M OTIVATIONS E NCODING SETS O PEN QUESTIONS Ramsey’s theorem under a computable perspective Ludovic PATEY 1 / 69 June 12, 2017
M OTIVATIONS E NCODING SETS O PEN QUESTIONS Motivations 2 / 69
M OTIVATIONS E NCODING SETS O PEN QUESTIONS R EVERSE MATHEMATICS Foundational program that seeks to determine the optimal axioms of ordinary mathematics. 3 / 69
M OTIVATIONS E NCODING SETS O PEN QUESTIONS R EVERSE MATHEMATICS Foundational program that seeks to determine the optimal axioms of ordinary mathematics. RCA 0 ⊢ A ↔ T in a very weak theory RCA 0 capturing computable mathematics 3 / 69
M OTIVATIONS E NCODING SETS O PEN QUESTIONS RCA 0 Robinson arithmetics m + 1 � = 0 m + 0 = m m + 1 = n + 1 → m = n m + ( n + 1 ) = ( m + n ) + 1 ¬ ( m < 0 ) m × 0 = 0 m < n + 1 ↔ ( m < n ∨ m = n ) m × ( n + 1 ) = ( m × n ) + m Σ 0 ∆ 0 1 induction scheme 1 comprehension scheme ϕ ( 0 ) ∧ ∀ n ( ϕ ( n ) ⇒ ϕ ( n + 1 )) ∀ n ( ϕ ( n ) ⇔ ψ ( n )) ⇒ ∀ n ϕ ( n ) ⇒ ∃ X ∀ n ( n ∈ X ⇔ ϕ ( n )) where ϕ ( n ) is Σ 0 where ϕ ( n ) is Σ 0 1 with free X , and ψ 1 is Π 0 1 . 4 / 69
M OTIVATIONS E NCODING SETS O PEN QUESTIONS Σ 0 n induction scheme ϕ ( 0 ) ∧ ∀ n ( ϕ ( n ) ⇒ ϕ ( n + 1 )) ⇒ ∀ n ϕ ( n ) where ϕ ( n ) is Σ 0 n bounded ∆ 0 n comprehension scheme ∀ t ∀ n ( ϕ ( n ) ⇔ ψ ( n )) ⇒ ∃ X ∀ n ( n ∈ X ⇔ ( x < t ∧ ϕ ( n ))) where ϕ ( n ) is Σ 0 n with free X , and ψ is Π 0 n . 5 / 69
M OTIVATIONS E NCODING SETS O PEN QUESTIONS R EVERSE MATHEMATICS Π 1 1 CA Mathematics are computationally ATR very structured ACA Almost every theorem is WKL empirically equivalent to one among five big subsystems. RCA 0 6 / 69
M OTIVATIONS E NCODING SETS O PEN QUESTIONS H ILBERT ’ S PROGRAM Justification of infinitary methods to prove finitistic mathematics Finitistic reductionnism: T ⊢ ϕ ⇒ PRA ⊢ ϕ where ϕ is a Π 0 1 formula “At least 85% of mathematics are reducible to finitistic methods” (Stephen Simpson 7 / 69
M OTIVATIONS E NCODING SETS O PEN QUESTIONS H ILBERT ’ S PROGRAM Justification of infinitary methods to prove finitistic mathematics Finitistic reductionnism: T ⊢ ϕ ⇒ PRA ⊢ ϕ where ϕ is a Π 0 1 formula “At least 85% of mathematics are reducible to finitistic methods” (Stephen Simpson) 7 / 69
M OTIVATIONS E NCODING SETS O PEN QUESTIONS R EVERSE MATHEMATICS Mathematics are Π 1 1 CA computationally ATR very structured ACA Almost every theorem is empirically equivalent to one WKL among five big subsystems. RCA 0 8 / 69
M OTIVATIONS E NCODING SETS O PEN QUESTIONS R EVERSE MATHEMATICS Mathematics are Π 1 1 CA computationally ATR very structured ACA Almost every theorem is RT 2 empirically equivalent to one WKL 2 among five big subsystems. RCA 0 Except for Ramsey’s theory... 8 / 69
M OTIVATIONS E NCODING SETS O PEN QUESTIONS 9 / 69
M OTIVATIONS E NCODING SETS O PEN QUESTIONS What is Ramsey’s theorem? 10 / 69
