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.

我们为\({\textsf{ZFA}}\)的 Fraenkel-Mostowski 模型提供了一个通用标准（即 Zermelo-Fraenkel 集合论被削弱以允许原子的存在），这意味着“每个线性有序的集合都可以很好地排序”（\({\textsf{LW}}\) )，然后查看\({\textsf{ZFA}}\)满足此标准的六个模型（因此\({\textsf{LW}}\)为真在这些模型中）和“每个 Dedekind 有限集都是有限的”（\({\textsf{DF}}={\textsf{F}}\) ）是正确的，并且还要考虑各种形式的有序族的选择这些模型中的有序集。在模型 1 中，可数无限集的可数无限族的多项选择公理 (\({\textsf{MC}}_{\aleph _{0}}^{\aleph _{0}}\) )是错误的。这种模型是否存在的悬而未决的问题（来自 Howard 和 Tachtsis“在没有选择公理的情况下某些产品的可度量性和紧凑性”）提供了本文的动机。在模型 2 中，首先在基础模型中选择一个不可数的正则基数，依赖选择的强形式为真，而有限集的良序族的选择公理 ( \({\textsf{AC } }^{{\textsf{WO}}}_{{\textsf{fin}}}\) ）是错误的。同样在这个模型中，有序集合的有序族的多项选择公理失败了。模型 3 与模型 2 类似，除了\({\textsf{AC}}^{{\textsf{WO}}}_{{\textsf{fin}}}\)这是未知的。模型 4 和 5 是模型 3 的变体。在模型 4 中，\({\textsf{AC}}_{\textrm{fin}}^{{\textsf{WO}}}\) 为真。Model 5 的构建始于在地面模型中选择一个常规的后继红衣主教。模型 6 是唯一一个其中\(2{\mathfrak {m}} = {\mathfrak {m}}\)对于每个无限基数\({\mathfrak {m}}\)。我们证明了良好有序集合的良好有序族的并集在模型 6 中是良好有序的，并且多重可数选择公理是错误的。