Download Set-generated Classes in Constructive Set Theory CZF

Set-generated Classes in Constructive Set Theory
CZF
Yasushi Sangu
(School of Information Science, JAIST)
March 10, 2011
Background
We use constructive Zermelo-Fraenkel set theory, CZF.
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CZF looks like ZF.
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CZF has a natural interpretation in Martin-Löf’s Intuitionistic
Type Theory, ITT.
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ITT is the base of a number of proof assistants, such as
NuPRL and Agda.
Constructive Set Theory CZF
The language of CZF are
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variables a, b, ....
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constant ω
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predicates =, ∈
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intuitionistic logic
and the following axioms:
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Extensionality
∀a∀b(∀x(x ∈ a ⇔ x ∈ b) ⇒ a = b)
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Pairing
∀a∀b∃c∀x(x ∈ c ⇔ x = a ∨ x = b)
Constructive Set Theory CZF
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Union
∀a∃b∀x(x ∈ b ⇔ ∃y ∈ a(x ∈ y ))
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Restricted Separation
∀a∃b∀x(x ∈ b ⇔ x ∈ b ∧ ϕ(x))
for any restricted formula ϕ(x); A formula is restricted if all
its quantifiers occurring in it are bounded, i.e. of the form
∃x ∈ y or ∀x ∈ y .
Constructive Set Theory CZF
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Strong Collection
∀a(∀x ∈ a∃y ϕ(x, y ) ⇒
∃b(∀x ∈ a∃y ∈ b ϕ(x, y ) ∧ ∀y ∈ b∃y ∈ a ϕ(x, y )))
for every formula ϕ(x, y ).
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Subset Collection
∀a∀b∃c∀u(∀x ∈ a∃y ∈ b ϕ(x, y , u) ⇒
∃d ∈ c(∀x ∈ a∃y ∈ d ϕ(x, y , u) ∧ ∀y ∈ d∃x ∈ a ϕ(x, y , u)))
for every formula ϕ(x, y , u).
Constructive Set Theory CZF
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Infinity
0 ∈ ω ∧ ∀x(x ∈ ω ⇒ x + 1 ∈ ω),
∀y (0 ∈ y ∧ ∀x(x ∈ y ⇒ x + 1 ∈ y ) ⇒ ∀z(z ∈ ω ⇒ z ∈ y )),
def
where x + 1 = x ∪ {x},
def
0 = ∅ = {x ∈ ω|x 6= x}.
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Set Induction
∀a(∀x ∈ a ϕ(x) ⇒ ϕ(a)) ⇒ ∀a ϕ(a)
for every formula ϕ(a).
Fullness
Let
r ∈ mv(a, b) ⇐⇒ r ⊆ a × b ∧ ∀x ∈ a∃y ∈ b(hx, y i ∈ r ).
Then Subset Collection is equivalent to the following.
Fullness
∀a, b∃c[c ⊆ mv(a, b) ∧ ∀r ∈ mv(a, b)∃s ∈ c(s ⊆ r )].
We have the following corollary.
Exponentiation
∀a∀b∃c∀f [f ∈ c ⇔ (f ∈ b a )]
Note that {0, 1}a 6= Pow(a) in intuitionistic logic.
Set-generated Class
Generally, given a set S, the collection
{α ∈ Pow(S) | ϕ(α)}
does not form a set in CZF.
Definition
Let S be a set, and X be a subclass of Pow(S). Then X is
set-generated if there exists a set Y ⊆ X such that
[
∀a ∈ X [a = {b ∈ Y | b ⊆ a}].
Example
Pow(S) is generated by Sin(S) = {{x} | x ∈ S}.
Choice Principles
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CZF + AC ⊢ φ ∨ ¬φ
where φ is restricted formula.
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CZF + AC ⊢ Powerset.
Relativized Dependent Choices, RDC: For arbitrary formulae φ
and ψ, Whenever
∀x[φ(x) ⇒ ∃y (φ(y ) ∧ ψ(x, y ))]
and φ(b0 ), then there exists a function f with domain ω such that
f (0) = b0 and
∀n ∈ ω[φ(f (n)) ∧ ψ(f (n), f (n + 1))].
Σ-closed set
Definition
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S : a set,
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B : a subset of Pow(S),
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Σ : B → Pow(Pow(S)).
Then a subset α of S is Σ-closed if
∀σ ∈ B[σ ⊆ α ⇒ ∃β ∈ Σ(σ)(β ⊆ α)].
Theorem
Definition
Fin(S) is the set of finitely enumerable subsets of S.
Theorem
Assume RDC. If B ⊆ Fin(S), then the class of Σ-closed sets is
set-generated; i.e
{α ∈ Pow(S) | α is Σ-closed }
is set-generated.
Another additional axiom, REA.
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there is a natural interpretation for RDC in ITT. But, in
sheaf semantics there is a model in which RDC fails.
So we use another axiom, REA( and its variants: uREA,
RRS-uREA, and RRS2 -uREA).
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REA is valid in sheaf semantics. But we need ITT with
W-types to interpret CZF + REA.
Regular Extension Axiom(REA)
Definition
A set A is regular if it is
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transitive, i.e. ∀a ∈ A(a ⊆ A), and
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for any a ∈ A and R ∈ mv(a, A) there exists b ∈ A such that
∀x ∈ a∃y ∈ b((x, y ) ∈ R) ∧ ∀y ∈ b∃x ∈ a((x, y ) ∈ R).
Regular Extension Axiom, REA: Every set is a subset of a
regular set.
Definition
S
A set A is union-closed if ∀a ∈ A( a ∈ A).
uREA: Every set is a subset of a union-closed regular set.
