On the strength of Ramsey's theorem for pairs

Cholak, Peter; Jockusch, Carl G.; Slaman, Theodore A.

doi:10.2307/2694910

Cited by 158 publications

(246 citation statements)

References 23 publications

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“…We give a proof based on that of the n = 2 case, beginning with an auxiliary lemma, which is an extension of Theorem 12.1 in [5], and is proved by a similar argument. We write deg(X) for the (Turing) degree of a set X.…”

Section: Wkl Coh Srtmentioning

confidence: 99%

On notions of computability-theoretic reduction between Π21 principles

Hirschfeldt

Jockusch

2016

J. Math. Log.

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Abstract. Several notions of computability theoretic reducibility between Π 1 2 principles have been studied. This paper contributes to the program of analyzing the behavior of versions of Ramsey's Theorem and related principles under these notions. Among other results, we show that for each n 3, there is an instance of RT n 2 all of whose solutions have PA degree over ∅ (n−2) , and use this to show that König's Lemma lies strictly between RT 2 2 and RT 3 2 under one of these notions. We also answer two questions raised by Dorais, Dzhafarov, Hirst, Mileti, and Shafer [ta] on comparing versions of Ramsey's Theorem and of the Thin Set Theorem with the same exponent but different numbers of colors. Still on the topic of the effect of the number of colors on the computable aspects of Ramsey theoretic properties, we show that for each m 2, there is an (m + 1)-coloring c of N such that every m-coloring of N has an infinite homogeneous set that does not compute any infinite homogeneous set for c, and connect this result with the notion of infinite information reducibility introduced by Dzhafarov and Igusa [ta]. Next, we introduce and study a new notion that provides a uniform version of the idea of implication with respect to ω-models of RCA 0 , and related notions that allow us to count how many applications of a principle P are needed to reduce another principle to P . Finally, we fill in a gap in the proof of Theorem 12.2 in Cholak, Jockusch, and Slaman [2001].

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Section: Wkl Coh Srtmentioning

confidence: 99%

On notions of computability-theoretic reduction between Π21 principles

Hirschfeldt

Jockusch

2016

J. Math. Log.

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“…In a restricted form, it was used even earlier by Soare [15], to build an infinite set with no subset of strictly higher Turing degree. In computability theory, it has subsequently become a prominent tool for constructing infinite homogeneous sets for computable colorings of pairs of integers, as in Seetapun and Slaman [12], Cholak, Jockusch, and Slaman [3], and Dzhafarov…”

Section: Definition 11 a Condition Is A Pair (D E) Where D Is A Fimentioning

confidence: 99%

“…, Corollary 6.7, or [3], Section 5.1 for a proof.) We shall extend this result in Theorem 4.5 below.…”

Section: Proposition 27 Every N-generic Real Is Weakly N-generic Amentioning

confidence: 99%

Generics for computable Mathias forcing

Cholak

Dzhafarov

Hirst

et al. 2014

Annals of Pure and Applied Logic

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Abstract. We study the complexity of generic reals for computable Mathias forcing in the context of computability theory. The n-generics and weak ngenerics form a strict hierarchy under Turing reducibility, as in the case of Cohen forcing. We analyze the complexity of the Mathias forcing relation, and show that if G is any n-generic with n ≥ 2 then it satisfies the jump property. We prove that every such G has generalized high Turing degree, and so cannot have even Cohen 1-generic degree. On the other hand, we show that every Mathias n-generic real computes a Cohen n-generic real.

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“…Diagram 1 illustrates the relationship of these principles below the system WKL 0 +CAC. (For more information on reverse mathematics and Ramsey's Theorem, see Cholak, Jockusch and Slaman [1]. )…”

Section: Introductionmentioning

confidence: 99%

Reverse mathematics and the equivalence of definitions for well and better quasi-orders

Cholak

Marcone

2004

J. symb. log.

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In reverse mathematics, one formalizes theorems of ordinary mathematics in second order arithmetic and attempts to discover which set theoretic axioms are required to prove these theorems. Often, this project involves making choices between classically equivalent definitions for the relevant mathematical concepts. In this paper, we consider a number of equivalent definitions for the notions of well quasi-order and better quasi-order and examine how difficult it is to prove the equivalences of these definitions.As usual in reverse mathematics, we work in the context of subsystems of second order arithmetic and take RCA0 as our base system. RCA0 is the subsystem formed by restricting the comprehension scheme in second order arithmetic to formulas and adding a formula induction scheme for formulas. For the purposes of this paper, we will be concerned with fairly weak extensions of RCA0 (indeed strictly weaker than the subsystem ACA0 which is formed by extending the comprehension scheme in RCA0 to cover all arithmetic formulas) obtained by adjoining certain combinatorial principles to RCA0. Among these, the most widely used in reverse mathematics is Weak König's Lemma; the resulting theory WKL0 is extensively documented in [11] and elsewhere.We give three other combinatorial principles which we use in this paper. In these principles, we use k to denote not only a natural number but also the finite set {0, …, k − 1}.

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On the strength of Ramsey's theorem for pairs

Cited by 158 publications

References 23 publications

On notions of computability-theoretic reduction between Π21 principles

On notions of computability-theoretic reduction between Π21 principles

Generics for computable Mathias forcing

Reverse mathematics and the equivalence of definitions for well and better quasi-orders

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