On the logical strengths of partial solutions to mathematical problems

Bienvenu, Laurent; Patey, Ludovic; Shafer, Paul

doi:10.1112/tlm3.12001

Cited by 7 publications

(102 citation statements)

References 33 publications

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“…The various consequences of Ramsey's theorem usually fail to coincide with the main five subsystems, and slight variations of their statements lead to different subsystems. The study of Ramsey-type statements has been a very active research subject in reverse mathematics over the past few years [2,6,14,19]. See Hirschfeldt [17] for a good introduction to recent reverse mathematics.…”

Section: Reverse Mathematicsmentioning

confidence: 99%

The weakness of being cohesive, thin or free in reverse mathematics

Patey¹

2016

Isr. J. Math.

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Informally, a mathematical statement is robust if its strength is left unchanged under variations of the statement. In this paper, we investigate the lack of robustness of Ramsey's theorem and its consequence under the frameworks of reverse mathematics and computable reducibility. To this end, we study the degrees of unsolvability of cohesive sets for different uniformly computable sequence of sets and identify different layers of unsolvability. This analysis enables us to answer some questions of Wang about how typical sets help computing cohesive sets.We also study the impact of the number of colors in the computable reducibility between coloring statements. In particular, we strengthen the proof by Dzhafarov that cohesiveness does not strongly reduce to stable Ramsey's theorem for pairs, revealing the combinatorial nature of this non-reducibility and prove that whenever k is greater than ℓ, stable Ramsey's theorem for n-tuples and k colors is not computably reducible to Ramsey's theorem for n-tuples and ℓ colors. In this sense, Ramsey's theorem is not robust with respect to his number of colors over computable reducibility. Finally, we separate the thin set and free set theorem from Ramsey's theorem for pairs and identify an infinite decreasing hierarchy of thin set theorems in reverse mathematics. This shows that in reverse mathematics, the strength of Ramsey's theorem is very sensitive to the number of colors in the output set. In particular, it enables us to answer several related questions asked by Cholak, Giusto, Hirst and Jockusch.

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Section: Reverse Mathematicsmentioning

confidence: 99%

The weakness of being cohesive, thin or free in reverse mathematics

Patey¹

2016

Isr. J. Math.

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“…This strict weakening of WKL 0 is sufficient in most applications of WKL 0 involved in proofs of RT 2 2 . The statement RWKL has been later studied by Bienvenu, Patey and Shafer [1] and by Flood and Towsner [11].…”

Section: Separating Principles In Reverse Mathematicsmentioning

confidence: 99%

The Strength of the Tree Theorem for Pairs in Reverse Mathematics

Patey¹

2016

J. symb. log.

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Abstract. No natural principle is currently known to be strictly between the arithmetic comprehension axiom (ACA0) and Ramsey's theorem for pairs (RT 2 2 ) in reverse mathematics. The tree theorem for pairs (TT 2 2 ) is however a good candidate. The tree theorem states that for every finite coloring over tuples of comparable nodes in the full binary tree, there is a monochromatic subtree isomorphic to the full tree. The principle TT 2 2 is known to lie between ACA0 and RT 2 2 over RCA0, but its exact strength remains open. In this paper, we prove that RT 2 2 together with weak König's lemma (WKL0) does not imply TT 2 2 , thereby answering a question of Montálban. This separation is a case in point of the method of Lerman, Solomon and Towsner for designing a computability-theoretic property which discriminates between two statements in reverse mathematics. We therefore put the emphasis on the different steps leading to this separation in order to serve as a tutorial for separating principles in reverse mathematics.

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“…Let f be any DNC function. By a classical theorem about DNC functions (see Bienvenu et al [2] for a proof), f computes a function g(·, ·, ·) such that whenever |W e | ≤ n, then g(e, n, i) ∈ X i W e . For each i, let e i be the index of the c.e.…”

Section: Definition 12 (Constant-bound Enumeration Avoidance) a K-enmentioning

confidence: 99%

Partial orders and immunity in reverse mathematics

Patey

2018

COM

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Abstract. We identify computability-theoretic properties enabling us to separate various statements about partial orders in reverse mathematics. We obtain simpler proofs of existing separations, and deduce new compound ones. This work is part of a larger program of unification of the separation proofs of various Ramsey-type theorems in reverse mathematics in order to obtain a better understanding of the combinatorics of Ramsey's theorem and its consequences. We also answer a question of Murakami, Yamazaki and Yokoyama about pseudo Ramsey's theorem for pairs.

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On the logical strengths of partial solutions to mathematical problems

Cited by 7 publications

References 33 publications

The weakness of being cohesive, thin or free in reverse mathematics

The weakness of being cohesive, thin or free in reverse mathematics

The Strength of the Tree Theorem for Pairs in Reverse Mathematics

Partial orders and immunity in reverse mathematics

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