∩ AV ⊆ A(U ↓ ∩V ↓), where U ↓=
x∈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
cover relation ⊳ where a ⊳ U
⇐ ⇒ a ∈ AU.
Up to isomorphism there is a one-one correspondence between formal topologies and set-generated locales:
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Each formal topology (A, A) determines a set-generated locale
Sat(A, A) = {U ∈ Pow(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 ∈ Pow(A).
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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 Pt(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 ∈ Pt(A, A) | a ∈ α}.
Constructive Set Theory – p.77/88
Classically there is a Galois adjunction between the category Top of topological spaces and the category
Loc of locales:
The maps of Top 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
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 sets in CZF .
Constructive Set Theory – p.78/88
Constructive Set Theory – p.79/88
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 ) 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
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.
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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
Constructive Set Theory – p.82/88
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 .
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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!.
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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 =
and for each class U the Φ-progressive class
J U =
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
Claim: 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.
Constructive Set Theory – p.86/88
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 = 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 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
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.
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