The metamathematics of Stable Ramsey’s Theorem for Pairs

Chong, Chai Tai; Slaman, Theodore A.; Yue, Yang

doi:10.1090/s0894-0347-2014-00789-x

Cited by 64 publications

(113 citation statements)

References 15 publications

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“…Proof. First we will prove that (1) implies (2). Reasoning in ACA 0 , let H be a hypergraph with finite edges represented by the sequence e i i∈N .…”

Section: Representations Of Edgesmentioning

confidence: 99%

“…To prove the converse, by Lemma III.1.3 of Simpson [7], it suffices to use item (2) to prove the existence of the range of an injection. Let g : N → N be an injection.…”

Section: Representations Of Edgesmentioning

confidence: 99%

“…Next, we will prove that (2) implies (1) for the case k = 2, indicating parenthetically how to modify the argument for (3) implies (1). By Lemma 8, it suffices to use (2) to determine which trees are well-founded in a list of trees with leaf sets. Let T i i∈N be a sequence of trees in N <N , and let L i i∈N be the corresponding leaf sets.…”

Section: Vertex Colorings and Infinite Edgesmentioning

confidence: 99%

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Reverse mathematics and colorings of hypergraphs

et al. 2018

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“…Proof. First we will prove that (1) implies (2). Reasoning in ACA 0 , let H be a hypergraph with finite edges represented by the sequence e i i∈N .…”

Section: Representations Of Edgesmentioning

confidence: 99%

“…To prove the converse, by Lemma III.1.3 of Simpson [7], it suffices to use item (2) to prove the existence of the range of an injection. Let g : N → N be an injection.…”

Section: Representations Of Edgesmentioning

confidence: 99%

Section: Vertex Colorings and Infinite Edgesmentioning

confidence: 99%

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Reverse mathematics and colorings of hypergraphs

et al. 2018

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“…(For a more complete proof, see Davis et al [5]). By Corollary 2.6 of Chong, Slaman, and Yang [3], SRT 2 2 cannot prove IΣ 0 2 , so neither can ERT. Thus although RCA 0 + IΣ 0 2 proves ERT, the full strength of IΣ 0 2 is not necessary.…”

mentioning

confidence: 97%

Combinatorial principles equivalent to weak induction

Davis

Hirschfeldt

Hirst

et al. 2020

COM

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We consider two combinatorial principles, ERT and ECT. Both are easily proved in RCA 0 plus Σ 0 2 induction. We give two proofs of ERT in RCA 0 , using different methods to eliminate the use of Σ 0 2 induction. Working in the weakened base system RCA * 0 , we prove that ERT is equivalent to Σ 0 1 induction and ECT is equivalent to Σ 0 2 induction. We conclude with a Weihrauch analysis of the principles, showing

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“…Given the fact that for RT 2 2 this question is still open, see [20], and for the weaker principle SRT 2 2 this was just recently resolved, see [6], this question might be very difficult to answer.…”

Section: Final Remarksmentioning

confidence: 99%

10.4115/jla.v4i0.178

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Let (X , d) be a complete metric space, m ∈ N \ {0}, and γ ∈ R with 0 ≤ γ < 1. A g-contraction is a mapping T : X −→ X such that for all x, y ∈ X there is anThe generalized Banach contractions principle states that each g-contraction has a fixed point. We show that this principle is a consequence of Ramsey's theorem for pairs over, roughly, Mathematics Subject Classification 03B30, 03F60, 47H10 (primary); 03F35 (secondary)Keywords: reverse mathematics, Ramsey's theorem for pairs, Banach contraction principleIn this paper we will show that a generalization of the Banach contraction mapping principle-the generalized Banach contractions principle-follows from Ramsey's theorem for pairs over a weak basis theory.Let (X , d) be a complete metric space, let m ∈ N \ {0}, and let γ ∈ R with 0 ≤ γ < 1. We call a functionThe ordinary Banach contraction mapping theorem states that every (1, γ)-g-contraction has a fixed point. The generalized Banach contraction mapping principle is the statement that every (m, γ)-g-contraction has a fixed point.First results on the generalized Banach contraction mapping principle have been established in Jachymski, Schroder and Stein [12], where it is shown that this principle is true for g-contractions where m = 2. In Jachymski and Stein [13] it was shown that the principle is true for all m if the g-contraction is uniformly continuous. Later in Merryfield, Rothschild and Stein [18] it was shown that this principle is true for all continuous g-contractions and for m = 3 without this continuity assumption. The proof of the former statement uses Ramsey's theorem. However, it also uses full arithmetical comprehension, which is-as we will see below-much stronger than this contraction principle. Therefore,

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The metamathematics of Stable Ramsey’s Theorem for Pairs

Abstract: Abstract. We show that, over the base theory RCA 0 , Stable Ramsey's Theorem for Pairs implies neither Ramsey's Theorem for Pairs nor Σ 0 2 -induction.

Cited by 64 publications

References 15 publications

Reverse mathematics and colorings of hypergraphs

Reverse mathematics and colorings of hypergraphs

Combinatorial principles equivalent to weak induction

10.4115/jla.v4i0.178

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