Models of \({{\textsf{ZFA}}}\) in which every linearly ordered set can be well ordered

Abstract

We provide a general criterion for Fraenkel–Mostowski models of \({\textsf{ZFA}}\) (i.e. Zermelo–Fraenkel set theory weakened to permit the existence of atoms) which implies “every linearly ordered set can be well ordered” (\({\textsf{LW}}\)), and look at six models for \({\textsf{ZFA}}\) which satisfy this criterion (and thus \({\textsf{LW}}\) is true in these models) and “every Dedekind finite set is finite” (\({\textsf{DF}}={\textsf{F}}\)) is true, and also consider various forms of choice for well-ordered families of well orderable sets in these models. In Model 1, the axiom of multiple choice for countably infinite families of countably infinite sets (\({\textsf{MC}}_{\aleph _{0}}^{\aleph _{0}}\)) is false. It was the open question of whether or not such a model exists (from Howard and Tachtsis “On metrizability and compactness of certain products without the Axiom of Choice”) that provided the motivation for this paper. In Model 2, which is constructed by first choosing an uncountable regular cardinal in the ground model, a strong form of Dependent choice is true, while the axiom of choice for well-ordered families of finite sets (\({\textsf{AC}}^{{\textsf{WO}}}_{{\textsf{fin}}}\)) is false. Also in this model the axiom of multiple choice for well-ordered families of well orderable sets fails. Model 3 is similar to Model 2 except for the status of \({\textsf{AC}}^{{\textsf{WO}}}_{{\textsf{fin}}}\) which is unknown. Models 4 and 5 are variations of Model 3. In Model 4 \({\textsf{AC}}_{\textrm{fin}}^{{\textsf{WO}}}\) is true. The construction of Model 5 begins by choosing a regular successor cardinal in the ground model. Model 6 is the only one in which \(2{\mathfrak {m}} = {\mathfrak {m}}\) for every infinite cardinal number \({\mathfrak {m}}\). We show that the union of a well-ordered family of well orderable sets is well orderable in Model 6 and that the axiom of multiple countable choice is false.