On the interrelation of a theorem of Juhász and certain weak axioms of choice
Tom 256 / 2022
Streszczenie
In set theory without the Axiom of Choice ($\mathsf{AC}$), we study the open problem of the set-theoretic strength (in terms of weak choice principles) of István Juhász’s topological generalization of Neumer’s Theorem from his paper [On Neumer’s theorem, Proc. Amer. Math. Soc. 54 (1976), 453–454], continuing the research initiated by Tachtsis [Juhász’s topological generalization of Neumer’s theorem may fail in $\mathsf{ZF}$, Proc. Amer. Math. Soc. 148 (2020), 1295–1310].
Among other results, we show that none of the two conjunctions
(1) “every partially ordered set has a cofinal well-founded subset” ($\mathsf{CWF}$) $\wedge $ “every Dedekind-finite set is finite” ($\mathsf{DF=F}$) and
(2) $\mathsf {CWF}$ $\wedge $ “the union of a well orderable collection of well orderable sets is well orderable”,
imply the axiom of multiple choice for countably infinite collections of countably infinite sets ($\mathsf {MC}_{\omega }^{\omega }$), and hence Juhász’s Theorem, in $\mathsf {ZFA}$ (i.e. in Zermelo–Fraenkel set theory with atoms). This settles the open problem whether either of (1) and (2) implies $\mathsf {MC}_{\omega }^{\omega }$, and substantially strengthens a result from Howard and Tachtsis [Models of $\mathsf {ZFA}$ in which every linearly ordered set can be well ordered].
We also show that “every linearly ordered set can be well ordered” ($\mathsf {LW}$) $\wedge $ the Axiom of Countable Choice ($\mathsf {AC}^{\omega }$) does not imply the Principle of Dependent Choices ($\mathsf {DC}$) in $\mathsf {ZFA}$. This settles an open problem from Howard and Rubin [Consequences of the Axiom of Choice, Amer. Math. Soc., 1998]. However, the question of whether $\mathsf {LW}\wedge \mathsf {AC}^{\omega }$ implies Juhász’s Theorem, which is formally weaker than $\mathsf {DC}$, is still open.
Furthermore, we introduce a new Fraenkel–Mostowski model of $\mathsf{ZFA}$ $\wedge $ Boolean Prime Ideal Theorem $\wedge $ $\neg (\mathsf {DF=F})$ $\wedge $ $\neg $(Juhász’s Theorem), and thus providing novel information on the construction of such $\mathsf {ZFA}$-models whose number in the literature is very limited.
