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.

1 Introduction: consequences of the axiom of choice

Definition 1

We use the following abbreviations for consequences of the axiom of choice.

• 1. AC (Form 1 in [5]) is the axiom of choice.

• 2. ${\textsf{MC}}_{\aleph _0}^{\aleph _0}$ (Form 350 in [5]) is the axiom of multiple choice for countably infinite families of countably infinite sets, i.e. the statement “for every family ${\mathcal {A}}=\{A_{n}:n\in \omega \}$ with $|{\mathcal {A}}|=|A_{n}|=\aleph _{0}$, there is a function on ${\mathcal {A}}$ such that $f(A_{n})$ is a non-empty finite subset of $A_{n}$ for all $n\in \omega$". (The function f is called a multiple choice function for ${\mathcal {A}}$.)

• 3. ${\textsf{MC}}_{{\textsf{WO}}}^{{\textsf{WO}}}$ (Form 330 in [5]) is the axiom of multiple choice for well orderable families of non-empty well orderable sets.

• 4. ${\textsf{MC}}^{\aleph _{0}}$ (Form 126 in [5]) is the axiom of multiple choice for countably infinite families of infinite sets.

• 5. LW (Form 90 in [5]) is the statement “every linearly ordered set can be well ordered”.

• 6. ${\textsf{DF}}= {\textsf{F}}$ (Form 9 in [5]) is the statement “every Dedekind finite set is finite”. (Where a set X is called Dedekind finite if there is no one-to-one mapping $f:\omega \rightarrow X$; otherwise, X is called Dedekind infinite.)

• 7. ${\textsf{DC}}$ (Form 43 in [5]) is the principle of dependent choices.

• 8. Let $\kappa$ be an infinite well-ordered cardinal number. ${\textsf{DC}}_{\kappa }$ (Form 87($\kappa$) in [5]) is the statement “if X is a non-empty set and $\mathrel R$ is a binary relation such that for every $\alpha <\kappa$ and every $\alpha$-sequence ${{{\textbf {x}}}}=(x_\xi )_{\xi <\alpha }$ of elements of X there exists $y\in X$ such that ${{\textbf {x}}}\mathrel R y$, then there is a function $f:\kappa \rightarrow X$ such that for every $\alpha <\kappa$, $(f\upharpoonright \alpha )\mathrel R f(\alpha )$". Note that ${\textsf{DC}}_{\aleph _0}$ is a reformulation of the principle of dependent choices.

• 9. ${\textsf{AC}}^{{\textsf{WO}}}_{{\textsf{fin}}}$ (Form 122 in [5]) is the statement “every well-ordered family of non-empty finite sets has a choice function".

• 10. ${\textsf{AC}}^{{\textsf{WO}}}_{{\textsf{WO}}}$ (Form 165 in [5]) is the statement “every well-ordered family of non-empty well orderable sets has a choice function".

• 11. $\forall {\mathfrak {m}}, 2{\mathfrak {m}} = {\mathfrak {m}}$ (Form 3 in [5]) is the statement “for every infinite cardinal ${\mathfrak {m}}$, $2\cdot {\mathfrak {m}}={\mathfrak {m}}$". (Form 3 is equivalent to “for every infinite set X, $|2\times X|=|X|$", i.e. for every infinite set X, there is a bijection $f:2\times X\rightarrow X$, where $2=\{0,1\}$.)

We recall that ${\textsf{LW}}$ is equivalent to ${\textsf{AC}}$ in ${\textsf{ZF}}$ (i.e. Zermelo–Fraenkel set theory minus the ${\textsf{AC}}$), but is not equivalent to ${\textsf{AC}}$ in ${\textsf{ZFA}}$ (see Jech [7, Theorems 9.1 and 9.2]). We also recall that $\forall \kappa ({\textsf{DC}}_{\kappa })$ (where the parameter $\kappa$ runs through the infinite well-ordered cardinal numbers) is equivalent to ${\textsf{AC}}$ in ${\textsf{ZFA}}$; see [7, Theorem 8.1(c)]. Furthermore, ${\textsf{DF}}= {\textsf{F}}$ is strictly weaker than $\forall {\mathfrak {m}}, 2{\mathfrak {m}} = {\mathfrak {m}}$ in ${\textsf{ZF}}$, and $\forall {\mathfrak {m}}, 2{\mathfrak {m}} = {\mathfrak {m}}$ does not imply ${\textsf{AC}}$ in ${\textsf{ZF}}$ (see [5, Forms 3 and 9, the ${\textsf{ZF}}$-model ${\mathcal {M}}6$ and the ${\textsf{ZFA}}$-model ${\mathcal {N}}9$]).

Our original motivation for the research presented in this paper was to provide an answer to the open question “Does ${\textsf{LW}}$ together with ${\textsf{DF}}= {\textsf{F}}$ imply ${\textsf{MC}}_{\aleph _0}^{\aleph _0}$ in the theory ${\textsf{ZFA}}$?”, which was stated in Howard and Tachtsis [6]. The status of the above implication is also mentioned as unknown in Howard and Rubin [5].

We will consider six models of ${\textsf{ZFA}}$ (Models 1 through 6 as described in the abstract). Of these models 4 and 5 are new. Model 1 was constructed in [10], Model 2 in [7] and Model 3 in [5]. We will show that in all six models both ${\textsf{LW}}$ and ${\textsf{DF}}= {\textsf{F}}$ are true. We will also show that ${\textsf{MC}}_{\aleph _0}^{\aleph _0}$ is false in Models 1 and 4, but true in the other four models. We also consider the truth or falsity of the statements listed above in these models.

Besides resolving certain open problems on the relationship between the above weak choice principles (and conjunctions of those principles), it is also a central goal of this paper to explore and develop the required machinery, both set-theoretic and group-theoretic, in order to establish the relative independence results in ${\textsf{ZFA}}$ set theory. In this direction, our aim is to provide as much information as possible on certain independence proofs and their techniques in the permutation models studied in this paper.

Table 1 summarizes the known relationships in ZFA between (most of) the statements listed in Definition 1. The symbol $\rightarrow$ in the body of the table indicates that the row heading form implies the column heading form in ZFA. The symbol $\not \rightarrow$ means that the implication does not hold. The fact that $\forall {\mathfrak {m}}, 2{\mathfrak {m}} = {\mathfrak {m}}$ does not imply ${\textsf{MC}}_{\aleph _0}^{\aleph _0}$ is a consequence of a result from Sageev [9] where a ${\textsf{ZF}}$-model is constructed in which $\forall {\mathfrak {m}}, 2{\mathfrak {m}} = {\mathfrak {m}}$ is true and there is a countable set of countable sets of reals without a choice function (and thus without a multiple choice function—the usual order on ${\mathbb {R}}$ is a linear order). References for all of the other entries in the table can be found in Howard and Rubin [5].

Table 1 Known relationships

2 A model-theoretic criterion which implies ${\textsf{LW}}$, and certain types of FM models satisfying this criterion

Definition 2

We will use the following notation assuming that Z is a set and S is a subset of Z.

• 1. $\textrm{Sym}(Z)$ is the set of all permutations of Z.

• 2. If H is a subgroup of $\textrm{Sym}(Z)$, then $\textrm{Sym}_H(S) = \{ \phi \in H : \phi (S) = S \}$ and $\textrm{fix}_H(S) = \{ \phi \in H : {\forall x \in S}( \phi (x) = x) \}$.

• 3. $\textrm{FSym}(Z)$ is the set of all finitary permutations of Z.

Our first result, Theorem 1 below, provides a general condition under which a permutation model of ${\textsf{ZFA}}$ satisfies ${\textsf{LW}}$. All of the ${\textsf{ZFA}}$-models in this paper satisfy this condition, and thus ${\textsf{LW}}$ holds in all those models.

Theorem 1

Let ${\mathcal {N}}$ be a permutation model which is determined by a group G of permutations of the set A of atoms, and a normal filter ${\mathscr {F}}$ of subgroups of G which is generated by some filter base ${\mathscr {B}}$ (of subgroups of G). If ${\mathcal {N}}$ satisfies the following condition:

(*):

for every $x\in {\mathcal {N}}$ and for every $B\in {\mathscr {B}}$ which does not support x (i.e. $B\setminus \textrm{Sym}_{G}(x)\ne \emptyset$), there exists $\gamma \in B\setminus \textrm{Sym}_{G}(x)$ of finite order,

then ${{\textsf{LW}}}$ is true in ${\mathcal {N}}$.

Proof

Let $(X,\le )$ be a linearly ordered set in ${\mathcal {N}}$. Let $K\in {\mathscr {F}}$ be such that $K \subseteq \textrm{Sym}_{G}((X,\le ))$. Since ${\mathscr {B}}$ is a filter base for ${\mathscr {F}}$, there exists $B\in {\mathscr {B}}$ with $B\subseteq K$, and thus $B \subseteq \textrm{Sym}_{G}((X,\le ))$.

By way of contradiction, we assume that X is not well orderable in ${\mathcal {N}}$. Then there exists $x\in X$ such that $B\setminus \textrm{Sym}_{G}(x)\ne \emptyset$. (Otherwise, if $B \subseteq \textrm{fix}_{G}(X)$, then $\textrm{fix}_{G}(X) \in {\mathscr {F}}$, and hence X is well orderable in ${\mathcal {N}}$—see Jech [7, Equation (4.2), p. 47]—which is a contradiction.)

By $(*)$, there exist $\gamma \in B\setminus \textrm{Sym}_{G}(x)$ and an integer $n\ge 2$ such that $\gamma ^{n}=\epsilon$, where $\epsilon$ is the identity permutation on A. (Note that for an element $\phi$ of G we tacitly use the same notation for the unique $\in$-automorphism of $({\mathcal {N}},\in )$ which extends $\phi$.)

