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Peter A. Cholak, Carl G. Jockusch & Theodore A. Slaman (2001). On the Strength of Ramsey's Theorem for Pairs.

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  1.  2
    A Weak Variant of Hindman’s Theorem Stronger Than Hilbert’s Theorem.Lorenzo Carlucci - 2018 - Archive for Mathematical Logic 57 (3-4):381-389.
    Hirst investigated a natural restriction of Hindman’s Finite Sums Theorem—called Hilbert’s Theorem—and proved it equivalent over \ to the Infinite Pigeonhole Principle for all colors. This gave the first example of a natural restriction of Hindman’s Theorem provably much weaker than Hindman’s Theorem itself. We here introduce another natural restriction of Hindman’s Theorem—which we name the Adjacent Hindman’s Theorem with apartness—and prove it to be provable from Ramsey’s Theorem for pairs and strictly stronger than Hirst’s Hilbert’s Theorem. The lower bound (...)
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  2.  5
    The Uniform Content of Partial and Linear Orders.Eric P. Astor, Damir D. Dzhafarov, Reed Solomon & Jacob Suggs - 2017 - Annals of Pure and Applied Logic 168 (6):1153-1171.
  3.  20
    The Strength of Infinitary Ramseyan Principles Can Be Accessed by Their Densities.Andrey Bovykin & Andreas Weiermann - 2017 - Annals of Pure and Applied Logic 168 (9):1700-1709.
  4.  5
    Unifying the Model Theory of First-Order and Second-Order Arithmetic viaWKL0⁎.Ali Enayat & Tin Lok Wong - 2017 - Annals of Pure and Applied Logic 168 (6):1247-1283.
  5.  1
    The Reverse Mathematics of Non-Decreasing Subsequences.Patey Ludovic - 2017 - Archive for Mathematical Logic 56 (5-6):491-506.
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  6.  1
    Dominating the Erdős–Moser Theorem in Reverse Mathematics.Ludovic Patey - 2017 - Annals of Pure and Applied Logic 168 (6):1172-1209.
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  7.  2
    Reverse Mathematical Bounds for the Termination Theorem.Silvia Steila & Keita Yokoyama - 2016 - Annals of Pure and Applied Logic 167 (12):1213-1241.
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  8.  13
    The Modal Logic of Reverse Mathematics.Carl Mummert, Alaeddine Saadaoui & Sean Sovine - 2015 - Archive for Mathematical Logic 54 (3-4):425-437.
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  9.  2
    Degrees Bounding Principles and Universal Instances in Reverse Mathematics.Ludovic Patey - 2015 - Annals of Pure and Applied Logic 166 (11):1165-1185.
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  10.  5
    Nonstandard Models in Recursion Theory and Reverse Mathematics.C. T. Chong, Wei Li & Yue Yang - 2014 - Bulletin of Symbolic Logic 20 (2):170-200.
  11.  1
    From Bolzano-Weierstraß to Arzelà-Ascoli.Alexander P. Kreuzer - 2014 - Mathematical Logic Quarterly 60 (3):177-183.
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  12.  9
    Cohesive Sets and Rainbows.Wei Wang - 2014 - Annals of Pure and Applied Logic 165 (2):389-408.
    We study the strength of RRT32, Rainbow Ramsey Theorem for colorings of triples, and prove that RCA0 + RRT32 implies neither WKL0 nor RRT42 source. To this end, we establish some recursion theoretic properties of cohesive sets and rainbows for colorings of pairs. We show that every sequence admits a cohesive set of non-PA Turing degree; and that every ∅′-recursive sequence admits a low3 cohesive set.
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  13.  1
    Separating Principles Below Ramsey's Theorem for Pairs.Manuel Lerman, Reed Solomon & Henry Towsner - 2013 - Journal of Mathematical Logic 13 (2):1350007.
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  14.  5
    On the Strength of Ramsey's Theorem Without S1-Induction.Keita Yokoyama - 2013 - Mathematical Logic Quarterly 59 (1-2):108-111.
    In this paper, we show that equation image is a equation image-conservative extension of BΣ1 + exp, thus it does not imply IΣ1.
