On some fixed point statements in KP Silvia Steila joint work with - - PowerPoint PPT Presentation

On some fixed point statements in KP

Silvia Steila

joint work with Gerhard J¨ ager

Universit¨ at Bern

Applied Proof Theory and the Computational Content of Mathematics ¨ OMG - DMV 2017, Salzburg September 14, 2017

Tarski Knaster’s theorem

Tarski Knaster’s theorem

Tarski Knaster’s theorem Let L be a complete lattice and let F : L → L be an order-preserving

• function. Then F has a least fixed point.

◮ This theorem holds in Kripke Platek Set Theory (KP). ◮ In ZFC, the powerset of a set is a complete lattice. ◮ Over ZFC, given any monotone function F : P(a) → P(a) for some

set a, there exists a set which is the least fixed point of F.

A first question

Over KP, given a set a and any monotone function F : P(a) → P(a), does there exist a set which is the least fixed point of F? First of all, could we say that? We can formalize this statement over KP, by using Barwise’s machinery

• f Σ function symbols, but this kind of formalization is rather clumsy.

So... we introduce a second order extension KPc of KP.

KPc

◮ Let Lc be the extension of L with countably many class variables. ◮ The atomic formulas comprise the ones of L and all expression of

the form “a ∈ U”.

◮ An Lc formula is elementary if it contains no class quantifiers. ◮ ∆c n, Σc n and Πc n are defined as usual, but permitting subformulas of

the form “a ∈ U”.

KPc

The theory KPc is formulated in Lc and consists of the following axioms:

◮ extensionality, pair, union, infinity, ◮ ∆c 0-Separation: i.e, for every ∆c 0 formula A in which x is not free

and any set a, ∃x(x = {y ∈ a : A[y]})

◮ ∆c 0-Collection: i.e, for every ∆c 0 formula A and any set a,

∀x ∈ a∃yA[x, y] → ∃b∀x ∈ a∃y ∈ bA[x, y]

◮ ∆c 1-Comprehension: i.e, for every Σc 1 formula A and every Πc 1

formula B, ∀x(A[x] ↔ B[x]) → ∃X∀x(x ∈ X ↔ A[x])

◮ Elementary ∈-induction: i.e, for every elementary formula A,

∀x((∀y ∈ xA[y]) → A[x]) → ∀xA[x]

Operators

◮ We call a class an operator if all its elements are ordered pairs and it

is right-unique (i.e. functional).

◮ We use F to denote operators. ◮ Given an operator F and a set a we write Mon[F, a] for:

∀x(F(x) ⊆ a) ∧ ∀x, y(x ⊆ y → F(x) ⊆ F(y)).

Least fixed point statements

FP Mon[F, a] → ∃x(F(x) = x) LFP Mon[F, a] → ∃x(F(x) = x ∧ ∀y(F(y) = y → x ⊆ y)

Σc

1-separation

Σc

1-separation

For every Σc

1 formula A in which x is not free and any set a,

∃x(x = {y ∈ a : A[y]}).

Σc

1-separation implies LFP

◮ Given any set a and any operator F, put

H[F, f , α] := Fun[f , α + 1] ∧ ∀β ≤ α(f (β) = F(

• ξ∈β

f (ξ)))

◮ Define by Σc 1-Separation, the set

z = {x ∈ a : ∃α∃f (H[F, f , α] ∧ x ∈ f (α))}.

◮ Σ-Reflection and monotonicity yield “z = F γ(∅)” for some ordinal γ. ◮ z is a set and it is the least fixed point.

Σc

1-separation implies LFP

Does the vice versa hold?

Bounded proper injections

BPI ∀x(F(x) ∈ a) → ∃x, y, (x = y ∧ F(x) = F(y))

Subset Bounded Separation

SBS For every ∆c

0 formula A and sets a and b,

∃z(z = {x ∈ a : ∃y ⊆ b(A[x, y])})

SBS BPI LFP

SBS implies BPI

◮ Given F and a as in BPI define by SBS the set

X = {x ∈ a : ∃z ⊆ a(F(z) = x)}.

◮ Suppose by contradiction that

∀y, z ⊆ a(F(y) = F(z)).

◮ Define h : X → V such that

h(x) := the unique z ⊆ a(F(z) = x).

◮ We can prove that ∀z(z ⊆ a ⇐

⇒ z ∈ h[X]).

◮ We can conclude with the usual Cantor’s argument.

SBS implies LFP

◮ Given F and a as in LFP, define

ClF[y] ⇐ ⇒ F(y) ⊆ y.

◮ By SBS we can define

z = {x ∈ a : ∀y ⊆ a(ClF[y] = ⇒ x ∈ y)}.

◮ We can prove that F(z) = z. ◮ Since every fixed point is closed under F, we have leastness.

Maximal Iteration

H[F, f , α] := Fun[f , α + 1] ∧ ∀β ≤ α(f (β) = F(

• ξ∈β

f (ξ))) MI ∀x(F(x) ⊆ a) → ∃α, f (H[F, f , α] ∧ f (α) ⊆

• ξ<α

f (ξ))

MI BPI LFP

Fixed point principles in KPc Σc

1-Sep

MI SBS BPI LFP FP

Working with the Axiom of Constructibility (V=L)

In KPc + (V=L) the following implications hold:

◮ BPI implies Σc 1-Separation. ◮ FP implies SBS.

We can conclude that all our principles are not provable in KPc + (V=L) since all of them are equivalent to Σ1-Separation in this setting.

Fixed point principles in KPc + (V=L) Σc

1-Sep

MI SBS BPI LFP FP (V=L) (V=L)

Fixed point principles in KPc + (V=L) Σc

1-Sep

MI SBS BPI LFP FP (V=L) (V=L) Thank you!