Since $B\subseteq \textrm{Sym}_{G}((X,\le ))$, $\gamma (x)\in X$ and ${\gamma (\le )} = {\le }$. Furthermore, since $\le$ is a linear order on X (in ${\mathcal {N}}$), either $\gamma (x) < x$ or $x < \gamma (x)$. If the first possibility occurs, then

$$\begin{aligned} x=\gamma ^{n}(x)<\gamma ^{n-1}(x)<\cdots<\gamma ^{2}(x)<\gamma (x)<x, \end{aligned}$$

and we thus arrived at a contradiction. In a similar manner, the second possibility also leads to a contradiction.

Thus X is well orderable in ${\mathcal {N}}$ as required. $\square$

Corollary 1

Let ${\mathcal {N}}$, A, G, ${\mathscr {F}}$, and ${\mathscr {B}}$, be as in the hypotheses of Theorem 1. If every element of G has finite order, or if G is a subgroup of $\textrm{FSym}(A)$, then ${\mathcal {N}}\models {{\textsf{LW}}}$.

Lemma 1 below will be a key result for the verification of condition (*) (of Theorem 1) in the majority of our models, except for Models 2 and 4 (see Sects. 4 and 6).

Lemma 1

Let Z be any infinite set and also let $\eta \in \textrm{Sym}(Z)$. Then there exists $\tau \in \textrm{Sym}(Z)$ such that

• 1. $\{ z \in Z : \tau (z) \ne z \} \subseteq \{ z \in Z : \eta (z) \ne z \}$ (that is, the support of $\tau$ is contained in the support of $\eta$);

• 2. $\tau ^2 = \epsilon$ (where $\epsilon$ is the identity element of $\textrm{Sym}(Z)$);

• 3. $(\eta \tau )^2 = \epsilon$.

Proof

We first consider the case where

$$\begin{aligned} \eta = (a_1, a_2, \ldots , a_n, a_{n+1}, a_{n+2} \ldots a_{2n}, a_{2n+1}) \end{aligned}$$

is a cycle of odd length. In this case, we let $\tau$ be the product of transpositions

$$\begin{aligned} \tau = (a_1,a_{2n+1})(a_2, a_{2n}) (a_3, a_{2n-1}) \cdots (a_n, a_{n+2}) = \prod _{i=1}^n (a_i, a_{2n+2-i}). \end{aligned}$$

Then, since $\tau$ is a product of disjoint transpositions, $\tau ^2 = \epsilon$. Also,

$$\begin{aligned} \eta \tau = (a_2, a_{2n+1})(a_3,a_{2n})\cdots (a_{n+1}, a_{n+2}) = \prod _{i=2}^{n+1} (a_i, a_{2n+3-i}) \end{aligned}$$

is a product of disjoint transpositions. Therefore, $(\eta \tau )^2 = \epsilon$.

Secondly, we assume that

$$\begin{aligned} \eta = (a_1, a_2, \ldots , a_n, a_{n+1}, \ldots , a_{2n}) \end{aligned}$$

is a cycle of even length. Let

$$\begin{aligned} \tau = (a_1,a_{2n})(a_2, a_{2n-1}) \cdots (a_n,a_{n+1}) = \prod _{i=1}^n (a_i, a_{2n+1-i}). \end{aligned}$$

As in the previous case, $\tau ^2 = \epsilon$ and

$$\begin{aligned} \eta \tau = (a_2,a_{2n}) (a_3, a_{2n-1}) \cdots (a_n, a_{n+2}) = \prod _{i=2}^n (a_i, a_{2n + 2 - i}) \end{aligned}$$

which is a product of disjoint transpositions. Hence, $(\eta \tau )^2 = \epsilon$.

Our third case is the case where

$$\begin{aligned} \eta = (\ldots , a_{-2}, a_{-1}, a_0, a_1, a_2, \ldots ) \end{aligned}$$

is an infinite cycle. Let

$$\begin{aligned} \tau = (a_1,a_{-1}) (a_2, a_{-2}) (a_3, a_{-3}) \cdots = \prod _{i=1}^{\infty } (a_i, a_{-i}). \end{aligned}$$

Then $\tau ^2 = \epsilon$ and

$$\begin{aligned} \eta \tau = (a_1, a_0) (a_2, a_{-1}) (a_3, a_{-2}) \cdots = \prod _{i = 1}^{\infty } (a_i, a_{1 - i}), \end{aligned}$$

so $(\eta \tau )^2 = \epsilon$.

Now we consider the general case where $\eta$ is any permutation of Z. Since any permutation can be written as a product of disjoint cycles, we assume that $\eta = \prod _{j \in J} \eta _j$ where the $\eta _j$’s are pairwise disjoint cycles and J is an (infinite or finite) index set. Using cases 1, 2 and 3, we know that for each $j \in J$ there exists $\tau _j\in \textrm{Sym}(Z)$ satisfying the three conditions of the lemma with $\eta$ and $\tau$ replaced by $\eta _j$ and $\tau _j$, respectively. By condition 1 (since the $\eta _j$’s are pairwise disjoint), the $\tau _j$’s are pairwise disjoint and for $j_1 \ne j_2$, $\tau _{j_1}$ is disjoint from $\eta _{j_2}$. Let $\tau = \prod _{j \in J} \tau _j$. Then $\tau ^2 = \prod _{j\in J} \tau _j^2 = \epsilon$ (since disjoint cycles commute). Further,

$$\begin{aligned} (\eta \tau )^2 = \left( \prod _{j \in J} \eta _j \prod _{j \in J} \tau _j \right) ^2 =\left( \prod _{j \in J} \eta _j \tau _j\right) ^2 =\prod _{j \in J} (\eta _j \tau _j)^2 = \epsilon \end{aligned}$$

(using the fact that disjoint cycles commute). This finishes the proof of the lemma. $\square$

2.1 Types of permutation models satisfying condition (*) of Theorem 1

The majority of the FM models of our paper—in particular, all models except for Model 6 of Sect. 8—are constructed as follows: Let $\mu$ be an infinite, well-ordered, regular, cardinal number, and also let $2\le \lambda \le \mu$ be a cardinal number. We start with a model M of ZFA + AC with a set A of atoms which is partitioned into a $\mu$-sized collection of sets each having cardinality $\lambda$. Say $A = \bigcup \{A_\alpha :\alpha \in \mu \}$, where for every $\alpha \in \mu$, $|A_\alpha |=\lambda$ and for $\alpha _1 \ne \alpha _2$, $A_{\alpha _1} \cap A_{\alpha _2} = \emptyset$.

For every $\alpha \in \mu$, let ${\mathscr {G}}_{\alpha }$ be a group of permutations of $A_{\alpha }$. (Usually, for every two distinct ordinals $\alpha ,\beta$ in $\mu$, ${\mathscr {G}}_{\alpha }$ is isomorphic to ${\mathscr {G}}_{\beta }$.) Let

$$\begin{aligned} G = \{ \phi \in \text{ Sym }(A) : \forall \alpha \in \mu (\phi \restriction {A_\alpha } \in {\mathscr {G}}_\alpha ) \} \end{aligned}$$

G is isomorphic to the unrestricted direct product of the ${\mathscr {G}}_{\alpha }$’s and therefore in subsequent sections we shall refer to G as the unrestricted direct product of the ${\mathscr {G}}_{\alpha }$’s.

Remark 1

For use in the forthcoming Sect. 11, let us note that, in a similar way, the weak direct product of the ${\mathscr {G}}_{\alpha }$’s will mean the group

$$\begin{aligned} \{ \phi \in \text{ Sym }(A) : \left( \forall \alpha \in \mu (\phi \restriction {A_\alpha } \in {\mathscr {G}}_\alpha )\right) \wedge \{ \alpha \in \mu : \phi \restriction {A_\alpha } \ne \epsilon _{A_\alpha } \} \text{ is } \text{ finite } \}, \end{aligned}$$

where $\epsilon _{A_\alpha }$ denotes the identity function on $A_\alpha$.

It is clear that for all $\alpha \in \mu$ and for all $\phi \in G$, $\phi (A_\alpha ) = A_\alpha$. Furthermore, for every $\alpha \in \mu$, ${\mathscr {G}}_{\alpha }$ is isomorphic to the subgroup of G,

$$\begin{aligned} G_{\alpha }= \{ \phi \in G : \forall a\not \in A_\alpha (\phi (a) = a) \}. \end{aligned}$$

(1)

• If $\lambda =\mu$, then the ideal I of supports is defined by

  $$\begin{aligned} I=\{S:\exists E\in [\mu ]^{<\omega }(S\subseteq \bigcup \{A_\alpha : \alpha \in E\})\}. \end{aligned}$$

• If $\lambda <\mu$, then the ideal I of supports is defined by

  $$\begin{aligned} I=\{S:\exists E\in [\mu ]^{<\mu }(S\subseteq \bigcup \{A_\alpha :\alpha \in E\})\}. \end{aligned}$$

In each of the above two cases, I is a normal ideal. The normal filter ${\mathscr {F}}$ of subgroups of G is the filter generated by the filter base $\{\textrm{fix}_{G}(S):S\in I\}$ (see [7, Chapter 4] for the definition of the terms “normal ideal” and “normal filter”). Note that, for $W\in \{ [\mu ]^{<\omega },[\mu ]^{<\mu }\}$, ${\mathscr {F}}$ is equal to the filter of subgroups of G generated by the filter base $\{\textrm{fix}_{G}(\bigcup \{A_{\alpha }:\alpha \in E\}):E\in W\}$ (the easy argument uses the fact that $S\subseteq S'$ implies $\textrm{fix}_{G}(S')\subseteq \textrm{fix}_{G}(S)$). It is the latter filter base that we use in order to check condition (*) of Theorem 1 for Models 1, 3, 4 and 5 (see the proof of the forthcoming Theorem 2).

