Ramsey Theorem for pairs as a classical principle in Intuitionistic - PowerPoint PPT Presentation

Introduction RT2 2 implies Σ0 3-LLPO Σ0 3-LLPO implies RT2 2 Conclusions Bibliography Ramsey Theorem for pairs as a classical principle in Intuitionistic Arithmetic. Silvia Steila, joint work with Stefano Berardi Universit` a degli studi di Torino British Logic Colloquium, Leeds September 6th, 2013 Silvia Steila, joint work with Stefano Berardi Ramsey Theorem as a classical principle in HA 1 / 18

Introduction RT2 2 implies Σ0 3-LLPO Σ0 3-LLPO implies RT2 2 Conclusions Bibliography Σ 0 n - LLPO Lesser Limited Principle of Omniscience. For any parameter a ∀ x , x ′ ( P ( x , a ) ∨ Q ( x ′ , a )) = ⇒ ∀ xP ( x , a ) ∨ ∀ xQ ( x , a ) . ( P , Q ∈ Σ 0 n − 1 ) Pigeonhole Principle for Π 0 n For any parameter a ∀ x ∃ z [ z ≥ x ∧ ( P ( z , a ) ∨ Q ( z , a ))] = ⇒ ∀ x ∃ z [ z ≥ x ∧ P ( z , a )] ∨ ∀ x ∃ z [ z ≥ x ∧ Q ( z , a )] . ( P , Q ∈ Π 0 n ) EM n Excluded Middle for Σ 0 n formulas. For any parameter a ∃ x P ( x , a ) ∨ ¬∃ x P ( x , a ) . ( P ∈ Π 0 n − 1 ) Silvia Steila, joint work with Stefano Berardi Ramsey Theorem as a classical principle in HA 2 / 18

Introduction RT2 2 implies Σ0 3-LLPO Σ0 3-LLPO implies RT2 2 Conclusions Bibliography Classical Logic . . . EM 2 Π 0 2 -EM Σ 0 2 -MARKOV Σ 0 2 -LLPO ∆ 0 2 -EM Thesis: RT 2 2 is equivalent EM 1 to Σ 0 3 -LLPO in HA. Π 0 1 -EM Σ 0 1 -MARKOV Σ 0 1 -LLPO ∆ 0 1 -EM HA EM 0 The purpose of this work is to study, from the viewpoint of first order arithmetic (no set variables, the only sets are the arithmetical sets), Ramsey Theorem for pairs for recursive assignement of two colors in order to find some principle of classical logic equivalent to it in HA. Silvia Steila, joint work with Stefano Berardi Ramsey Theorem as a classical principle in HA 3 / 18

Introduction RT2 2 implies Σ0 3-LLPO Σ0 3-LLPO implies RT2 2 Conclusions Bibliography If X is a set, [ X ] 2 = { Y ⊆ X | | Y | = 2 } . We can think of [ X ] 2 as the complete graph on X . We only consider arithmetically definable sets. RT 2 2 ( Σ 0 n ). Ramsey Theorem for graphs and Σ 0 n 2-colorings For any coloring c a : [ ω ] 2 → 2 with a parameter a, there exists an infinite subset X of ω homogeneous for the given coloring, i.e. [ X ] 2 is painted with only one color. ( c a ∈ Σ 0 n ) . Silvia Steila, joint work with Stefano Berardi Ramsey Theorem as a classical principle in HA 4 / 18

Introduction RT2 2 implies Σ0 3-LLPO Σ0 3-LLPO implies RT2 2 Conclusions Bibliography In this work we formalize Ramsey Theorem for two colors, for pairs and for recursive colorings by the following schema: {∀ a ( B ( ., c a ) infinite homogeneous black ∨ W ( ., c a ) infinite homogeneous white ) | for some B , W arithmetical predicates } . Here c = { c a | a ∈ ω } denotes any recursive family of recursive assignment of two colors, black and white. We call this a disjunctive schema and we prove it if we prove some instance of it. Silvia Steila, joint work with Stefano Berardi Ramsey Theorem as a classical principle in HA 5 / 18

Introduction RT2 2 implies Σ0 3-LLPO Σ0 3-LLPO implies RT2 2 Conclusions Bibliography First part: RT 2 2 implies Σ 0 3 -LLPO We will prove that in the first order intuitionistic arithmetic, RT 2 2 (Σ 0 0 ) is equivalent to the classical principle Σ 0 3 -LLPO. By definition of disjunctive schema, Ramsey Theorem for graphs and a recursive 2-coloring implies Σ 0 3 -LLPO if the following holds: for each P in Σ 0 3 -LLPO, there exist a finite number of recursive family of recursive colorings c a , 0 , . . . , c a , j − 1 such that, fixed any W i ( ., c a , i ) and B i ( ., c a , i ), if we assume {∀ a ( W i ( ., c a , i ) is inf. and hom. ∨ B i ( ., c a , i ) is inf. and hom. ) | i ∈ j } then we deduce P . Silvia Steila, joint work with Stefano Berardi Ramsey Theorem as a classical principle in HA 6 / 18

Introduction RT2 2 implies Σ0 3-LLPO Σ0 3-LLPO implies RT2 2 Conclusions Bibliography Proof sketch of RT 2 2 implies Σ 0 3 -LLPO Lemma RT 2 2 (Σ 0 0 ) implies EM 1 ; 1 EM 1 implies that, for any family F = { s ( n , · ) | n ∈ ω } of recursive 2 monotone and bounded above sequences enumerated by a binary primitive recursive function s : ω × ω → ω , each sequence in F is stationary; EM 1 implies that, for any family G = { t ( n , · ) | n ∈ ω } of recursive 3 sequences enumerated by a binary primitive recursive function t : ω × ω → ω for which there are at most k values of x such that t ( n , x ) � = t ( n , x + 1) , each sequence in G is stationary. Silvia Steila, joint work with Stefano Berardi Ramsey Theorem as a classical principle in HA 7 / 18