Regular Extension Axiom(REA)
Definition
A regular set A is RRS-regular if for each A′ ⊆ A and
R ∈ mv(A′ , A′ ), if a0 ∈ A′ , then there exists a set A0 ∈ A such
that a0 ∈ A0 ⊆ A′ and ∀x ∈ A0 ∃y ∈ A0 (hx, y i ∈ R).
RRS-uREA: Every set is a subset of a union-closed RRS-regular
set.
Definition
A regular set A is RRS2 -regular if for each A′ ⊆ A and
R ∈ mv(A′ × A′ , A′ ), if a0 ∈ A′ , then there exists a set A0 ∈ A such
that a0 ∈ A0 ⊆ A′ and ∀x ∈ A0 ∀y ∈ A0 ∃z ∈ A0 (hhx, y i, zi ∈ R).
RRS2 -uREA: Every set is a subset of a union-closed RRS2 -regular
set.
Theorems
Theorem
Assume RRS-uREA. If B ⊆ Sin(S), then the class of Σ-closed
sets is set-generated.
Theorems
Theorem
Assume RRS-uREA. If B ⊆ Sin(S), then the class of Σ-closed
sets is set-generated.
Theorem
Assume RRS2 -uREA. If B ⊆ Fin(S), the class of Σ-closed sets is
set-generated.
Application :(Γ, ∆)-closed
Let X , Y be sets, and let Γ : Pow(X ) → Pow(Y ) and
∆ : Y → Pow(Pow(X )).
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Γ is B-based if
∃B ⊆ Pow(X )[Γ(α) =
[
{Γ(β) | β ∈ B ∧ β ⊆ α}]
for any α ∈ Pow(X ).
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α ∈ Pow(X ) is (Γ, ∆)-closed if
∀a ∈ Γ(α)∃b ∈ ∆(a)(b ⊆ α).
Proposition
Assume RRS2 -uREA or RDC. If Γ is B-based and B ⊆ Fin(X ),
then the class of (Γ, ∆)-closed sets is set-generated.
Application
Let (A, ∩, ∪,′ , 0, 1) be a Boolean algebra. Then a subset F of A is
a filter if
1. 1 ∈ F ,
2. x, y ∈ F ⇒ x ∩ y ∈ F ,
3. x ∈ F and x ≤ y ⇒ y ∈ F .
A filter F is prime if
x ∪ y ∈ F ⇒ x ∈ F or y ∈ F .
Proposition
Assume RRS2 -uREA or RDC. Then the class of prime filters is
set-generated.
Proof
Proof.
Let B = {∅} ∪ {{x, y } | x, y ∈ A} and X = A + {{x, y } | x, y ∈ A}.
Define B-based Γ : Pow(A) → Pow(X ) and
∆ : X → Pow(Pow(A)) by
Γ(∅) = A,
Γ({x, y }) = {{x, y }},
and
∆(x) = {{1, x}, {1, x ′ }},
∆({x, y }) = {{x ∩ y }∪ ↑ x∪ ↑ y }.
a subset F of A is a prime filter ⇐⇒ F is (Γ, ∆)-closed.
Application
A filter F is proper if 0 ∈
/ F.
Proposition
Let X be a class of inhabited subsets of S, and let
Min(X ) = {x ∈ X | ∀y ∈ X (y ⊆ x ⇒ y = x)}. If X is
set-generated, then Min(X ) is a set.
Proposition
Assume RRS2 -uREA or RDC. Then the class of proper prime
filters is a set; i.e.
{F ∈ Pow(A) | F is a proper prime filter }
is a set.
Application to Topology
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A neighborhood space is a pair (X , τ ) consisting of a set X
and a subset τ of Pow(X ) such that
1.∀x ∈ X ∃U ∈ τ (x ∈ U),
2.∀x ∈ X ∀U, V ∈ τ [x ∈ U ∩ V =⇒ ∃W ∈ τ (x ∈ W ⊆ U ∩ V )].
We say that τ is an open base on X .
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A subset A of X is open if
∀x ∈ A∃U ∈ τ (x ∈ U ⊆ A).
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A function f between (X , τ ) and (Y , σ) is continuous if
f −1 (V ) is open for each V ∈ σ.
Application to Topology
Let
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X be a set,
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{(Xi , τi ) | i ∈ I } a family of neighborhood spaces,
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{fi : Xi → X | i ∈ I } a family of functions.
Then a open base τ on X is final for the family
{fi : Xi → X | i ∈ I } if
g is continuous ⇐⇒ g ◦ fi is continuous for each i ∈ I .
for any neighborhood space (Y , σ) and any function g : X → Y ,
Application to Topology
Proposition
Assume RRS-uREA or RDC. Then the class
C = {U ∈ Pow(X ) | fi −1 (U) is open for each i ∈ I }.
is set-generated, and the generating set is a final open base on X .
Application to Topology
Proof.
P
Let B = Sin(X ) and Z = i ∈I Xi . Define B-based
Γ : Pow(X ) → Pow(Z ) and ∆ : Z → Pow(Pow(X )) by
Γ({x}) = {(i , y ) | i ∈ I , y ∈ Xi , fi (y ) = x},
∆(i , y ) = {fi (V ) | V ∈ τi , y ∈ V }.
Then a subset U of X is in C ⇐⇒ U is (Γ, ∆)-closed.
Further Work
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Find more applications.
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Prove that the class of Σ-closed sets is set-generated in CZF
+ RRS.
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RRS is valid in sheaf semantics,
CZF + RDC ⊢ RRS.
Fin