Let ${\mathcal {N}}$ be the permutation model determined by M, G, and I, or equivalently, by M, G, and ${\mathscr {F}}$. (A set x in M is in ${\mathcal {N}}$ if and only if x and all elements in its transitive closure, TC(x), are supported by some element of I, that is, if and only if for every $y\in \{x\}\cup \textrm{TC}(x)$ there exists $S\in I$ such that for all $\phi \in \textrm{fix}_{G}(S)$, $\phi (y)=y$—equivalently, if and only if for every $y\in \{x\}\cup \textrm{TC}(x)$, $\textrm{Sym}_{G}(y)\in {\mathscr {F}}$.)

Note that if $x \in {\mathcal {N}}$ and S is a support of x of the form $\bigcup \{A_\alpha :\alpha \in E\}$, then by the fact that for all $\phi \in G$ and for all $\alpha \in \mu$, $\phi (A_{\alpha })=A_{\alpha }$, it follows that

$$\begin{aligned} \forall \phi \in G (\phi (S) = S), \text{ so } S \text{ is } \text{ a } \text{ support } \text{ of } \phi (x). \end{aligned}$$

(2)

Another lemma which will be useful for the proofs of ${\textsf{LW}}$ and ${\textsf{DF}}= {\textsf{F}}$ in our models, is the following one.

Lemma 2

Let ${\mathcal {N}}$ be a permutation model determined by M, G, and I as in the previous paragraph. Assume

• 1. $S = \bigcup \{A_\alpha :\alpha \in E\} \in I$,

• 2. $\eta \in \textrm{fix}_G(S)$ and

• 3. x is an element of ${\mathcal {N}}$ for which $\eta (x) \ne x$ (and hence x is not supported by S).

Then there exists $E'\subset \mu$ which is disjoint from E, and for each $\alpha \in E'$ there exists $\eta _{\alpha }\in G_{\alpha }$ (where $G_{\alpha }$ is given by (1)) such that for $\eta '=\prod _{\alpha \in E'}\eta _{\alpha }$ (and hence $\eta '\in \textrm{fix}_{G}(S)$), we have $\eta '(x)\ne x$.

In particular, if $\lambda =\mu$ (so that $E\in [\mu ]^{<\omega }$), then for some $\alpha \in E'$ there is $\eta _{\alpha } \in G_\alpha$ (and hence $\eta _{\alpha }\in \textrm{fix}_{G}(S)$) such that $\eta _{\alpha }(x) \ne x$.

Proof

Let $S \cup S'$ be a support of x where $S' = \bigcup \{A_\alpha :\alpha \in E'\}$ for some $E'\subset \mu$ with $E' \cap E = \emptyset$. We let $\eta '$ be the permutation of A which agrees with $\eta$ on $S'$ and is the identity outside of $S'$. Hence, $\eta '\in \textrm{fix}_{G}(A\setminus S')$, and so $\eta '\in \textrm{fix}_{G}(S)$. By the definition of the group G, it follows that for each $\alpha \in E'$, there exists $\eta _{\alpha }\in G_{\alpha }$ such that $\eta '=\prod _{\alpha \in E'}\eta _{\alpha }$. Since $\eta$ and $\eta '$ agree on the support $S\cup S'$ of x, we have $\eta '(x) = \eta (x)$, and since $\eta$ does not fix x, neither does $\eta '$.

The second assertion of the lemma follows from the proof of the first one and the facts that $E'$ is finite and $\eta '(x)\ne x$ (and note that since $E'$ is finite, $\eta '\upharpoonright A_{\alpha }=\epsilon _{A_{\alpha }}$ for all but finitely many $\alpha \in \mu$). $\square$

The following theorem essentially establishes the validity of ${\textsf{LW}}$ in the forthcoming Models 1, 3, 4, and 5 (of Sects. 3, 5, 6, and 7, respectively).

Theorem 2

Let ${\mathcal {N}}$ be a permutation model determined by M, G, and I as in the opening paragraph of Sect. 2.1. We assume that $\omega \le \lambda \le \mu$.

• (i) If for every $\alpha \in \mu$, ${{\mathscr {G}}}_{\alpha }$ is the group of even permutations of $A_{\alpha }$ (i.e. ${{\mathscr {G}}}_{\alpha }$ consists of all elements $\gamma$ of $\textrm{FSym}(A_{\alpha })$ which are an even permutation of their (finite) support $\{a\in A_{\alpha }:\gamma (a)\ne a\}$), then ${{\mathcal {N}}\models {{\textsf{LW}}}}$.

• (ii) If for every $\alpha \in \mu$, ${{\mathscr {G}}}_{\alpha }=\textrm{Sym}(A_{\alpha })$, then ${\mathcal {N}}\models {{\textsf{LW}}}$.

Proof

We first consider the case where $\lambda <\mu$, so that

$$\begin{aligned} I=\{S:\exists E\in [\mu ]^{<\mu }(S\subseteq \bigcup \{A_\alpha :\alpha \in E\})\}. \end{aligned}$$

(i) By Theorem 1, it suffices to show that ${\mathcal {N}}$ satisfies condition (*). To this end, let $x\in {\mathcal {N}}$ and also let $S = \bigcup \{ A_\alpha :\alpha \in E\}$ (for some $E\in [\mu ]^{<\mu }$) which does not support x, i.e. there exists $\eta \in \textrm{fix}_{G}(S)$ such that $\eta (x)\ne x$. By Lemma 2, there exists $E'\subset \mu$ which is disjoint from E, and for each $\alpha \in E'$ there exists $\eta _{\alpha }\in G_{\alpha }$ such that $\eta '(x)\ne x$, where $\eta '=\prod _{\alpha \in E'}\eta _{\alpha }$. Hence, $\eta _{\alpha }'=\eta _{\alpha }\upharpoonright A_{\alpha }\in {{\mathscr {G}}}_{\alpha }$.

For each $\alpha \in E'$, we apply Lemma 1 to $\eta _{\alpha }'$ to obtain a permutation $\tau _{\alpha }'$ on $A_{\alpha }$ with the following properties:

• 1. $(\tau _\alpha ')^2 = \epsilon$ and $(\eta _{\alpha }' \tau _{\alpha }')^2 = \epsilon$.

• 2. $\tau _{\alpha }$, $\eta _{\alpha }$ and $\eta _{\alpha } \tau _{\alpha }$ (i.e. the elements of $G_{\alpha }$ which extend $\tau _{\alpha }'$, $\eta _{\alpha }'$ and $\eta _{\alpha }' \tau _{\alpha }'$, respectively) are all in $\textrm{fix}_G(S)$ (since $\alpha \not \in E$).

• 3. $\tau _{\alpha }'\in \textrm{FSym}(A_{\alpha })$ (by condition 1 of Lemma 1).

We may also assume that $\tau _{\alpha }'\in {{\mathscr {G}}}_{\alpha }$, i.e. that $\tau _{\alpha }'$ is an even permutation of $A_{\alpha }$. If not, then we choose two elements a and $a'$ of $A_\alpha$ which are fixed by $\eta _{\alpha }'$ (and therefore fixed by $\tau _{\alpha }'$) and replace $\tau _{\alpha }'$ by the product $\tau _{\alpha }' (a,a')$ of $\tau _{\alpha }'$ and the transposition $(a,a')$.

Let $\tau = \prod _{\alpha \in E'} \tau _\alpha$. Then,

$$\begin{aligned} \tau ^2&= \left( \prod _{\alpha \in E'} \tau _\alpha \right) ^2 =\prod _{\alpha \in E'} \tau _\alpha ^2 = \epsilon ; \end{aligned}$$

(3)

$$\begin{aligned} (\eta ' \tau )^2&= \left( \prod _{\alpha \in E'} \eta _\alpha \prod _{\alpha \in E'} \tau _\alpha \right) ^2 = \prod _{\alpha \in E'} (\eta _\alpha \tau _\alpha )^2 =\epsilon . \end{aligned}$$

(4)

Formulas (3) and (4) use the fact that for $\alpha _1 \ne \alpha _2$, $\eta _{\alpha _1}$ and $\tau _{\alpha _1}$ both commute with $\eta _{\alpha _2}$ and $\tau _{\alpha _2}$.

Since $\eta '(x) \ne x$, it follows that either $\tau (x) \ne x$ or $\eta ' \tau (x) \ne x$. Since $\tau$ and $\eta '\tau$ are both elements of $\textrm{fix}_{G}(S)$, and $\tau ^{2}=(\eta '\tau )^{2}=\epsilon$, we conclude that condition (*) is satisfied.

Part (ii) (for the case where $\lambda <\mu$) can be proved in much the same way as (i), and so we leave it to the reader.

Now we assume that $\lambda =\mu$, so that

$$\begin{aligned} I=\{S:\exists E\in [\mu ]^{<\omega }(S\subseteq \bigcup \{A_\alpha :\alpha \in E\})\}. \end{aligned}$$

(i) Again, it suffices to show that ${\mathcal {N}}$ satisfies condition (*) of Theorem 1. Let $x\in {\mathcal {N}}$ and also let $S = \bigcup \{ A_\alpha :\alpha \in E\}$ (for some $E\in [\mu ]^{<\omega }$) which does not support x. By (the second assertion of) Lemma 2, there exist $E'\in [\mu ]^{<\omega }$ which is disjoint from E, and $\alpha \in E'$ such that for some $\eta _{\alpha } \in G_\alpha$ (and hence $\eta _{\alpha }\in \textrm{fix}_{G}(S)$), $\eta _{\alpha }(x) \ne x$. Since $\eta _{\alpha }$ moves only finitely many atoms, $\eta _{\alpha }$ has finite order. Thus (*) is satisfied, finishing the proof.

