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 of Σ function symbols, but this kind of formalization is rather clumsy. So... we introduce a second order extension KP c of KP.

KP c ◮ Let L c 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 L c formula is elementary if it contains no class quantifiers. ◮ ∆ c n , Σ c n and Π c n are defined as usual, but permitting subformulas of the form “ a ∈ U ”.

KP c The theory KP c is formulated in L c 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 Σ c 1 formula A and every Π c 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 )

Σ c 1 -separation Σ c 1 -separation For every Σ c 1 formula A in which x is not free and any set a , ∃ x ( x = { y ∈ a : A [ y ] } ) .

Σ c 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 ( f ( ξ ))) ξ ∈ β ◮ Define by Σ c 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.

Σ c 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 Cl F [ y ] ⇐ ⇒ F ( y ) ⊆ y . ◮ By SBS we can define z = { x ∈ a : ∀ y ⊆ a (Cl F [ 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 ( f ( ξ ))) ξ ∈ β MI � ∀ x ( F ( x ) ⊆ a ) → ∃ α, f ( H [ F , f , α ] ∧ f ( α ) ⊆ f ( ξ )) ξ<α MI BPI LFP

Fixed point principles in KP c Σ c 1 -Sep MI SBS BPI LFP FP

Working with the Axiom of Constructibility (V=L) In KP c + (V=L) the following implications hold: ◮ BPI implies Σ c 1 -Separation. ◮ FP implies SBS. We can conclude that all our principles are not provable in KP c + (V=L) since all of them are equivalent to Σ 1 -Separation in this setting.

Fixed point principles in KP c + (V=L) Σ c 1 -Sep (V=L) MI SBS (V=L) BPI LFP FP

Fixed point principles in KP c + (V=L) Σ c 1 -Sep (V=L) MI SBS (V=L) BPI LFP FP Thank you!