Fraenkel-Mostowski (FM) permutation models for ZFA
On Ramsey's theorem and the existence of infinite chains or infinite anti-chains in infinite posets
\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.
scientific article; zbMATH DE number 6578139