Part (ii) can be proved in a similar manner, using the second assertion of Lemma 2, and Lemma 1. We thus take the liberty to leave the details to the interested reader. $\square$

3 Model 1: ${\mathcal {M}}$

3.1 Motivation

We use Model 1 to establish that ${\textsf{LW}}+ {\textsf{DF}}= {\textsf{F}}$ does not imply ${\textsf{MC}}_{\aleph _0}^{\aleph _0}$ in ${\textsf{ZFA}}$. This answers in the negative the relative open question in Howard and Tachtsis [6], and also fills the gap in information in Howard and Rubin [5].

We note that this model has been considered in Tachtsis [10, proof of Theorem 4(iv)], where it was shown that ${\textsf{DF}}= {\textsf{F}}$ is true in the model. In the interest of making our paper self-contained, we will provide our own proof of ${\textsf{DF}}= {\textsf{F}}$ in the model.

3.2 The description of ${\mathcal {M}}$

We construct a model ${\mathcal {M}}$ of ZFA starting with a model ${\mathcal {M}}'$ of ZFA + AC with a countably infinite set of atoms A which is partitioned into a countably infinite collection of countably infinite sets. Say $A=\bigcup \{A_k:k \in \omega \}$ where for every $k\in \omega$, $|A_k|=\aleph _{0}$ and for $k_1 \ne k_2$, $A_{k_1} \cap A_{k_2} =\emptyset$. Let G be the unrestricted direct product of ${{\mathscr {G}}}_{n}=\textrm{Sym}(A_{n})$ ($n\in \omega$). The ideal of supports is defined (according to Subsect. 2.1) by

$$\begin{aligned} I = \{ C : \exists E \in [\omega ]^{<\omega } (C \subseteq \bigcup \{A_k : k \in E \})\}. \end{aligned}$$

${\mathcal {M}}$ is the permutation model determined by ${\mathcal {M}}'$, G, and I.

3.3 Versions of ${\textsf{AC}}$ in ${\mathcal {M}}$

We first give a (known) group-theoretic result which will be useful for the verification of ${\textsf{DF}}= {\textsf{F}}$ and ${\textsf{AC}}_{{\textsf{fin}}}^{{\textsf{WO}}}$ in ${\mathcal {M}}$.

Definition 3

Let Z be any set. If H is a subgroup of $\textrm{Sym}(Z)$, then $|\textrm{Sym}(Z) : H|$ denotes the index of H in $\textrm{Sym}(Z)$.

Theorem 3 below, is due to Dixon, Neumann, and Thomas (see [2, Theorem 1]).

Theorem 3

Let Z be countably infinite and let K be a subgroup of $\textrm{Sym}(Z)$ for which $| \textrm{Sym}(Z) : K | < 2^{\aleph _0}$. Then there is a finite subset $\Delta$ of Z such that $\textrm{fix}_{\textrm{Sym}(Z)} (\Delta ) \le K \le \textrm{fix}_{\textrm{Sym}(Z)}(\{\Delta \})$.

The subsequent Lemma 3 was originally proved by Onofri [8], and is also a consequence of Theorem 3. For the reader’s convenience, we include the short proof of the lemma (using Theorem 3).

Lemma 3

Let Z be a countably infinite set and let K be a proper subgroup of $\textrm{Sym}(Z)$. Then $|\textrm{Sym}(Z) : K|$ is infinite.

Proof

Let $\phi _{0}\in \textrm{Sym}(Z)\setminus K$. Toward a proof by contradiction, we assume that $|\textrm{Sym}(Z):K|$ is finite, so let $\textrm{Sym}(Z)/K=\{K,\phi _{0}K,\ldots ,\phi _{n}K\}$ for some $n\in \omega$ and $\phi _{i}\in \textrm{Sym}(Z)\setminus K$ ($i\le n$). Then, by Theorem 3, there exists a finite subset $\Delta \subset Z$ such that $\textrm{fix}_{\textrm{Sym}(Z)} (\Delta ) \le K \le \textrm{fix}_{\textrm{Sym}(Z)}(\{\Delta \})$. Since $\phi _{0}\not \in K$, there is $\delta \in \Delta$ such that $\phi _{0}(\delta )\ne \delta$. As $\Delta$ is finite, we have $|Z'|=\aleph _{0}$, where $Z'=Z\setminus (\Delta \cup (\bigcup \{\phi _{i}[\Delta ]:i\le n\}))$, and thus we may let $\psi \in \textrm{Sym}(Z)$ such that $\psi (\delta )\in Z'$ (for example, let $\psi$ be the transposition $(\delta ,z')$ for any $z'\in Z'$). Then $\psi \not \in \textrm{fix}_{\textrm{Sym}(Z)}(\{\Delta \})$ and $\phi _{i}^{-1}\psi \not \in \textrm{fix}_{\textrm{Sym}(Z)}(\{\Delta \})$ for all $i\le n$, and hence $\psi \not \in K$ and $\phi _{i}^{-1}\psi \not \in K$ for all $i\le n$. Therefore, $\psi K\not \in \{K,\phi _{0}K,\ldots , \phi _{n}K\}=\textrm{Sym}(Z)/K$, a contradiction. $\square$

Theorem 4

In ${\mathcal {M}}$, ${{\textsf{LW}}}$, ${{\textsf{DF}}} = {{\textsf{F}}}$, and ${{\textsf{AC}}}^{{{\textsf{WO}}}}_{{{\textsf{fin}}}}$ are true, but ${{\textsf{MC}}}_{\aleph _0}^{\aleph _0}$ and $\forall {\mathfrak {m}},2{\mathfrak {m}} = {\mathfrak {m}}$ are false.

Proof

By Theorem 2(ii), we immediately have that ${\textsf{LW}}$ is true in ${\mathcal {M}}$.

Now, it is reasonably clear that the set ${\mathscr {A}}= \{A_k : k \in \omega \}$ is a countably infinite set of countably infinite sets in ${\mathcal {M}}$, which has no multiple choice function in ${\mathcal {M}}$. (If $f: {\mathscr {A}}\rightarrow {\mathcal {P}}(A)$ is a multiple choice function for ${\mathscr {A}}$ with support $S = \bigcup \{A_k:k \in E\}$ (for some finite $E\subset \omega$), then choose an integer $k_0 \not \in E$; then $f(A_{k_0})$ is a finite subset of $A_{k_0}$. Then it is possible to choose $\phi \in G$ such that $\phi \in \textrm{fix}_G(S)$ and $\phi (f(A_{k_0})) \ne f(A_{k_0})$. But, since $\phi$ fixes both f and $A_{k_0}$, $\phi (f(A_{k_0})) = f(A_{k_0})$. This gives a contradiction, so ${\textsf{MC}}_{\aleph _0}^{\aleph _0}$ is false in ${\mathcal {M}}$.)

Claim

$\forall {\mathfrak {m}},2{\mathfrak {m}}={\mathfrak {m}}$ is false in ${\mathcal {M}}$.

Proof

Indeed, there is no one-to-one mapping $f:2\times A\rightarrow A$ in ${\mathcal {M}}$. Assuming the contrary, let f be such a function in ${\mathcal {M}}$ with support $S = \bigcup \{A_i:i \in E\}$ for some finite $E \subset \omega$.

Now let $k\in \omega \setminus E$, and also let $x\in A_{k}$. Since $f((0,x))\ne f((1,x))$, there exists $i\in 2$ such that $f((i,x))=y$ with $y\ne x$ (and note that similarly to the following argument, y is necessarily an element of $A_{k}$). Let $z\in A_{k}\setminus \{x,y\}$ (recall that $A_{k}$ is (countably) infinite) and also let $\psi =(x,z)$ (i.e. $\psi$ transposes x and z but fixes all the other atoms of A). Then $\psi \in \textrm{fix}_{G}(S)$, so $\psi (f)=f$. However,

$$\begin{aligned} ((i,x),y)\in f\Rightarrow (\psi ((i,x)),\psi (y)) \in \psi (f) \Rightarrow ((i,z),y)\in f, \end{aligned}$$

so that $(i,x)\ne (i,z)$ and $f((i,x))=f((i,z))$, contrary to the fact that f is one-to-one. Hence, $|2\times A|\ne |A|$ in ${\mathcal {M}}$. $\square$

Now we prove that ${\textsf{DF}}= {\textsf{F}}$ is true in ${\mathcal {M}}$. The following lemma will be useful for the proof.

Lemma 4

Assume $x \in {\mathcal {M}}$ and $m \in \omega$. Let H be the subgroup $H = \{ \phi \in G_m : \phi (x) = x \}$ of $G_m$ (where $G_{m}$ is given by (1) in Sect. 2.1). Then for all $\phi _1$ and $\phi _2$ in $G_m$, $\phi _1(x) = \phi _2(x)$ if and only if $\phi _1 H = \phi _2 H$.

Proof

The proof uses the standard properties of right cosets. Indeed, we have $\phi _1(x) = \phi _2(x)$, if and only if, $\phi _2^{-1} \phi _1(x) = x$, if and only if, $\phi _2^{-1} \phi _1 \in H$, if and only if, $\phi _2^{-1} \phi _1 H = H$, if and only if, $\phi _1 H = \phi _2 H$. $\square$

Claim

${\textsf{DF}}= {\textsf{F}}$ is true in ${\mathcal {M}}$.

