The strength of Ramsey theorem for coloring large sets Konrad - PowerPoint PPT Presentation

Basic notions Ramsey principle for coloring n –tuples Ramsey principle for coloring α –large sets Open problems The strength of Ramsey theorem for coloring ω –large sets Konrad Zdanowski (join work with Lorenzo Carlucci, La Sapienza) University of Cardinal Stefan Wyszy´ nski, Warsaw Kotlarski–Ratajczyk conference 2012 Konrad Zdanowski (join work with Lorenzo Carlucci, La Sapienza) Ramsey theorem for ω –large sets

Basic notions Ramsey principle for coloring n –tuples Ramsey principle for coloring α –large sets Open problems Outline Basic notions 1 Ramsey principle for coloring n –tuples 2 Ramsey principle for coloring α –large sets 3 Open problems 4 Konrad Zdanowski (join work with Lorenzo Carlucci, La Sapienza) Ramsey theorem for ω –large sets

Basic notions Second order arithmetics Ramsey principle for coloring n –tuples Ordinals and large sets Ramsey principle for coloring α –large sets Recursion theory Open problems Outline Basic notions 1 Second order arithmetics Ordinals and large sets Recursion theory Ramsey principle for coloring n –tuples 2 RT ( n ) ∀ n RT ( n ) Ramsey principle for coloring α –large sets 3 Farmaki theorem Coloring ω –large sets Arithmetics with truth predicates Open problems 4 Konrad Zdanowski (join work with Lorenzo Carlucci, La Sapienza) Ramsey theorem for ω –large sets

Basic notions Second order arithmetics Ramsey principle for coloring n –tuples Ordinals and large sets Ramsey principle for coloring α –large sets Recursion theory Open problems Outline Basic notions 1 Second order arithmetics Ordinals and large sets Recursion theory Ramsey principle for coloring n –tuples 2 Ramsey principle for coloring α –large sets 3 Open problems 4 Konrad Zdanowski (join work with Lorenzo Carlucci, La Sapienza) Ramsey theorem for ω –large sets

Basic notions Second order arithmetics Ramsey principle for coloring n –tuples Ordinals and large sets Ramsey principle for coloring α –large sets Recursion theory Open problems We consider theories of second order arithmetic. First order formulas in the usual hierarchy Σ 0 n , Π 0 n may contain second order parameters. Basic axioms: n + 1 � = 0, n + 1 = m + 1 → n = m , m + 0 = m , m + ( n + 1 ) = ( m + n ) + 1, m · 0 = 0, m · ( n + 1 ) = ( m · n ) + m , ¬ m < 0, m < n + 1 → ( m < n ∨ m = n ) . Konrad Zdanowski (join work with Lorenzo Carlucci, La Sapienza) Ramsey theorem for ω –large sets

Basic notions Second order arithmetics Ramsey principle for coloring n –tuples Ordinals and large sets Ramsey principle for coloring α –large sets Recursion theory Open problems For a set of formulas F , by F comprehension scheme we define the set of formulas ∃ X ∀ n ( n ∈ X ↔ ϕ ( n )) , for ϕ ∈ F . By ∆ 0 1 comprehension scheme we define ∀ n ( ϕ ( n ) ↔ ψ ( n )) → ∃ X ∀ n ( n ∈ X ↔ ϕ ( n )) , for ϕ ∈ Σ 0 1 , ψ ∈ Π 0 1 . Konrad Zdanowski (join work with Lorenzo Carlucci, La Sapienza) Ramsey theorem for ω –large sets

Basic notions Second order arithmetics Ramsey principle for coloring n –tuples Ordinals and large sets Ramsey principle for coloring α –large sets Recursion theory Open problems Definition 1 By RCA 0 we denote arithmetic containing basic axioms, Σ 0 1 induction and ∆ 0 1 comprehension. Definition 2 By ACA 0 we denote RCA 0 extended by first order comprehension. Definition 3 By ATR 0 we denote RCA 0 extended by definitions of sets by transfinite recursion. Konrad Zdanowski (join work with Lorenzo Carlucci, La Sapienza) Ramsey theorem for ω –large sets

Basic notions Second order arithmetics Ramsey principle for coloring n –tuples Ordinals and large sets Ramsey principle for coloring α –large sets Recursion theory Open problems Outline Basic notions 1 Second order arithmetics Ordinals and large sets Recursion theory Ramsey principle for coloring n –tuples 2 Ramsey principle for coloring α –large sets 3 Open problems 4 Konrad Zdanowski (join work with Lorenzo Carlucci, La Sapienza) Ramsey theorem for ω –large sets

Basic notions Second order arithmetics Ramsey principle for coloring n –tuples Ordinals and large sets Ramsey principle for coloring α –large sets Recursion theory Open problems Definition 4 For each ordinal α < ω 1 let us fixed a sequence { α } ( x ) , for x ∈ ω such that { β + 1 } ( x ) = β , { α } ( x ) ≤ { α } ( y ) , for x ≤ y, lim x ∈ ω { alpha } ( x ) = α . Definition 5 Let λ be limit and let λ 0 = { λ } ( a ) , λ i + 1 = { λ i } ( a ) . By { λ } ∗ ( a ) we denote the first successor ordinal in the sequence λ 0 , λ 1 , . . . Konrad Zdanowski (join work with Lorenzo Carlucci, La Sapienza) Ramsey theorem for ω –large sets

