Reverse Mathematics

2018 ◽  
Author(s):  
John Stillwell
Keyword(s):  
Reverse Mathematics

scholarly journals Filling cages. Reverse mathematics and combinatorial principles

2020 ◽  
Vol 26 (3-4) ◽  
pp. 300-300
Author(s):  
Marta Fiori Carones
Keyword(s):  
Reverse Mathematics ◽  
Combinatorial Principles

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 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 Reverse-engineering Reverse Mathematics

2013 ◽  
Vol 164 (5) ◽  
pp. 528-541 ◽  
Author(s):  
Sam Sanders
Keyword(s):  
Reverse Engineering ◽  
Reverse Mathematics

Reverse mathematics and marriage problems with finitely many solutions

2016 ◽  
Vol 55 (7-8) ◽  
pp. 1015-1024
Author(s):  
Jeffry L. Hirst ◽  
Noah A. Hughes
Keyword(s):  
Reverse Mathematics ◽  
Marriage Problems

scholarly journals REVERSE MATHEMATICS, YOUNG DIAGRAMS, AND THE ASCENDING CHAIN CONDITION

2017 ◽  
Vol 82 (2) ◽  
pp. 576-589 ◽  
Author(s):  
KOSTAS HATZIKIRIAKOU ◽  
STEPHEN G. SIMPSON
Keyword(s):  
Theorem Proving ◽  
Chain Condition ◽  
Reverse Mathematics ◽  
Partial Ordering ◽  
Young Diagrams ◽  
Equivalence Proof ◽  
Image Position ◽  
Countably Infinite ◽  
Infinite Set ◽  
Ascending Chain Condition

AbstractLetSbe the group of finitely supported permutations of a countably infinite set. Let$K[S]$be the group algebra ofSover a fieldKof characteristic 0. According to a theorem of Formanek and Lawrence,$K[S]$satisfies the ascending chain condition for two-sided ideals. We study the reverse mathematics of this theorem, proving its equivalence over$RC{A_0}$(or even over$RCA_0^{\rm{*}}$) to the statement that${\omega ^\omega }$is well ordered. Our equivalence proof proceeds via the statement that the Young diagrams form a well partial ordering.

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