Proof

Assume that Y is an infinite, non-well-orderable set in ${\mathcal {M}}$ with support $S = \bigcup \{A_i:i \in E\}$, where $E \subset \omega$ is finite. Then for some $x \in Y$, S is not a support of x. Let $S \cup S'$ be a suppport of x, where $S' = \bigcup \{ A_i:i \in E'\}$ with $E \cap E' = \emptyset$. By (the second assertion of) Lemma 2, we obtain an $m \in E'$ and a $\beta \in G_m$ such that $\beta (x) \ne x$.

It follows that the set

$$\begin{aligned} H = \{ \phi \in G_m : \phi (x) = x \} \end{aligned}$$

is a proper subgroup of $G_m$. Since $G_m$ is isomorphic to $\textrm{Sym}(\omega )$ (for $G_{m}\simeq \textrm{Sym}(A_{m})\simeq \textrm{Sym}(\omega )$), we may apply Lemma 3 to conclude that the set of left cosets $\{ \phi H : \phi \in G_m \}$ is infinite. Let

$$\begin{aligned} W = \{ \phi (x) : \phi \in G_m \}. \end{aligned}$$

Then we know the following about W:

• 1. By Lemma 4 and the fact that the set of left cosets of H in G is infinite, W is infinite.

• 2. $W \subseteq Y$ since for all $\phi \in G_m$, $\phi \in \textrm{fix}_G(S)$.

• 3. Every element of W has support $S \cup S'$, by (2) of Sect. 2.1.

So W is an infinite subset of Y which can be well ordered in ${\mathcal {M}}$. Therefore, Y has a countably infinite subset in ${\mathcal {M}}$. $\square$

Claim

${\textsf{AC}}^{{\textsf{WO}}}_{{\textsf{fin}}}$ is true in ${\mathcal {M}}$.

Proof

Let ${\mathscr {X}}=\{X_{\alpha }:\alpha \in \kappa \}$ be an infinite well-ordered set in ${\mathcal {M}}$ ($\kappa$ is an infinite well-ordered cardinal and the mapping $\alpha \mapsto X_{\alpha }$ is a bijection) such that $X_{\alpha }$ is non-empty and finite for all $\alpha \in \kappa$. Let $S = \bigcup \{A_i:i \in E\}$ (for some finite $E\subset \omega$) be a support of $X_{\alpha }$ for all $\alpha \in \kappa$. We will show that S supports every element of $\bigcup {\mathscr {X}}$; hence, $\bigcup {\mathscr {X}}$ will be well orderable in the model.

Assume on the contrary that there exist $\alpha \in \kappa$ and $x\in X_{\alpha }$ such that S is not a support of x. By (the second assertion of) Lemma 2, there exist $m\in \omega \setminus E$ and $\eta \in G_{m}$ such that $\eta (x)\ne x$. Let

$$\begin{aligned} Z=\{\phi (x):\phi \in G_{m}\}. \end{aligned}$$

Then $Z\subseteq X_{\alpha }$ (since $x\in X_{\alpha }$, S is a support of $X_{\alpha }$, and $G_{m}\subseteq \textrm{fix}_{G}(S)$). Hence, Z is finite (and has at least two elements). Furthermore, since $\eta \in G_{m}$ and $\eta (x)\ne x$, the group

$$\begin{aligned} H=\{\phi \in G_{m}:\phi (x)=x\} \end{aligned}$$

is a proper subgroup of $G_{m}$. Since Z is finite, the index $|G_{m}:H|$ of H in $G_{m}$ is finite. But this contradicts Lemma 3, since $G_{m}$ is isomorphic to $\textrm{Sym}(\omega )$ and H is a proper subgroup of $G_{m}$. $\square$

The above arguments complete the proof of the theorem. $\square$

By Theorem 4, we obtain that ${\textsf{LW}}+ {\textsf{DF}}= {\textsf{F}}\nRightarrow {\textsf{MC}}^{{\textsf{WO}}}_{{\textsf{WO}}}$ in ${\textsf{ZFA}}$.

4 Model 2: ${\mathscr {V}}$

4.1 Motivation

We recall that ${\textsf{DC}}$ implies the axiom of countable choice (i.e. “Every countably infinite family of non-empty sets has a choice function"), which in turn implies both ${\textsf{DF}}= {\textsf{F}}$ and ${\textsf{MC}}_{\aleph _0}^{\aleph _0}$. Furthermore, ${\textsf{DC}}$ does not imply ${\textsf{MC}}^{{\textsf{WO}}}_{{\textsf{WO}}}$ in ${\textsf{ZF}}$; in the Brunner/Howard permutation model ${\mathcal {N}}15$ in [5], ${\textsf{DC}}$ is true but ${\textsf{MC}}^{{\textsf{WO}}}_{\aleph _{0}}$ (the axiom of multiple choice for well-ordered families of countably infinite sets) is false (see Sect. 5). The result can be transferred to ${\textsf{ZF}}$ using Pincus’ transfer theorems (see [5, Note 103, third theorem, p. 286]), and notice that $\lnot {\textsf{MC}}^{{\textsf{WO}}}_{\aleph _{0}}$ is a boundable, and hence injectively boundable, statement (see [5, Note 103] for the definitions of those terms).

So in view of Theorem 4 and the above discussion, the natural question that emerges here is whether ${\textsf{LW}}+ {\textsf{DC}}$ implies ${\textsf{MC}}^{{\textsf{WO}}}_{{\textsf{WO}}}$ in ${\textsf{ZFA}}$ (recall that ${\textsf{LW}}$ is equivalent to ${\textsf{AC}}$ in ${\textsf{ZF}}$, so the above implication is vacuously true in ${\textsf{ZF}}$). The status of this implication is mentioned as unknown in [5]. Therein, it is also mentioned as unknown whether ${\textsf{LW}}+ {\textsf{DC}}$ implies ${\textsf{AC}}^{{\textsf{WO}}}_{{\textsf{fin}}}$. In the model ${\mathscr {V}}$ of this section, we address these open questions and prove that their respective answers are in the negative. In fact, we prove a much stronger result than “${\textsf{LW}}+ {\textsf{DC}}$ implies neither ${\textsf{MC}}^{{\textsf{WO}}}_{{\textsf{WO}}}$ nor ${\textsf{AC}}^{{\textsf{WO}}}_{{\textsf{fin}}}$ in ${\textsf{ZFA}}$", as Theorem 5 below clarifies.

4.2 The description of ${\mathscr {V}}$

We use a permutation model constructed in the proof of Jech’s Theorem 8.3 in [7] (see also [5, Model ${\mathcal {N}}2(\aleph _{\alpha })$, p. 180]). The construction starts with a ground model M of ${\textsf{ZFA}}+ {\textsf{AC}}$ which has a set A of atoms of cardinality $\aleph _\alpha$, where $\aleph _\alpha$ is an uncountable regular cardinal in M. We partition A into a disjoint union of $\aleph _\alpha$ pairs, so that $A=\bigcup \{P_{\xi }:\xi <\aleph _{\alpha }\}$ ($|P_{\xi }|=2$ for all $\xi <\aleph _{\alpha }$, and for $\xi \ne \xi '$, $P_{\xi }\cap P_{\xi '}=\emptyset$). Let G be the group of all permutations of A which fix $P_{\xi }$ for all $\xi <\aleph _{\alpha }$. (Note that G is essentially the unrestricted direct product of $\textrm{Sym}(P_{\xi })$ ($\xi <\aleph _{\alpha }$), and that every element of G has order 2.) Let ${\mathscr {F}}$ be the filter on G generated by the groups $\textrm{fix}_{G}(E)$, where $E\subset A$, $|E|<\aleph _{\alpha }$. Let ${\mathscr {V}}$ be the permutation model determined by M, G and ${\mathscr {F}}$.

4.3 Versions of ${\textsf{AC}}$ in ${\mathscr {V}}$

Theorem 5

In ${\mathscr {V}}$, ${{\textsf{LW}}}$ and ${{\textsf{DC}}}_{\xi }$ for all infinite cardinals $\xi < \aleph _{\alpha }$ are true but ${{\textsf{MC}}}^{{{\textsf{WO}}}}_{{{\textsf{WO}}}}$, ${{\textsf{AC}}}^{{{\textsf{WO}}}}_{{{\textsf{fin}}}}$ and $\forall {\mathfrak {m}}, 2{\mathfrak {m}} = {\mathfrak {m}}$ are false.

Proof

By Corollary 1, we have ${\mathscr {V}}\models {\textsf{LW}}$. Furthermore, in [7, Lemma 8.4, p. 123], it is shown that for every $\xi <\aleph _{\alpha }$, ${\textsf{DC}}_{\xi }$ is true in ${\mathscr {V}}$, and that the family ${\mathscr {A}}=\{P_{\xi }:\xi <\aleph _{\alpha }\}$ (which is in ${\mathscr {V}}$ and has cardinality $\aleph _{\alpha }$ in ${\mathscr {V}}$) has no choice function in the model; hence, ${\textsf{AC}}^{{\textsf{WO}}}_{{\textsf{fin}}}$ is false in ${\mathscr {V}}$.

Claim

${\textsf{MC}}^{{\textsf{WO}}}_{{\textsf{WO}}}$ is false in ${\mathscr {V}}$¹.