M OTIVATIONS E NCODING SETS O PEN QUESTIONS R AMSEY ’ S THEOREM [ X ] n is the set of unordered n -tuples of elements of X A k -coloring of [ X ] n is a map f : [ X ] n → k A set H ⊆ X is homogeneous for f if | f ([ X ] n ) | = 1. RT n Every k -coloring of [ N ] n admits k an infinite homogeneous set. 11 / 69
M OTIVATIONS E NCODING SETS O PEN QUESTIONS P IGEONHOLE PRINCIPLE RT 1 Every k -partition of N admits k an infinite part. 12 / 69
M OTIVATIONS E NCODING SETS O PEN QUESTIONS R AMSEY ’ S THEOREM FOR PAIRS RT 2 Every k -coloring of the infinite clique admits k an infinite monochromatic subclique. 13 / 69
M OTIVATIONS E NCODING SETS O PEN QUESTIONS Reverse mathematics from a computational viewpoint. 14 / 69
M OTIVATIONS E NCODING SETS O PEN QUESTIONS S TANDARD MODELS OF RCA 0 An ω -structure is a structure M = { ω, S , <, + , ·} where (i) ω is the set of standard natural numbers (ii) < is the natural order (iii) + and · are the standard operations over natural numbers (iv) S ⊆ P ( ω ) 15 / 69
M OTIVATIONS E NCODING SETS O PEN QUESTIONS S TANDARD MODELS OF RCA 0 An ω -structure is a structure M = { ω, S , <, + , ·} where (i) ω is the set of standard natural numbers (ii) < is the natural order (iii) + and · are the standard operations over natural numbers (iv) S ⊆ P ( ω ) An ω -structure is fully specified by its second-order part S . 15 / 69
M OTIVATIONS E NCODING SETS O PEN QUESTIONS Turing ideal M � ( ∀ X ∈ M )( ∀ Y ≤ T X )[ Y ∈ M ] � ( ∀ X , Y ∈ M )[ X ⊕ Y ∈ M ] Examples � { X : X is computable } � { X : X ≤ T A ∧ X ≤ T B } for some sets A and B 16 / 69
M OTIVATIONS E NCODING SETS O PEN QUESTIONS Let M = { ω, S , <, + , ·} be an ω -structure M | = RCA 0 ≡ S is a Turing ideal 17 / 69
M OTIVATIONS E NCODING SETS O PEN QUESTIONS Many theorems can be seen as problems. Intermediate value theorem For every continuous function f over an a interval [ a , b ] such that f ( a ) · f ( b ) < 0, there b is a real x ∈ [ a , b ] such that f ( x ) = 0. K¨ onig’s lemma Every infinite, finitely branching tree admits an infinite path. 18 / 69
M OTIVATIONS E NCODING SETS O PEN QUESTIONS Let M be a Turing ideal and P , Q be problems. Computable entailment Satisfaction M | = P P | = c Q if every P-instance in M if every Turing ideal has a solution in M . satisfying P satisfies Q. 19 / 69
M OTIVATIONS E NCODING SETS O PEN QUESTIONS RT 2 2 �| = c ACA (Seetapun and Slaman, 1995) 2 with ∅ ′ �∈ M = RT 2 � Build M | = ACA then ∅ ′ ∈ M � If M | ∅ ′ = { e : ( ∃ s )Φ e ( e ) halts after s steps } 20 / 69
M OTIVATIONS E NCODING SETS O PEN QUESTIONS 2 with ∅ ′ �∈ M . = RT 2 Build M | Thm (Seetapun and Slaman) Suppose A �≤ T Z . Then every Z -computable f : [ ω ] 2 → 2 has an infinite f -homogeneous set H such that A �≤ T Z ⊕ H . Start with M 0 = { Z : Z is computable } . In particular ∅ ′ �∈ M 0 . Given a Turing ideal M n = { Z : Z ≤ T U } where ∅ ′ �≤ T U , 21 / 69