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  15.  7
    Linear Extensions of Partial Orders and Reverse Mathematics.Emanuele Frittaion & Alberto Marcone - 2012 - Mathematical Logic Quarterly 58 (6):417-423.
    We introduce the notion of τ-like partial order, where τ is one of the linear order types ω, ω*, ω + ω*, and ζ. For example, being ω-like means that every element has finitely many predecessors, while being ζ-like means that every interval is finite. We consider statements of the form “any τ-like partial order has a τ-like linear extension” and “any τ-like partial order is embeddable into τ” . Working in the framework of reverse mathematics, we show that these (...)
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  16.  12
    The Cohesive Principle and the Bolzano‐Weierstraß Principle.Alexander P. Kreuzer - 2011 - Mathematical Logic Quarterly 57 (3):292-298.
    The aim of this paper is to determine the logical and computational strength of instances of the Bolzano-Weierstraß principle and a weak variant of it.We show that BW is instance-wise equivalent to the weak König’s lemma for Σ01-trees . This means that from every bounded sequence of reals one can compute an infinite Σ01-0/1-tree, such that each infinite branch of it yields an accumulation point and vice versa. Especially, this shows that the degrees d ≫ 0′ are exactly those containing (...)
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  17.  10
    The Maximal Linear Extension Theorem in Second Order Arithmetic.Alberto Marcone & Richard A. Shore - 2011 - Archive for Mathematical Logic 50 (5-6):543-564.
    We show that the maximal linear extension theorem for well partial orders is equivalent over RCA 0 to ATR 0. Analogously, the maximal chain theorem for well partial orders is equivalent to ATR 0 over RCA 0.
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  18.  60
    Reverse Mathematics: The Playground of Logic.Richard A. Shore - 2010 - Bulletin of Symbolic Logic 16 (3):378-402.
    This paper is essentially the author's Gödel Lecture at the ASL Logic Colloquium '09 in Sofia extended and supplemented by material from some other papers. After a brief description of traditional reverse mathematics, a computational approach to is presented. There are then discussions of some interactions between reverse mathematics and the major branches of mathematical logic in terms of the techniques they supply as well as theorems for analysis. The emphasis here is on ones that lie outside the usual main (...)
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  19.  7
    On a Question of Andreas Weiermann.Henryk Kotlarski & Konrad Zdanowski - 2009 - Mathematical Logic Quarterly 55 (2):201-211.
    We prove that for each β, γ < ε0 there existsα < ε0 such that whenever A ⊆ ω is α -large and G: A → β is such that ) ≤ a), then there exists a γ -large C ⊆ A on which G is nondecreasing. Moreover, we give upper bounds for α for small ordinals β ≤ ωmath image.
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  20.  2
    More on Lower Bounds for Partitioning Α-Large Sets.Henryk Kotlarski, Bożena Piekart & Andreas Weiermann - 2007 - Annals of Pure and Applied Logic 147 (3):113-126.
    Continuing the earlier research from [T. Bigorajska, H. Kotlarski, Partitioning α-large sets: some lower bounds, Trans. Amer. Math. Soc. 358 4981–5001] we show that for the price of multiplying the number of parts by 3 we may construct partitions all of whose homogeneous sets are much smaller than in [T. Bigorajska, H. Kotlarski, Partitioning α-large sets: some lower bounds, Trans. Amer. Math. Soc. 358 4981–5001]. We also show that the Paris–Harrington independent statement remains unprovable if the number of colors is (...)
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  21.  6
    On the Equimorphism Types of Linear Orderings.Antonio Montalbán - 2007 - Bulletin of Symbolic Logic 13 (1):71-99.
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  22.  17
    Partition Theorems and Computability Theory.Joseph R. Mileti - 2005 - Bulletin of Symbolic Logic 11 (3):411-427.
  23. Forcing in Proof Theory.Jeremy Avigad - 2004 - Bulletin of Symbolic Logic 10 (3):305-333.
    Paul Cohen’s method of forcing, together with Saul Kripke’s related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also has a place in traditional Hilbert-style proof theory, where the goal is to formalize portions of ordinary mathematics in restricted axiomatic theories, and study those theories in constructive or syntactic terms. I will discuss the aspects (...)
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