Proof

Fix an infinite cardinal number $\kappa <\aleph _{\alpha }$. Let

$$\begin{aligned} {\mathscr {U}}=\{U_{\xi }:\xi <\aleph _{\alpha }\} \end{aligned}$$

be a partition of $\aleph _{\alpha }$ ($\xi \mapsto U_{\xi }$ is a bijection) into sets each of which has cardinality $\kappa$. (And note that ${\mathscr {U}}$ gives rise to an $\aleph _{\alpha }$-sized partition of ${\mathscr {A}}=\{P_{\xi }:\xi <\aleph _{\alpha }\}$ into $\kappa$-sized sets, namely $\{\{P_{\gamma }:\gamma \in U_{\xi }\}:\xi <\aleph _{\alpha }\}$, which clearly has a choice function in the model.) For each $\xi <\aleph _{\alpha }$, we let

$$\begin{aligned} W_{\xi }=\prod _{\gamma \in U_{\xi }}P_{\gamma }. \end{aligned}$$

Then for every $\xi <\aleph _{\alpha }$, $W_{\xi }\in {\mathscr {V}}$ (any permutation of A in G fixes $W_{\xi }$), and furthermore, the subset $E=\bigcup \{P_{\gamma }:\gamma \in U_{\xi }\}$ of A (which has cardinality $\kappa <\aleph _{\alpha }$) is a support of every element of $W_{\xi }$. Thus the infinite sets $W_{\xi }$ ($\xi <\aleph _{\alpha }$) are well orderable in ${\mathscr {V}}$.

Now we let

$$\begin{aligned} {\mathscr {W}}=\{W_{\xi }:\xi <\aleph _{\alpha }\}. \end{aligned}$$

Then ${\mathscr {W}}\in {\mathscr {V}}$ and has cardinality $\aleph _{\alpha }$ in ${\mathscr {V}}$, so ${\mathscr {W}}$ is well orderable in ${\mathscr {V}}$. However, ${\mathscr {W}}$ has no multiple choice function in ${\mathscr {V}}$. Assume the contrary and let F be a multiple choice function for ${\mathscr {W}}$, which is in ${\mathscr {V}}$. Let $E\subset A$, $|E|<\aleph _{\alpha }$, be a support of F. Since ${\mathscr {U}}$ is an $\aleph _{\alpha }$-sized partition of $\aleph _{\alpha }$ and $|E|<\aleph _{\alpha }$, it follows that for some $\xi _{0}<\aleph _{\alpha }$,

$$\begin{aligned} E\cap \left( \bigcup \{P_{\gamma }:\gamma \in U_{\xi _{0}}\}\right) =\emptyset . \end{aligned}$$

(5)

Let $f\in F(W_{\xi _{0}})$ ($F(W_{\xi _{0}})\ne \emptyset$, since F is a multiple choice function for ${\mathscr {W}}$), and also let $g\in W(\xi _{0})\setminus F(W_{\xi _{0}})$ ($W_{\xi _{0}}$ is infinite, whereas $F(W_{\xi _{0}})$ is a finite subset of $W_{\xi _{0}}$). Then $g\ne f$, so the set $J=\{\gamma \in U_{\xi _{0}}:{g(\gamma )\ne f(\gamma )}\}$ is nonempty. For each $\gamma \in J$, let $\psi _{\gamma }$ be the permutation of A which interchanges the two elements of $P_{\gamma }$ but fixes every atom in $A\setminus P_{\gamma }$. Let

$$\begin{aligned} \pi =\prod _{\gamma \in J}\psi _{\gamma }. \end{aligned}$$

By the definition of $\pi$ and (5), we have $\pi \in \textrm{fix}_{G}(E)$, so $\pi (F)=F$, and since $\pi (W_{\xi _{0}})=W_{\xi _{0}}$ and F is a function, we also have $\pi (F(W_{\xi _{0}}))=F(W_{\xi _{0}})$. Furthermore, it is clear that $\pi (f)=g$. Thus we have

$$\begin{aligned} f\in F(W_{\xi _{0}})\Rightarrow \pi (f)\in \pi (F(W_{\xi _{0}})) \Rightarrow g\in F(W_{\xi _{0}}), \end{aligned}$$

contradicting the fact that $g\not \in F(W_{\xi _{0}})$. Therefore, ${\textsf{MC}}^{{\textsf{WO}}}_{{\textsf{WO}}}$ is false in ${\mathscr {V}}$ as required. $\square$

In [5], it is mentioned that $\forall {\mathfrak {m}},2{\mathfrak {m}}={\mathfrak {m}}$ is false in ${\mathscr {V}}$. The argument is similar to the one given for the proof of Claim 3.3. (Assuming that there is a one-to-one mapping $f:2\times A\rightarrow A$ which is in ${\mathscr {V}}$, let E be a support of f and $\gamma <\aleph _{\alpha }$ such that $E\cap P_{\gamma }=\emptyset$. It is easy to see that $f[2\times P_{\gamma }]\subseteq P_{\gamma }$, which is a contradiction since f is one-to-one and $|P_{\gamma }|<|2\times P_{\gamma }|$.)

The above arguments complete the proof of the theorem. $\square$

By Theorem 5, we immediately obtain the following corollary.

Corollary 2

${{\textsf{LW}}} + {{\textsf{DC}}}$ implies neither ${{\textsf{MC}}}^{{{\textsf{WO}}}}_{{{\textsf{WO}}}}$ nor ${{\textsf{AC}}}^{{{\textsf{WO}}}}_{{{\textsf{fin}}}}$ in ${{\textsf{ZFA}}}$.

Clearly the above result readily yields that ${\textsf{LW}}+ {\textsf{DF}}= {\textsf{F}}$ does not imply ${\textsf{MC}}^{{\textsf{WO}}}_{{\textsf{WO}}}$ in ${\textsf{ZFA}}$ (and neither does it imply ${\textsf{AC}}^{{\textsf{WO}}}_{{\textsf{fin}}}$). Therefore, Theorem 4 is an essential strengthening of the latter non-implication in ${\textsf{ZFA}}$ (and note again that the model of the proof of Theorem 5 (or the model ${\mathcal {N}}15$ in [5]) satisfies ${\textsf{MC}}_{\aleph _0}^{\aleph _0}\wedge \lnot {\textsf{MC}}^{{\textsf{WO}}}_{{\textsf{WO}}}$).

5 Model 3: ${\mathcal {N}}15$

5.1 Motivation

As already mentioned in Sect. 4, this model satisfies ${\textsf{DC}}\wedge \lnot {\textsf{MC}}^{{\textsf{WO}}}_{\aleph _{0}}$ (see the forthcoming Theorem 6). Therefore, the next natural question that comes up is whether ${\textsf{LW}}$ is true in ${\mathcal {N}}15$. We note that the status of ${\textsf{LW}}$ in ${\mathcal {N}}15$ is not specified in [5].

The answer to this open question is in the affirmative; thus filling the gap in information in [5] and providing further insight to the reader.

5.2 The description of ${\mathcal {N}}15$

The set A of atoms has cardinality $\aleph _{1}$, and is written as a union of an $\aleph _{1}$-sized family of pairwise disjoint countably infinite sets,

$$\begin{aligned} A=\bigcup \{B_{\alpha }:\alpha <\aleph _{1}\}, \text { where}\ B_{\alpha }=\{a_{i,\alpha }:i\in \omega \}. \end{aligned}$$

For each $\alpha <\aleph _{1}$, let ${{\mathscr {G}}}_{\alpha }$ be the group of even permutations on $B_{\alpha }$. Let G be the unrestricted direct product of ${{\mathscr {G}}}_{\alpha }$ ($\alpha <\aleph _{1}$).

Let I be the ideal of all countable subsets of A. Note that I is equal to the ideal generated by all sets of the form $\bigcup \{B_{\alpha }:\alpha \in E\}$, where E is a countable subset of $\aleph _{1}$. ${\mathcal {N}}15$ is the permutation model determined by A, G and I.²

5.3 Versions of ${\textsf{AC}}$ in ${\mathcal {N}}15$

Theorem 6

In ${\mathcal {N}}15$, ${{\textsf{LW}}}$ and ${{\textsf{DC}}}$ are true, but ${{\textsf{MC}}}^{{{\textsf{WO}}}}_{\aleph _{0}}$ is false.

Proof

By Theorem 2(i), we have ${\mathcal {N}}15\models {\textsf{LW}}$.

Furthermore, ${\textsf{DC}}$ is true in ${\mathcal {N}}15$ (for the normal ideal I comprises all countable subsets of A, and $\aleph _{1}$ is a regular cardinal—the argument in the proof of [7, Lemma 8.4, p. 123], and in the paragraph following this lemma, can be adapted in our case by making the obvious minor changes).

It is also easy to see that ${\textsf{MC}}^{{\textsf{WO}}}_{\aleph _{0}}$ is false in ${\mathcal {N}}15$. Indeed, let ${\mathcal {B}}=\{B_{\alpha }:\alpha <\aleph _{1}\}$. Clearly $|{\mathcal {B}}|=\aleph _{1}$ in ${\mathcal {N}}15$ (every permutation of A in G fixes ${\mathcal {B}}$ pointwise and $|{\mathcal {B}}|=\aleph _{1}$ in the ground model) and for every $\alpha <\aleph _{1}$, $|B_\alpha |=\aleph _{0}$ in ${\mathcal {N}}15$ ($B_{\alpha }$ is a support of each of its elements, and is countable in the ground model). Now, ${\mathcal {B}}$ has no multiple choice function in ${\mathcal {N}}15$. Assuming the contrary, let f be such a function in ${\mathcal {N}}15$. Let $S=\bigcup \{B_{\alpha }:\alpha \in E\}$, where $E\subset \aleph _{1}$ is countable, be a support of f. Let $\alpha _0\in \aleph _{1}\setminus E$ and u be any element of $f(B_{\alpha _0})$. Let $Z=\{z_{1},z_{2},z_{3}\}$ be a 3-element subset of $B_{\alpha _0}$ which is disjoint from $f(B_{\alpha _0})$, and also let $\pi =(u,z_1)(z_2,z_3)$. Then $\pi \in \textrm{fix}_{G}(S)$, and thus $\pi (f)=f$. However, $z_{1}\in \pi (f(B_{\alpha _0}))\setminus f(B_{\alpha _0})$ (so $\pi (f(B_{\alpha _0}))\ne f(B_{\alpha _0})$), contradicting f’s being supported by S. Hence, ${\mathcal {B}}$ has no multiple choice function in ${\mathcal {N}}15$. $\square$

Remark 2

We note that, similarly to the proof of Claim 3.3 (of the proof of Theorem 4), $\forall {\mathfrak {m}},2{\mathfrak {m}} ={\mathfrak {m}}$ is false in ${\mathcal {N}}15$.

