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.

Appendix

The following theorem provides general information about certain types of permutation models, and thus it is interesting in its own right. Furthermore, the theorem shows that Models 1 and 4 (of Sects. 3 and 6) are respectively equal to the models determined (by the same set of atoms and the same normal ideal, and) by weak direct products of groups (see Remark 1 of Subsect. 2.1) rather than unrestricted direct products. Obvious adjustments to this theorem can be made so that to obtain analogous results for Model 5 (of Sect. 7) and for generalizations of Models 1 and 4.

Theorem 11

Assume that the set A of atoms of the ground model M (of ${{\textsf{ZFA}}}+{{\textsf{AC}}}$) is a union of a disjoint, denumerable family $\{A_{n}:n\in \omega \}$, where each $A_{n}$ is denumerable. For each $n\in \omega$, let ${{\mathscr {G}}}_{n}$ be a group of permutations of $A_{n}$, and also let G be the weak direct product of the ${{\mathscr {G}}}_{n}$’s. Let I be the ideal which is generated by all unions $\bigcup \{A_{n}:n\in E\}$, $E\in [\omega ]^{<\omega }$. Let ${\mathcal {M}}$ be the permutation model determined by M, G, and I.

Let ${{\mathscr {G}}}$ be the unrestricted direct product of ${{\mathscr {G}}}_{n}$ ($n\in \omega$), and also let ${\mathcal {N}}$ be the permutation model determined by M, ${{\mathscr {G}}}$, and I. Then ${\mathcal {N}}={\mathcal {M}}$.

Proof

We prove by $\in$-induction that for every $x\in M$, $\Phi (x)$ is true, where

$$\begin{aligned} \Phi (x):\ x \in {\mathcal {M}} \Longleftrightarrow x \in {\mathcal {N}}. \end{aligned}$$

Clearly $\Phi (x)$ is true, if $x = \emptyset$, or if $x\in A$. Assume that $y \in M$ and that for all $x \in y$, $\Phi (x)$ is true. We will show that $\Phi (y)$ is true. Assume that $y \in {\mathcal {M}}$. Then the following hold:

• (1) y has a (countable) support $E\subset A$ relative to the group G (i.e., for every $\psi \in \textrm{fix}_{G}(E)$, $\psi (y)=y$);

• (2) for every $x \in y$, $x \in {\mathcal {M}}$ (${\mathcal {M}}$ is a transitive class);

• (3) for every $x \in y$, $x \in {\mathcal {N}}$ (by (2) and the induction hypothesis).

We assert that E is a support of y relative to the group ${{\mathscr {G}}}$. It suffices to show that for all $\phi \in \textrm{fix}_{{{\mathscr {G}}}}(E)$ and for all $x \in y$, $\phi (x) \in y$ (since then $\phi (y) =y$ follows from “$\phi (y) \subseteq y$ and $\phi ^{-1}(y) \subseteq y$").

To this end, let $\phi \in \textrm{fix}_{{{\mathscr {G}}}}(E)$ and let $x \in y$. By (3), x has a (countable) support $E'\subset A$ relative to ${{\mathscr {G}}}$. The permutation $\phi$ may not be in G, but we construct a permutation $\phi ' \in \textrm{fix}_{G}(E)$ which agrees with $\phi$ on $E'$ as follows: For each $a \in E'$, the set $\{ \phi ^n(a) : n \in {\mathbb {Z}} \}$ is countable. Therefore, since $E'$ is countable, so is $D = \bigcup \{\{ \phi ^n(a) : n \in {\mathbb {Z}} \}:a \in E'\}$. Furthermore, D contains $E'$ and is closed under $\phi$.

We define a mapping $\phi ': A \rightarrow A$ by

$$\begin{aligned} \phi '(a) = {\left\{ \begin{array}{ll} \phi (a), &{} \text{ if }\ a \in D; \\ a, &{} \text{ otherwise. } \end{array}\right. } \end{aligned}$$

Then the following hold:

• (4) $\phi '\in G$;

• (5) $\phi '$ fixes E pointwise (since $\phi$ fixes E pointwise); and

• (6) $\phi '$ agrees with $\phi$ on $E'$.

By (4) and (5), $\phi ' \in \textrm{fix}_{G}(E)$ so $\phi '(y) = y$. It follows that $\phi '(x) \in y$. Now, (6) gives $\phi '(x) = \phi (x)$, and hence $\phi (x) \in y$.

Conversely, assume that $y \in {\mathcal {N}}$ and that y has a support $E'$ relative to ${{\mathscr {G}}}$. Then $E'$ is a support of y relative to G since $G \subset {{\mathscr {G}}}$. By the induction hypothesis, every element of y is in ${\mathcal {M}}$, and so $y \in {\mathcal {M}}$. $\square$