M OTIVATIONS E NCODING SETS O PEN QUESTIONS 2 with ∅ ′ �∈ M . = RT 2 Build M | Thm (Seetapun and Slaman) Suppose A �≤ T Z . Then every Z -computable f : [ ω ] 2 → 2 has an infinite f -homogeneous set H such that A �≤ T Z ⊕ H . Start with M 0 = { Z : Z is computable } . In particular ∅ ′ �∈ M 0 . Given a Turing ideal M n = { Z : Z ≤ T U } where ∅ ′ �≤ T U , 1. pick some f : [ ω ] 2 → 2 in M n 21 / 69
M OTIVATIONS E NCODING SETS O PEN QUESTIONS 2 with ∅ ′ �∈ M . = RT 2 Build M | Thm (Seetapun and Slaman) Suppose A �≤ T Z . Then every Z -computable f : [ ω ] 2 → 2 has an infinite f -homogeneous set H such that A �≤ T Z ⊕ H . Start with M 0 = { Z : Z is computable } . In particular ∅ ′ �∈ M 0 . Given a Turing ideal M n = { Z : Z ≤ T U } where ∅ ′ �≤ T U , 1. pick some f : [ ω ] 2 → 2 in M n 2. let H be f -homogeneous set such that ∅ ′ �≤ T U ⊕ H 21 / 69
M OTIVATIONS E NCODING SETS O PEN QUESTIONS 2 with ∅ ′ �∈ M . = RT 2 Build M | Thm (Seetapun and Slaman) Suppose A �≤ T Z . Then every Z -computable f : [ ω ] 2 → 2 has an infinite f -homogeneous set H such that A �≤ T Z ⊕ H . Start with M 0 = { Z : Z is computable } . In particular ∅ ′ �∈ M 0 . Given a Turing ideal M n = { Z : Z ≤ T U } where ∅ ′ �≤ T U , 1. pick some f : [ ω ] 2 → 2 in M n 2. let H be f -homogeneous set such that ∅ ′ �≤ T U ⊕ H 3. let M n + 1 = { Z : Z ≤ T U ⊕ H } 21 / 69
M OTIVATIONS E NCODING SETS O PEN QUESTIONS Non-implications over RCA 0 often involve purely computability-theoretic arguments. 22 / 69
M OTIVATIONS E NCODING SETS O PEN QUESTIONS For m , n ≥ 3, RCA 0 � RT m 2 ↔ RT n 2 (Jockusch) Theorem (Jockusch) For every n ≥ 3 , there is a computable coloring f : [ ω ] n → 2 such that every infinite f-homogeneous set computes ∅ ( n − 2 ) . Let f ( x , y , z ) = 1 if the approximation of ∅ ′ ↾ x at stage y and at stage z coincide. 23 / 69
M OTIVATIONS E NCODING SETS O PEN QUESTIONS Σ 0 Π 0 3 3 RT 3 k Fix some n ≥ 2. ∆ 0 3 Thm (Jockusch) Σ 0 Π 0 2 2 Every computable instance of RT n k RT 2 has a Π 0 n solution. k ∆ 0 2 Thm (Jockusch) Σ 0 Π 0 1 1 There is a computable instance of RT n k with no Σ 0 n solution. ∆ 0 1 RT 1 k 24 / 69
M OTIVATIONS E NCODING SETS O PEN QUESTIONS For k , ℓ ≥ 2, RCA 0 � RT n k ↔ RT n ℓ Given a coloring f : [ ω ] n → { red , green , blue } � Define g : [ ω ] n → { red , grue } by merging green and blue 2 on g to obtain H such that g [ H ] n = { red } or � Apply RT n g [ H ] n = { grue } 2 on f [ H ] n → { green , blue } to obtain � In the latter case, apply RT n G such that f [ G ] n = { green } or f [ G ] n = { blue } 25 / 69
M OTIVATIONS E NCODING SETS O PEN QUESTIONS We use more than once the premise for RCA 0 � RT n 2 → RT n + 1 2 RCA 0 � RT n k → RT n k + 1 Can we do it in one step? 26 / 69
M OTIVATIONS E NCODING SETS O PEN QUESTIONS C OMPUTABLE REDUCTION P solver Computable Computable Q solver transformation transformation P ≤ c Q Every P-instance I computes a Q-instance J such that for every solution X to J , X ⊕ I computes a solution to I . 27 / 69
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