Ramsey’s theorem for pairs and provable recursive functions Alexander Kreuzer (joint work with Ulrich Kohlenbach) TU Darmstadt RaTLoCC 2009 RT 2 and provable recursive functions A. Kreuzer (TU Darmstadt) RaTLoCC 2009 1 / 20
Outline Introduction 1 Reverse Mathematics Strength of Ramsey’s theorem Theorem 2 Elimination of Skolem functions for monotone formulas Comparison of proof-techniques Recent work 3 RT 2 and provable recursive functions A. Kreuzer (TU Darmstadt) RaTLoCC 2009 2 / 20
Ramsey’s Theorem Let [ N ] k be the set of unordered k -tuples of natural numbers. A n -coloring of [ N ] k is a map of [ N ] k into n . Definition ( RT k n ) For every n -coloring of [ N ] k exists an infinite homogeneous set H ⊆ N (i.e. the coloring is constant on [ H ] k ). Lemma for all n, n ′ ∈ N \ { 1 } . RT k n ↔ RT k n ′ RT k < ∞ is defined as ∀ n RT 2 n . RT 2 and provable recursive functions A. Kreuzer (TU Darmstadt) RaTLoCC 2009 3 / 20
Reverse Mathematics Seeks to find which axioms over seconder order arithmetic are needed to prove theorems. Usually consider the big five subsystems of second order arithmetic: RCA 0 Σ 0 1 - IA + recursive comprehension 1st order part: Σ 0 1 -induction 2nd order part: recursive sets WKL 0 RCA 0 + Weak K¨ onigs Lemma 1st order part: like RCA 0 2nd order part: low sets ACA 0 RCA 0 + arithmetic comprehension 1st order part: arithmetic induction 2nd order part: arithmetic sets (includes WKL ) ATR 0 ACA 0 + arithmetical transfinite recursion Π 1 1 - CA 0 ACA 0 + Π 1 1 -comprehension Weaker systems: RCA ∗ 0 QF - IA + recursive comprehension + exponential function WKL ∗ 0 RCA ∗ 0 + Weak K¨ onigs Lemma RT 2 and provable recursive functions A. Kreuzer (TU Darmstadt) RaTLoCC 2009 4 / 20
Reverse Mathematics Seeks to find which axioms over seconder order arithmetic are needed to prove theorems. Usually consider the big five subsystems of second order arithmetic: RCA 0 Σ 0 1 - IA + recursive comprehension 1st order part: Σ 0 1 -induction 2nd order part: recursive sets WKL 0 RCA 0 + Weak K¨ onigs Lemma 1st order part: like RCA 0 2nd order part: low sets ACA 0 RCA 0 + arithmetic comprehension 1st order part: arithmetic induction 2nd order part: arithmetic sets (includes WKL ) ATR 0 ACA 0 + arithmetical transfinite recursion Π 1 1 - CA 0 ACA 0 + Π 1 1 -comprehension Weaker systems: RCA ∗ 0 QF - IA + recursive comprehension + exponential function WKL ∗ 0 RCA ∗ 0 + Weak K¨ onigs Lemma RT 2 and provable recursive functions A. Kreuzer (TU Darmstadt) RaTLoCC 2009 4 / 20
Strength of Ramsey’s theorem RT 1 n is derivable in pure logic. RT 1 < ∞ is the infinite pigeonhole principle . RT 1 < ∞ ↔ Π 0 1 - CP Π 0 3 -conservative over RCA 0 . (Hirst, Friedman) Especially RT 1 < ∞ cause only primitive recursive growth. RT k n ↔ ACA 0 for k ≥ 3 and n ≥ 2 . (Simpson) RT 2 and provable recursive functions A. Kreuzer (TU Darmstadt) RaTLoCC 2009 5 / 20