6 Model 4: ${\mathscr {U}}$, a variant of ${\mathcal {N}}15$

6.1 Motivation

In view of the preceding study of the model ${\mathcal {N}}15$, it is natural to consider a variation of this model which witnesses “${\textsf{LW}}+ {\textsf{DF}}= {\textsf{F}}+ {\textsf{AC}}^{{\textsf{WO}}}_{{\textsf{fin}}}$ $\nRightarrow$ ${\textsf{MC}}^{\aleph _{0}}_{\aleph _{0}}$” in ${\textsf{ZFA}}$. Indeed, our ${\textsf{ZFA}}$-model ${\mathscr {U}}$ of this section appeals to this consideration. We note that ${\mathscr {U}}$ does not appear in either of [1] and [5].

6.2 The description of ${\mathscr {U}}$

The set A of atoms is countably infinite, and is written as a union of a countably infinite family of pairwise disjoint countably infinite sets,

$$\begin{aligned} A=\bigcup \{B_{n}:n\in \omega \}, \text { where}\ B_{n}=\{a_{i,n}:i\in \omega \}. \end{aligned}$$

For every $n\in \omega$, let ${{\mathscr {G}}}_n$ be the group of even permutations of $B_n$. Let G be the unrestricted direct product of the ${{\mathscr {G}}}_n$’s. Let I be the ideal of subsets of A which is generated by all finite unions of $B_n$ ($n\in \omega$). Let ${\mathscr {U}}$ be the Fraenkel–Mostowski model determined by A, G, and I.

Let us point out here that $\mathscr {U}$ can be generalized. Indeed, for any infinite regular cardinal number $\kappa$, we may similarly construct a permutation model ${\mathscr {U}}_{\kappa }$: The set of atoms, $A=\bigcup \{B_{\alpha }:\alpha <\kappa \}$ (where each of the $B_{\alpha }$’s has cardinality $\lambda$, where $\omega \le \lambda \le \kappa$, and $\{B_{\alpha }:\alpha <\kappa \}$ is disjoint), ${{\mathscr {G}}}_{\alpha }$ is the group of even permutations of $B_{\alpha }$, $G=\prod _{\alpha <\kappa }{{\mathscr {G}}}_{\alpha }$ (the unrestricted direct product of the ${{\mathscr {G}}}_{\alpha }$’s), and I is the (normal) ideal generated by $\{\bigcup \{B_{\alpha }:\alpha \in E\}: E \in [\kappa ]^{<\omega }\}$.

6.3 Versions of ${\textsf{AC}}$ in ${\mathscr {U}}$

Theorem 7

In ${\mathscr {U}}$, ${{\textsf{LW}}}$, ${{\textsf{DF}}} = {{\textsf{F}}}$, and ${{\textsf{AC}}}^{{{\textsf{WO}}}}_{{{\textsf{fin}}}}$ are true, but ${{\textsf{MC}}}^{\aleph _{0}}_{\aleph _{0}}$ is false.

Proof

By Theorem 2(i), we have ${\mathscr {U}}\models {\textsf{LW}}$.

Furthermore, the proof that ${\textsf{MC}}^{\aleph _{0}}_{\aleph _{0}}$ is false in ${\mathscr {U}}$ is almost identical to the proof that ${\textsf{MC}}^{{\textsf{WO}}}_{\aleph _{0}}$ is false in ${\mathcal {N}}15$ (see the proof of Theorem 6), and we thus skip it.

Claim

${\textsf{DF}}= {\textsf{F}}$ is true in ${\mathscr {U}}$.

Proof

Assume that Y is an infinite, non-well-orderable set in ${\mathscr {U}}$ with support $S = \bigcup \{B_i:i \in E\}$ for some finite $E \subset \omega$. Then for some $x \in Y$, S is not a support of x. Let $S \cup S'$ be a suppport of x, where $S' = \bigcup \{ B_i:i \in E'\}$ with $E' \cap E = \emptyset$. By (the second assertion of) Lemma 2, we obtain an $m \in E'$ and a $\beta \in G_m$ (where $G_{m}$ is given by (1) of Sect. 2.1) such that $\beta (x) \ne x$.

Consider the $G_m$-orbit of x, i.e. the set

$$\begin{aligned} \textrm{Orb}_{G_m}(x)=\{\pi (x):\pi \in G_m\}. \end{aligned}$$

Since S is a support of Y, $x\in Y$, and for all $\pi \in G_{m}$, $\pi \in \textrm{fix}_{G}(S)$, we conclude that $\textrm{Orb}_{G_m}(x)\subseteq Y$. Furthermore, $\textrm{Orb}_{G_m}(x)$ is well orderable in the model since $S\cup S'$ is a support of every element of $\textrm{Orb}_{G_m}(x)$.

We assert that $\textrm{Orb}_{G_m}(x)$ is infinite. If not, then $\textrm{Sym}(\textrm{Orb}_{G_m}(x))$ is also finite. Let $\phi :G_m\rightarrow \textrm{Sym}(\textrm{Orb}_{G_m}(x))$ be defined by

$$\begin{aligned} \phi (\eta )(y)=\eta (y)\ (y\in \textrm{Orb}_{G_m}(x)). \end{aligned}$$

Then $\phi$ is a homomorphism, and hence $\ker (\phi )$ is a normal subgroup of $G_m$ and the quotient group $G_{m}/\ker (\phi )$ embeds into $\textrm{Sym}(\textrm{Orb}_{G_m}(x))$. However, $\ker (\phi )$ is a proper subgroup of $G_m$ (for $\beta \in G_{m}\setminus \ker (\phi )$), and since $G_m$ is a simple group (for $G_{m}\simeq {{\mathscr {G}}}_{m}$), we obtain that $\ker (\phi )=\{\epsilon \}$.

Thus, $G_{m}/\ker (\phi )$ is isomorphic to $G_m$, and so $\textrm{Sym}(\textrm{Orb}_{G_m}(x))$ contains a copy of $G_m$. This is a contradiction, since $\textrm{Sym}(\textrm{Orb}_{G_m}(x))$ is finite and $G_m$ is infinite. Therefore, $\textrm{Orb}_{G_m}(x)$ is infinite, and thus Y is Dedekind infinite. $\square$

Claim

${\textsf{AC}}^{{\textsf{WO}}}_{{\textsf{fin}}}$ is true in ${\mathscr {U}}$.

Proof

Letting ${\mathcal {V}}=\{V_\alpha :\alpha <\kappa \}$ be an infinite well-ordered family ($\kappa$ is an infinite well-ordered cardinal number) of non-empty finite sets in ${\mathscr {U}}$ and $S=\bigcup \{B_{i}:i\in E\}$ (where $E\in [\omega ]^{<\omega }$) be a support of every $V_{\alpha }$, we may work similarly to the proof of Claim 6.3 in order to show that every element of $\bigcup {\mathcal {V}}$ is supported by S, so that $\bigcup {\mathcal {V}}$ is well orderable in ${\mathscr {U}}$. Thus ${\mathcal {V}}$ has a choice function in ${\mathscr {U}}$. $\square$

The above arguments complete the proof of the theorem. $\square$

Remark 3

As with the model ${\mathcal {N}}15$, $\forall {\mathfrak {m}},2{\mathfrak {m}}={\mathfrak {m}}$ is false in ${\mathscr {U}}$.

7 Model 5: ${\mathcal {U}}$

7.1 The description of ${\mathcal {U}}$

Suitable adjustments to the construction of the model ${\mathcal {N}}15$ yield the result of Theorem 5, modulo the assertion about ${\textsf{AC}}^{{\textsf{WO}}}_{{\textsf{fin}}}$. Indeed, fix any regular cardinal number $\aleph _{\alpha +1}$. We start with a model M of ${\textsf{ZFA}}+ {\textsf{AC}}$ with a set A of atoms which has cardinality $\aleph _{\alpha +1}$ and is written as a disjoint union,

$$\begin{aligned} A=\bigcup \{B_{\beta }:\beta<\aleph _{\alpha +1}\}, \text { where}\ B_{\beta }=\{a_{\mu ,\beta }:\mu < \aleph _{\alpha }\}. \end{aligned}$$

For each $\beta <\aleph _{\alpha +1}$, let ${{\mathscr {G}}}_{\beta }$ be the group of even permutations of $B_{\beta }$. Let G be the unrestricted direct product of the ${{\mathscr {G}}}_{\beta }$’s. Let I be the ideal of all subsets of A having cardinality less than $\aleph _{\alpha +1}$. Let ${\mathcal {U}}$ be the permutation model determined by A, G and I.

