On Ramsey's theorem and the existence of infinite chains or infinite anti-chains in infinite posets
type: Item
instance of: scholarly article
zbMATH Open document ID: 1342.03036
author: Eleftherios Tachtsis
publication date: 2016-05-09T00:00:00Z
Mathematics Subject Classification ID: 03E25
Mathematics Subject Classification ID: 03E35
Mathematics Subject Classification ID: 05D10
zbMATH DE Number: 6578139
zbMATH Keywords: axiom of choice
zbMATH Keywords: weak forms of the axiom of choice
zbMATH Keywords: Ramsey's theorem
zbMATH Keywords: partially ordered sets
zbMATH Keywords: chains in partially ordered sets
zbMATH Keywords: anti-chains in partially ordered sets
zbMATH Keywords: Fraenkel-Mostowski (FM) permutation models for ZFA
zbMATH Keywords: Pincus' transfer theorems
MaRDI profile type: Publication
full work available at URL: jsl.2015.47
OpenAlex ID: W2467421883
cites work: Set theory. An introduction to large cardinals
cites work: Q5573370
cites work: Ramsey's theorem and König's lemma
cites work: On Ramsey's theorem and the axiom of choice
cites work: Partial choice functions for families of finite sets
cites work: The axiom of choice
DOI: 10.1017/JSL.2015.47
title: On Ramsey's theorem and the existence of infinite chains or infinite anti-chains in infinite posets
published in: The Journal of Symbolic Logic
review text: \noindent It is a well known consequence of Ramsey's theorem (for pairs) that every infinite partially ordered set has either an infinite chain or an infinite antichain, which is addressed as the Chain-AntiChain (CAC) principle in [\textit{D. R. Hirschfeldt} and \textit{R. A. Shore}, J. Symb. Log. 72, No. 1, 171--206 (2007; Zbl 1118.03055)]. The reverse direction is an open problem listed in [\textit{P. Howard} and \textit{J. E. Rubin}, Consequences of the axiom of choice. Providence, RI: American Mathematical Society (1998; Zbl 0947.03001)].NEWLINENEWLINEThis article shows that the reverse implication is not a theorem of ZF. More precisely, via a suitable Mostowski permutation model of ZFA, the author produced a ZF-model in which CAC is true but Ramsey's theorem is false (Theorem 2.3). In addition, the author also showed (Theorem 2.4) that both statements are true in Mostowski's linearly ordered model (the model \(\mathcal N3\) in Howard-Rubin's book). In the closing remarks, the author suggested the direction of studying CAC from computability theory and reverse mathematics' points of view.
reviewed by: Xianghui Shi
description: scientific article; zbMATH DE number 6578139