Introduction RT2 2 implies Σ0 3-LLPO Σ0 3-LLPO implies RT2 2 Conclusions Bibliography Proof sketch of RT 2 2 implies Σ 0 3 -LLPO Let a be a parameter. We assume the hypothesis of Σ 0 3 -LLPO: ∀ x , x ′ ( H 0 ( x , a ) ∨ H 1 ( x ′ , a )) , where H 0 ( x , a ) := ∃ y ∀ z P 0 ( x , y , z , a ) H 1 ( x , a ) := ∃ y ∀ z P 1 ( x , y , z , a ) for some P 0 , P 1 primitive recursive predicates. Silvia Steila, joint work with Stefano Berardi Ramsey Theorem as a classical principle in HA 8 / 18

Introduction RT2 2 implies Σ0 3-LLPO Σ0 3-LLPO implies RT2 2 Conclusions Bibliography Proof sketch of RT 2 2 implies Σ 0 3 -LLPO Our thesis is ∀ x H 0 ( x , a ) ∨ ∀ x H 1 ( x , a ) we define a recursive 2-coloring such that: if there are infinitely many white edges from x , then for all y ≤ x H 0 ( y , a ) holds; if there are infinitely many black edges from x , then for all y ≤ x H 1 ( y , a ) holds. Applying RT 2 2 (Σ 0 0 ), there exists an infinite homogeneous set X . We prove that if there is some infinite set X homogeneous in color c , then H c ( x , a ) holds for infinitely many x . We obtain ∀ x H c ( x , a ) . Silvia Steila, joint work with Stefano Berardi Ramsey Theorem as a classical principle in HA 9 / 18

Introduction RT2 2 implies Σ0 3-LLPO Σ0 3-LLPO implies RT2 2 Conclusions Bibliography Second part: Σ 0 3 -LLPO implies RT 2 2 Now we modify Jockusch proof of Ramsey Theorem in order to obtain a proof in HA of Σ 0 ⇒ RT 2 3 -LLPO = 2 . Given a coloring c : [ ω ] 2 → 2 we say that X ⊆ ω defines a 1-coloring if for all x ∈ X , any two edges from x to some y , z ∈ X have the same color. If X is infinite and defines a 1-coloring, thanks to the Pigeonhole Principle we may define an infinite arithmetical subset Y of X whose points all have the same color. Y is homogeneous for c . So we need to find an infinite set that defines a 1-coloring. Silvia Steila, joint work with Stefano Berardi Ramsey Theorem as a classical principle in HA 10 / 18

Introduction RT2 2 implies Σ0 3-LLPO Σ0 3-LLPO implies RT2 2 Conclusions Bibliography Proof Sketch of Σ 0 3 -LLPO implies RT 2 2 A tree T included in a graph ω defines a 1-coloring w.r.t. T if for all x ∈ T for any two proper descendants y , z of x in T , the edges from x to y , z have the same color. Assume there exists some infinite binary tree T defining a 1-coloring w.r.t. T . Then T has an infinite branch B by K¨ onig’s Lemma. B defines an infinite 1-coloring and so proves RT 2 2 . Therefore a sufficient condition for RT 2 2 is the existence of an infinite binary tree defining a 1-coloring. Silvia Steila, joint work with Stefano Berardi Ramsey Theorem as a classical principle in HA 11 / 18

Introduction RT2 2 implies Σ0 3-LLPO Σ0 3-LLPO implies RT2 2 Conclusions Bibliography Proof Sketch of Σ 0 3 -LLPO implies RT 2 2 Jockusch proof: PP(Π 0 K¨ onig 2 ) tree homogeneous set infinite branch Our work (Σ 0 3 -LLPO): infinite branch (Π 0 2 ) (it is unique) M 2 E tree (Π 0 1 ) P P ( Π 0 ) color of homogeneous set 1 Silvia Steila, joint work with Stefano Berardi Ramsey Theorem as a classical principle in HA 12 / 18

Introduction RT2 2 implies Σ0 3-LLPO Σ0 3-LLPO implies RT2 2 Conclusions Bibliography Proof Sketch of Σ 0 3 -LLPO implies RT 2 2 Given any set T we may equip it by the following ancestor/descendant relation 0 ≺ T 1, x ≺ T y iff x ∈ T , y ∈ ω , x < y and ∀ z ( z ≺ T x ( c ( { z , x } = c { z , y } ))) . Definition (Inductive definition of the set T in HA) Define T n by induction on n . If n = 0 then T 0 = x 0 := 0. For n + 1, if Chosen( x n +1 , T n ), then T n +1 = T n ∪ { x n +1 } . � T = T n . n ∈ ω Chosen is some suitable arithmetical predicate. Silvia Steila, joint work with Stefano Berardi Ramsey Theorem as a classical principle in HA 13 / 18

Introduction RT2 2 implies Σ0 3-LLPO Σ0 3-LLPO implies RT2 2 Conclusions Bibliography Proof Sketch of Σ 0 3 -LLPO implies RT 2 2 We proved (using a part of Σ 0 3 -LLPO) that T is a Π 0 1 binary tree, T has a unique infinite branch r such that if T has infinitely many edges with color c , then r has infinitely many edges with color c . Moreover using Σ 0 Pigeonhole Principle for Π 0 3 -LLPO = ⇒ 1 predicates , we obtain that T has infinitely many edges of color c , so r has infinite many edges of color c ; their smaller nodes define a monochromatic set for the original graph. Silvia Steila, joint work with Stefano Berardi Ramsey Theorem as a classical principle in HA 14 / 18