On the set-theoretic strength of the existence of disjoint cofinal sets in posets without maximal elements
Zorn's lemma is a well-known statement about partially ordered sets. Moreover, Zorn's lemma is provably equivalent, in ZF, to the axiom of choice (AC). In the paper under review, the authors investigate the deductive strength (without AC) of the following statements about partially ordered sets. {\parindent=1.3cm \begin{itemize} \item[CS:] Every partially ordered set without a maximal element has two disjoint cofinal subsets. \item [{\(\text{CS}_{\aleph_0}\):}] For each partially ordered set without a maximal element, there is a countably infinite set of disjoint cofinal subsets.\item [LCS:] Every linearly ordered set without a maximal element has two disjoint cofinal subsets. \item [{\(\text{LCS}_{\aleph_0}\):}] For each linearly ordered set without a maximal element, there is a countably infinite set of disjoint cofinal subsets. NEWLINENEWLINE\end{itemize}} The authors show that none of the principles CS, \(\text{CS}_{\aleph_0}\), LCS, \(\text{LCS}_{\aleph_0}\) can be established without some form of the axiom of choice.NEWLINENEWLINERecall that ZFA identifies Zermelo-Fraenkel theory where the extensionality axiom is modified to allow for the existence of atoms. Also recall that MC denotes the \textit{multiple axiom of choice}, that is, MC is an abbreviation of the assertion ``for every indexed family of nonempty sets \(\{X_i : i\in I\}\), there is a function \(F\) with domain \(I\) such that \(F(i)\) is a nonempty finite subset of \(A_i \), for each \(i\in I\).'' In ZF, the axiom of choice is equivalent to MC; however, in ZFA, AC implies MC, but the converse does not hold. After showing (in ZF) that CS is equivalent to \(\text{CS}_{\aleph_0}\) and that \(\text{DC} + \text{LCS}\) implies \(\text{LCS}_{\aleph_0}\) (DC is the axiom of dependent choices), the authors show that LCS does not imply AC in ZF. The authors also show, in ZFA, that MC implies CS. They then conclude that MC and CS are not equivalent in ZFA. Many other such results are also established in this article. The paper ends with five open questions. The first two of which follow: ``Is CS equivalent to AC in ZF?'' and ``Does LCS imply \(\text{LCS}_{\aleph_0}\)?''
scientific article; zbMATH DE number 6598139