Strength of Ramsey’s theorem for pairs Theorem (Hirst) RT 2 2 → Π 0 1 - CP RT 2 < ∞ → Π 0 2 - CP Theorem (Jockusch) There exists a computable coloring, which has no in 0 ′ computable infinite homogeneous set. Especially WKL 0 � RT 2 2 . RT 2 and provable recursive functions A. Kreuzer (TU Darmstadt) RaTLoCC 2009 6 / 20
Strength of Ramsey’s theorem for pairs Theorem (Hirst) RT 2 2 → Π 0 1 - CP RT 2 < ∞ → Π 0 2 - CP Theorem (Jockusch) There exists a computable coloring, which has no in 0 ′ computable infinite homogeneous set. Especially WKL 0 � RT 2 2 . RT 2 and provable recursive functions A. Kreuzer (TU Darmstadt) RaTLoCC 2009 6 / 20
Theorem (Cholak, Jockusch, Slaman) Every computable coloring has an infinite homogeneous H set which is low 2 , i.e. H ′′ ≤ T 0 ′′ . RCA 0 + Σ 0 2 - IA + RT 2 2 is Π 1 1 -conservative over RCA 0 + Σ 0 2 - IA . WKL 0 + Σ 0 3 - IA + RT 2 < ∞ is Π 1 1 -conservative over RCA 0 + Σ 0 3 - IA . Question (Cholak, Jockusch, Slaman) Does RT 2 2 imply Σ 0 2 - IA or the totality of the Ackermann-Function? Does RT 2 < ∞ imply Σ 0 3 - IA ? We show 0 + instances of RT 2 2 + instances of Σ 0 WKL ∗ 1 - IA does not prove the totality of the Ackermann-Function. RT 2 and provable recursive functions A. Kreuzer (TU Darmstadt) RaTLoCC 2009 7 / 20
Theorem (Cholak, Jockusch, Slaman) Every computable coloring has an infinite homogeneous H set which is low 2 , i.e. H ′′ ≤ T 0 ′′ . RCA 0 + Σ 0 2 - IA + RT 2 2 is Π 1 1 -conservative over RCA 0 + Σ 0 2 - IA . WKL 0 + Σ 0 3 - IA + RT 2 < ∞ is Π 1 1 -conservative over RCA 0 + Σ 0 3 - IA . Question (Cholak, Jockusch, Slaman) Does RT 2 2 imply Σ 0 2 - IA or the totality of the Ackermann-Function? Does RT 2 < ∞ imply Σ 0 3 - IA ? We show 0 + instances of RT 2 2 + instances of Σ 0 WKL ∗ 1 - IA does not prove the totality of the Ackermann-Function. RT 2 and provable recursive functions A. Kreuzer (TU Darmstadt) RaTLoCC 2009 7 / 20
Theorem (Cholak, Jockusch, Slaman) Every computable coloring has an infinite homogeneous H set which is low 2 , i.e. H ′′ ≤ T 0 ′′ . RCA 0 + Σ 0 2 - IA + RT 2 2 is Π 1 1 -conservative over RCA 0 + Σ 0 2 - IA . WKL 0 + Σ 0 3 - IA + RT 2 < ∞ is Π 1 1 -conservative over RCA 0 + Σ 0 3 - IA . Question (Cholak, Jockusch, Slaman) Does RT 2 2 imply Σ 0 2 - IA or the totality of the Ackermann-Function? Does RT 2 < ∞ imply Σ 0 3 - IA ? We show 0 + instances of RT 2 2 + instances of Σ 0 WKL ∗ 1 - IA does not prove the totality of the Ackermann-Function. RT 2 and provable recursive functions A. Kreuzer (TU Darmstadt) RaTLoCC 2009 7 / 20
Main Result For a schema S let S − denote the schema restricted to instances which only have number parameters. Theorem (K., Kohlenbach) For every fixed n G ∞ A ω + QF - AC + WKL + Π 0 1 - CA − + RT 2 − n is Π 0 2 -conservative over PRA , Π 0 3 -conservative over PRA + Σ 0 1 - IA and Π 0 4 -conservative over PRA + Π 0 1 - CP . RT 2 and provable recursive functions A. Kreuzer (TU Darmstadt) RaTLoCC 2009 8 / 20
Grzegorczyk Arithmetic in all finite types ( G ∞ A ω ) Corresponds to the (full) Grzegorczyk hierarchy. Contains quantifier free induction, bounded primitive recursion with function parameters, all primitive recursive functions, but not all primitive recursive functionals. The function iterator is not contained. Remark 0 can be embedded into G A ω + QF - AC . The system RCA ∗ RT 2 and provable recursive functions A. Kreuzer (TU Darmstadt) RaTLoCC 2009 9 / 20
