scholarly journals Reverse mathematics and a Ramsey-type König's Lemma

2012 ◽  
Vol 77 (4) ◽  
pp. 1272-1280 ◽  
Author(s):  
Stephen Flood
Keyword(s):  
Reverse Mathematics ◽  
Ramsey's Theorem ◽  
Weak Regularity ◽  
Ramsey’S Theorem ◽  
Ramsey Type ◽  
Weak König’S Lemma ◽  
König’S Lemma

AbstractIn this paper, we propose a weak regularity principle which is similar to both weak König's lemma and Ramsey's theorem. We begin by studying the computational strength of this principle in the context of reverse mathematics. We then analyze different ways of generalizing this principle.

scholarly journals THE STRENGTH OF THE TREE THEOREM FOR PAIRS IN REVERSE MATHEMATICS

2016 ◽  
Vol 81 (4) ◽  
pp. 1481-1499 ◽  
Author(s):  
LUDOVIC PATEY
Keyword(s):  
Binary Tree ◽  
Reverse Mathematics ◽  
Theoretic Property ◽  
Ramsey's Theorem ◽  
Ramsey’S Theorem ◽  
Image Position ◽  
Weak König’S Lemma ◽  
König’S Lemma ◽  
Comprehension Axiom

AbstractNo 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.

scholarly journals OPEN QUESTIONS ABOUT RAMSEY-TYPE STATEMENTS IN REVERSE MATHEMATICS

2016 ◽  
Vol 22 (2) ◽  
pp. 151-169 ◽  
Author(s):  
LUDOVIC PATEY
Keyword(s):  
Reverse Mathematics ◽  
Ramsey's Theorem ◽  
The Past ◽  
Open Questions ◽  
Ramsey’S Theorem ◽  
Ramsey Type ◽  
Infinite Set

AbstractRamsey’s theorem states that for any coloring of then-element subsets of ℕ with finitely many colors, there is an infinite setHsuch that alln-element subsets ofHhave the same color. The strength of consequences of Ramsey’s theorem has been extensively studied in reverse mathematics and under various reducibilities, namely, computable reducibility and uniform reducibility. Our understanding of the combinatorics of Ramsey’s theorem and its consequences has been greatly improved over the past decades. In this paper, we state some questions which naturally arose during this study. The inability to answer those questions reveals some gaps in our understanding of the combinatorics of Ramsey’s theorem.

scholarly journals ON THE UNIFORM COMPUTATIONAL CONTENT OF RAMSEY’S THEOREM

2017 ◽  
Vol 82 (4) ◽  
pp. 1278-1316 ◽  
Author(s):  
VASCO BRATTKA ◽  
TAHINA RAKOTONIAINA
Keyword(s):  
Finite Number ◽  
Ramsey's Theorem ◽  
Borel Hierarchy ◽  
Ramsey’S Theorem ◽  
Homogeneous Set ◽  
Colored Version ◽  
Output Information ◽  
Image Position ◽  
Weak König’S Lemma ◽  
König’S Lemma

AbstractWe study the uniform computational content of Ramsey’s theorem in the Weihrauch lattice. Our central results provide information on how Ramsey’s theorem behaves under product, parallelization, and jumps. From these results we can derive a number of important properties of Ramsey’s theorem. For one, the parallelization of Ramsey’s theorem for cardinalityn≥ 1 and an arbitrary finite number of colorsk≥ 2 is equivalent to then-th jump of weak Kőnig’s lemma. In particular, Ramsey’s theorem for cardinalityn≥ 1 is${\bf{\Sigma }}_{n + 2}^0$-measurable in the effective Borel hierarchy, but not${\bf{\Sigma }}_{n + 1}^0$-measurable. Secondly, we obtain interesting lower bounds, for instance then-th jump of weak Kőnig’s lemma is Weihrauch reducible to (the stable version of) Ramsey’s theorem of cardinalityn+ 2 forn≥ 2. We prove that with strictly increasing numbers of colors Ramsey’s theorem forms a strictly increasing chain in the Weihrauch lattice. Our study of jumps also shows that certain uniform variants of Ramsey’s theorem that are indistinguishable from a nonuniform perspective play an important role. For instance, the colored version of Ramsey’s theorem explicitly includes the color of the homogeneous set as output information, and the jump of this problem (but not the uncolored variant) is equivalent to the stable version of Ramsey’s theorem of the next greater cardinality. Finally, we briefly discuss the particular case of Ramsey’s theorem for pairs, and we provide some new separation techniques for problems that involve jumps in this context. In particular, we study uniform results regarding the relation of boundedness and induction problems to Ramsey’s theorem, and we show that there are some significant differences with the nonuniform situation in reverse mathematics.

scholarly journals The Brouwer invariance theorems in reverse mathematics

2020 ◽  
Vol 8 ◽  
Author(s):  
Takayuki Kihara
Keyword(s):  
Reverse Mathematics ◽  
Base System ◽  
Open Problems ◽  
Explicit Algorithm ◽  
Topological Embedding ◽  
Weak König’S Lemma ◽  
König’S Lemma

Abstract In [12], John Stillwell wrote, ‘finding the exact strength of the Brouwer invariance theorems seems to me one of the most interesting open problems in reverse mathematics.’ In this article, we solve Stillwell’s problem by showing that (some forms of) the Brouwer invariance theorems are equivalent to the weak König’s lemma over the base system ${\sf RCA}_0$ . In particular, there exists an explicit algorithm which, whenever the weak König’s lemma is false, constructs a topological embedding of $\mathbb {R}^4$ into $\mathbb {R}^3$ .