Basic notions Second order arithmetics Ramsey principle for coloring n –tuples Ordinals and large sets Ramsey principle for coloring α –large sets Recursion theory Open problems Example { ω } ( a ) = a , { α + β } ( a ) = α + { β } ( a ) , { ω α + 1 } ( a ) = ω α a , { ω λ } ( a ) = ω { λ } ( a ) . Konrad Zdanowski (join work with Lorenzo Carlucci, La Sapienza) Ramsey theorem for ω –large sets

Basic notions Second order arithmetics Ramsey principle for coloring n –tuples Ordinals and large sets Ramsey principle for coloring α –large sets Recursion theory Open problems Let h be a, possibly finite, function from N to N . We define the Hardy hierarchy of functions: h 0 ( x ) = x , h α + 1 ( x ) = h α ( h ( x )) , h λ ( x ) = h { λ } ( x ) ( x ) = h { λ } ∗ ( x ) − 1 ( h ( x )) . Konrad Zdanowski (join work with Lorenzo Carlucci, La Sapienza) Ramsey theorem for ω –large sets

Basic notions Second order arithmetics Ramsey principle for coloring n –tuples Ordinals and large sets Ramsey principle for coloring α –large sets Recursion theory Open problems Example Let h ( x ) = x + 1. h n ( x ) = h n ( x ) , h ω ( x ) = h x ( x ) = 2 x , h ω 2 ( x ) = h ω + x ( x ) = h ω ( 2 x ) = 2 2 x , h ω 2 ( x ) = h ω x ( x ) = 2 x x , h ω ω ( x ) is ackermanian. Konrad Zdanowski (join work with Lorenzo Carlucci, La Sapienza) Ramsey theorem for ω –large sets

Basic notions Second order arithmetics Ramsey principle for coloring n –tuples Ordinals and large sets Ramsey principle for coloring α –large sets Recursion theory Open problems Let X = { x 0 , . . . , x k } . Let h be a successor in the sense of X : h ( x i ) = x i + 1 . Thus, h ( max X ) is undefined. Definition 6 We say that X is α –large if h α ( min X ) is defined. We say that X is exactly α –large if h α ( min X ) = max ( X ) . Definition 7 For a given X, by [ X ] ! α we denote its exactly α –large subsets. Konrad Zdanowski (join work with Lorenzo Carlucci, La Sapienza) Ramsey theorem for ω –large sets

Basic notions Second order arithmetics Ramsey principle for coloring n –tuples Ordinals and large sets Ramsey principle for coloring α –large sets Recursion theory Open problems Example Let X be a finite set. X is 0–large if h 0 ( min X ) ↓ , X is nonempty, X is n –large if h n ( min X ) ↓ , X has n + 1 elements, X is ω –large if h ω ( min X ) = h x ( min X ) ↓ , X has min ( X ) + 1 elements. Konrad Zdanowski (join work with Lorenzo Carlucci, La Sapienza) Ramsey theorem for ω –large sets

Basic notions Second order arithmetics Ramsey principle for coloring n –tuples Ordinals and large sets Ramsey principle for coloring α –large sets Recursion theory Open problems Outline Basic notions 1 Second order arithmetics Ordinals and large sets Recursion theory Ramsey principle for coloring n –tuples 2 Ramsey principle for coloring α –large sets 3 Open problems 4 Konrad Zdanowski (join work with Lorenzo Carlucci, La Sapienza) Ramsey theorem for ω –large sets

Basic notions Second order arithmetics Ramsey principle for coloring n –tuples Ordinals and large sets Ramsey principle for coloring α –large sets Recursion theory Open problems Definition 8 For a Turing machine e, by { e } ( x ) ↓ we denote the fact that e stops on the input x. By { e } z ( x ) ↓ we denote the fact that e stops on the input x with a computation less than z. Konrad Zdanowski (join work with Lorenzo Carlucci, La Sapienza) Ramsey theorem for ω –large sets

Basic notions Second order arithmetics Ramsey principle for coloring n –tuples Ordinals and large sets Ramsey principle for coloring α –large sets Recursion theory Open problems Definition 9 The jump of the set X is defined as X ′ = { e : { e } X ( 0 ) ↓} . The ( n + 1 ) -th jump of X is defined as X ( n + 1 ) = ( X ( n ) ) ′ . The ω –jump of X is defined as X ω = { ( i , j ): j ∈ X ( i ) } . The above notions can be easily generalized to higher ordinals α ’s provided (recursive) fundamental sequences up to α are fixed. Konrad Zdanowski (join work with Lorenzo Carlucci, La Sapienza) Ramsey theorem for ω –large sets

Basic notions Second order arithmetics Ramsey principle for coloring n –tuples Ordinals and large sets Ramsey principle for coloring α –large sets Recursion theory Open problems Theorem 10 ACA 0 can be characterized as RCA 0 + ∀ X X ′ exists.. Definition 11 0 is RCA 0 + ∀ X X ω exists.. ACA + Konrad Zdanowski (join work with Lorenzo Carlucci, La Sapienza) Ramsey theorem for ω –large sets