Grzegorczyk Arithmetic in all finite types ( G ∞ A ω ) Corresponds to the (full) Grzegorczyk hierarchy. Contains quantifier free induction, bounded primitive recursion with function parameters, all primitive recursive functions, but not all primitive recursive functionals. The function iterator is not contained. Remark 0 can be embedded into G A ω + QF - AC . The system RCA ∗ RT 2 and provable recursive functions A. Kreuzer (TU Darmstadt) RaTLoCC 2009 9 / 20
G ∞ A ω + QF - AC + WKL + Π 0 1 - CA − Lemma G ∞ A ω + QF - AC + WKL + Π 0 1 - CA − proves Π 0 1 - IA − , Σ 0 1 - IA − , Π 0 1 - AC − , Π 0 1 - CP − , Σ 0 1 - WKL − , BW − (instances of Bolzano-Weierstrass). All these principles cannot be nested. RT 2 and provable recursive functions A. Kreuzer (TU Darmstadt) RaTLoCC 2009 10 / 20
Proof Theorem For every fixed n G ∞ A ω + QF - AC + WKL + Π 0 1 - CA − + RT 2 − n is Π 0 2 -conservative over PRA , Π 0 3 -conservative over PRA + Σ 0 1 - IA and Π 0 4 -conservative over PRA + Π 0 1 - CP . Proof. 1 Show G ∞ A ω + QF - AC + WKL + Π 0 1 - CA − proves RT 2 − . n 2 Use elimination of Skolem functions to obtain conservation result. RT 2 and provable recursive functions A. Kreuzer (TU Darmstadt) RaTLoCC 2009 11 / 20
Reduction step os’ and Rado’s proof of RT 2 Analyze Erd˝ n based on full K¨ onig’s Lemma. Theorem (K., Kohlenbach) G ∞ A ω + Π 0 1 - IA − ⊢ Σ 0 1 - WKL − → RT 2 − n for every fixed n . Corollary G ∞ A ω + QF - AC + WKL + Π 0 1 - CA − proves RT 2 − , for every fixed n . n RT 2 and provable recursive functions A. Kreuzer (TU Darmstadt) RaTLoCC 2009 12 / 20
Reduction step os’ and Rado’s proof of RT 2 Analyze Erd˝ n based on full K¨ onig’s Lemma. Theorem (K., Kohlenbach) G ∞ A ω + Π 0 1 - IA − ⊢ Σ 0 1 - WKL − → RT 2 − n for every fixed n . Corollary G ∞ A ω + QF - AC + WKL + Π 0 1 - CA − proves RT 2 − , for every fixed n . n RT 2 and provable recursive functions A. Kreuzer (TU Darmstadt) RaTLoCC 2009 12 / 20
Elimination of Skolem functions for monotone formulas Theorem (Kohlenbach) G ∞ A ω + QF - AC + WKL + Π 0 1 - CA − is Π 0 2 -conservative over PRA , Π 0 3 -conservative over PRA + Σ 0 1 - IA and Π 0 4 -conservative over PRA + Π 0 1 - CP . RT 2 and provable recursive functions A. Kreuzer (TU Darmstadt) RaTLoCC 2009 13 / 20
Elimination of Skolem functions for monotone formulas Let T ω := G ∞ A ω + QF - AC + WKL . Theorem (Kohlenbach) For every closed term ξ : T ω ⊢ ∀ f : N N � Π 0 � 1 - CA( ξ ( f )) → ∃ x ∈ N A qf ( f, x ) ⇒ one can extract a (Kleene-)primitive recursive functional Φ s.t. PRA ω ⊢ ∀ f : N N A qf ( f, Φ( f )) . Experience from proof-mining shows that many theorems from mathematics can be proved in this system. RT 2 and provable recursive functions A. Kreuzer (TU Darmstadt) RaTLoCC 2009 14 / 20
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