Combinatorial principles weaker than Ramsey's Theorem for pairs

2007 ◽  
Vol 72 (1) ◽  
pp. 171-206 ◽  
Author(s):  
Denis R. Hirschfeldt ◽  
Richard A. Shore
Keyword(s):  
Reverse Mathematics ◽  
The Other ◽  
Base System ◽  
Ramsey's Theorem ◽  
Open Questions ◽  
Ramsey’S Theorem ◽  
Combinatorial Principles ◽  
Points Of View ◽  
The One ◽  
Infinite Linear

AbstractWe investigate the complexity of various combinatorial theorems about linear and partial orders, from the points of view of computability theory and reverse mathematics. We focus in particular on the principles ADS (Ascending or Descending Sequence), which states that every infinite linear order has either an infinite descending sequence or an infinite ascending sequence, and CAC (Chain-AntiChain), which states that every infinite partial order has either an infinite chain or an infinite antichain. It is wellknown that Ramsey's Theorem for pairs () splits into a stable version () and a cohesive principle (COH). We show that the same is true of ADS and CAC, and that in their cases the stable versions are strictly weaker than the full ones (which is not known to be the case for and ). We also analyze the relationships between these principles and other systems and principles previously studied by reverse mathematics, such as WKL0, DNR, and BΣ2. We show, for instance, that WKL0 is incomparable with all of the systems we study. We also prove computability-theoretic and conservation results for them. Among these results are a strengthening of the fact, proved by Cholak, Jockusch, and Slaman, that COH is -conservative over the base system RCA0. We also prove that CAC does not imply DNR which, combined with a recent result of Hirschfeldt, Jockusch. Kjos-Hanssen, Lempp, and Slaman, shows that CAC does not imply (and so does not imply ). This answers a question of Cholak, Jockusch, and Slaman.Our proofs suggest that the essential distinction between ADS and CAC on the one hand and on the other is that the colorings needed for our analysis are in some way transitive. We formalize this intuition as the notions of transitive and semitransitive colorings and show that the existence of homogeneous sets for such colorings is equivalent to ADS and CAC, respectively. We finish with several open questions.

scholarly journals Comparing DNR and WWKL

2004 ◽  
Vol 69 (4) ◽  
pp. 1089-1104 ◽  
Author(s):  
Klaus Ambos-Spies ◽  
Bjørn Kjos-Hanssen ◽  
Steffen Lempp ◽  
Theodore A. Slaman
Keyword(s):  
Reverse Mathematics ◽  
Axiom System ◽  
Recursive Functions ◽  
Weak König’S Lemma ◽  
König’S Lemma

Abstract.In Reverse Mathematics, the axiom system DNR. asserting the existence of diagonally non-recursive functions, is strictly weaker than WWKL0 (weak weak König's Lemma).

scholarly journals Reverse mathematics, computability, and partitions of trees

2009 ◽  
Vol 74 (1) ◽  
pp. 201-215 ◽  
Author(s):  
Jennifer Chubb ◽  
Jeffry L. Hirst ◽  
Timothy H. McNicholl
Keyword(s):  
Binary Tree ◽  
Reverse Mathematics ◽  
Computability Theory ◽  
Ramsey's Theorem ◽  
Ramsey’S Theorem

AbstractWe examine the reverse mathematics and computability theory of a form of Ramsey's theorem in which the linear n-tuples of a binary tree are colored.

scholarly journals ERNA and Friedman's Reverse Mathematics

2011 ◽  
Vol 76 (2) ◽  
pp. 637-664 ◽  
Author(s):  
Sam Sanders
Keyword(s):  
Nonstandard Analysis ◽  
Reverse Mathematics ◽  
Consistency Proof ◽  
Transfer Principles ◽  
Weak König’S Lemma ◽  
König’S Lemma

AbstractElementary Recursive Nonstandard Analysis, in short ERNA, is a constructive system of nonstandard analysis with a PRA consistency proof, proposed around 1995 by Patrick Suppes and Richard Sommer. Recently, the author showed the consistency of ERNA with several transfer principles and proved results of nonstandard analysis in the resulting theories (see [12] and [13]). Here, we show that Weak König's lemma (WKL) and many of its equivalent formulations over RCA0 from Reverse Mathematics (see [21] and [22]) can be ‘pushed down’ into the weak theory ERNA, while preserving the equivalences, but at the price of replacing equality with equality ‘up to infinitesimals’. It turns out that ERNA plays the role of RCA0 and that transfer for universal formulas corresponds to WKL.

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