7.2 Versions of ${\textsf{AC}}$ in ${\mathcal {U}}$

Theorem 8

In ${\mathcal {U}}$, ${{\textsf{LW}}}$ and ${{\textsf{DC}}}_{\xi }$ for all infinite cardinals $\xi < \aleph _{\alpha + 1}$ are true, but ${{\textsf{MC}}}^{{{\textsf{WO}}}}_{{{\textsf{WO}}}}$ is false.

Proof

By Theorem 2, ${\mathcal {U}}\models {\textsf{LW}}$. Furthermore, the well orderable family ${\mathcal {B}}=\{B_{\beta }:\beta <\aleph _{\alpha +1}\}$, which is in ${\mathcal {U}}$ (any permutation of A in ${\mathscr {H}}$ fixes ${\mathcal {B}}$ pointwise), and which comprises sets that are well orderable in ${\mathcal {U}}$ (for every $\beta <\aleph _{\alpha +1}$, $B_{\beta }$ is a support of each of its elements), has no multiple choice function in ${\mathcal {U}}$, and thus ${\textsf{MC}}^{{\textsf{WO}}}_{{\textsf{WO}}}$ is false in ${\mathcal {U}}$. The fact that ${\textsf{DC}}_{\xi }$ is true in ${\mathcal {U}}$ for all infinite cardinals $\xi <\aleph _{\alpha +1}$ can be established as in the proof of Jech’s Lemma 8.4 (p. 123) in [7]. $\square$

Remark 4

As with the model ${\mathcal {N}}15$, $\forall {\mathfrak {m}},2{\mathfrak {m}}={\mathfrak {m}}$ is false in ${\mathcal {U}}$.

8 Model 6: ${\mathcal {N}}9$

8.1 Motivation

As mentioned in Sect. 1, $\forall {\mathfrak {m}},2{\mathfrak {m}}={\mathfrak {m}}$ does not imply ${\textsf{MC}}^{\aleph _{0}}_{\aleph _{0}}$ in ${\textsf{ZF}}$; in Sageev’s ${\textsf{ZF}}$-model ${\mathcal {M}}6$ of [5], $\forall {\mathfrak {m}},2{\mathfrak {m}}={\mathfrak {m}}$ is true, but there is a countably infinite family of countably infinite sets of reals without a choice function in the model (see [9]). Thus ${\textsf{MC}}^{\aleph _{0}}_{\aleph _{0}}$ is false in ${\mathcal {M}}6$. Furthermore, ${\textsf{LW}}$ is also false in ${\mathcal {M}}6$, since ${\textsf{LW}}$ is equivalent to ${\textsf{AC}}$ in ${\textsf{ZF}}$.

It is an intriguing open problem whether ${\textsf{LW}}+ \forall {\mathfrak {m}}, 2{\mathfrak {m}}={\mathfrak {m}}$ (which is stronger than ${\textsf{LW}}+ {\textsf{DF}}= {\textsf{F}}$ in ${\textsf{ZFA}}$) implies ${\textsf{MC}}^{\aleph _{0}}_{\aleph _{0}}$ in ${\textsf{ZFA}}$. In this direction, note that the Halpern/Howard permutation model ${\mathcal {N}}9$ in [5] satisfies $\forall {\mathfrak {m}}, 2{\mathfrak {m}}={\mathfrak {m}}$ (and thus satisfies ${\textsf{DF}}= {\textsf{F}}$, see [3, Theorem 2.3]); however, the status of ${\textsf{LW}}$, ${\textsf{MC}}^{\aleph _{0}}_{\aleph _{0}}$, and ${\textsf{MC}}^{\aleph _{0}}$ in ${\mathcal {N}}9$, were open problems until now. We settle these problems by showing next that ${\textsf{LW}}$ and ${\textsf{AC}}^{{\textsf{WO}}}_{{\textsf{WO}}}$ are both true in ${\mathcal {N}}9$, whereas ${\textsf{MC}}^{\aleph _{0}}$ is false in ${\mathcal {N}}9$.

8.2 The description and basic properties of ${\mathcal {N}}9$

We start with a model M of ${\textsf{ZFA}}+{\textsf{AC}}$ with a set A of atoms which has the structure of the set

$$\begin{aligned} \omega ^{(\omega )}=\{s:s:\omega \rightarrow \omega \wedge (\exists n\in \omega ) (\forall j>n)(s_{j}=0)\}. \end{aligned}$$

We identify A with the latter set to simplify the description of the group G.

For $s\in A$, the pseudo length of s is the least natural number k such that for all $\ell \ge k$, $s_{\ell }=0$. A subset of A is called bounded there is an upper bound for the pseudo lengths of the elements of A. G is the group of all permutations $\phi$ of A such that the support of $\phi$, $\{a \in A : \phi (a) \ne a \}$, is bounded.

For every $s\in A$ and every $n\in \omega$, let

$$\begin{aligned} A_{s}^{n}=\{t\in A:\forall j\ge n(t_{j}=s_{j})\}. \end{aligned}$$

Definition 4

(Mostly from [4]) Assume $s \in A$ and $n \in \omega$; then

• 1. $A_{s}^{n}$ is called the n-block containing s.

• 2. For any $t \in A_{s}^{n}$, the n-block code of t is the sequence

  $$\begin{aligned} (t_n, t_{n+1}, t_{n+2}, \ldots ) = (s_n, s_{n+1}, s_{n+2}, \ldots ). \end{aligned}$$

  The n-block code of $A_{s}^{n}$ is the n-block code of any of its elements. We will denote the n-block code of an element $t\in A$ or an n-block B by $\textrm{bc}^{n}(t)$ or $\textrm{bc}^{n}(B)$, respectively.

• 3. For any $t \in A_{s}^{n}$, the finite sequence $(t_0, t_1, t_2, \ldots , t_{n-1}) = t \upharpoonright n$ is called the n-location of t (in $A_{s}^{n}$).

Note the following

• 1. $A_{0}^n$ is the set of all elements of A with pseudo length less than or equal to n. (In the expression $A_{0}^n$, 0 denotes the constant sequence all of whose terms are 0.)

• 2. For $s\in A$ and $n,m\in \omega$ with $n\le m$, $A_{s}^{n}\subseteq A_{s}^{m}$.

• 3. If $n \le m$, B is an n-block, $B'$ is an m-block and $B \cap B' \ne \emptyset$ then $B \subseteq B'$. (This follows from the previous item.)

• 4. Any $t \in A$ is the concatenation $(t \upharpoonright n)^{\frown } \textrm{bc}^n(t)$ of the n-location of t and the n-block code t.

For each $n \in \omega$, $G_n$ is the subgroup of G consisting of all permutations $\phi \in G$ such that

• 1. $\phi$ fixes $A^n_0$ pointwise,

• 2. $\phi$ fixes the set of n-blocks, that is, $A_{s}^{n} = A_{t}^{n}$ if and only if $A_{\phi (s)}^{n} = A_{\phi (t)}^{n}$,

• 3. for each $s \in A$, the n-location of $\phi (s)$ is the same as the n-location of s.

(Note that if $n\le m$, then $G_{m}\subseteq G_{n}$.) J is the filter of subgroups of G generated by the groups $G_n$, $n \in \omega$. That is, $H \in J$ if and only if H is a subgroup of G and there exists $n \in \omega$ such that $G_n \subseteq H$. It is shown in [4] that J is a normal filter, that is, closed under conjugation. ${\mathcal {N}}9$ is the Fraenkel–Mostowski model of ${\textsf{ZFA}}$ which is determined by M, G, and J.

Lemma 5

Assume that f is a one-to-one function from a subset of A into A and n is a natural number such that

• 1. $A_{0}^{n}$ is a subset of the domain of f and f fixes $A_{0}^{n}$ pointwise.

• 2. The domain of f is the union of n-blocks and $f(A_{s}^{n})$ is an n-block for any n-block $A_{s}^{n}$ contained in the domain of f.

• 3. f fixes n-locations.

• 4. The domain and range of f are bounded. That is, there is an upper bound for the pseudo-lengths of the elements of $\textrm{dom}(f) \cup \textrm{ran}(f)$.

Then there is a $\phi \in G_n$ that extends f.

Proof

By assumption (4) there is an $m \in \omega$ such that $\textrm{dom}(f) \cup \textrm{ran}(f) \subseteq A_{0}^{m}$. If we let

$$\begin{aligned}&{\mathcal {B}}_0 = \{ B : B \text{ is } \text{ an } n\text{-block } \text{ and } B \subseteq A_{0}^{m} \} \\&{\mathcal {B}}_1 = \{ B : B \text{ is } \text{ an } n\text{-block } \text{ and } B \subseteq A_{0}^{m + 1} \} \\&{\mathcal {B}}_2 = \{ B : B \text{ is } \text{ an } n\text{-block } \text{ and } B \subseteq \textrm{dom}(f) \} \\&{\mathcal {B}}_3 = \{ B : B \text{ is } \text{ an } n\text{-block } \text{ and } B \subseteq \textrm{ran}(f) \} \end{aligned}$$

then, since ${\mathcal {B}}_2 \subseteq {\mathcal {B}}_0$ and ${\mathcal {B}}_1 \setminus {\mathcal {B}}_0$ is countably infinite, we have ${\mathcal {B}}_1 \setminus {\mathcal {B}}_2$ is countably infinite. Similarly, ${\mathcal {B}}_1 \setminus {\mathcal {B}}_3$ is countably infinite. Let G be a one-to-one function from ${\mathcal {B}}_1 \setminus {\mathcal {B}}_2$ onto ${\mathcal {B}}_1 \setminus {\mathcal {B}}_3$ and define $F: A_{0}^{m+1} \rightarrow A_{0}^{m+1}$ by

$$\begin{aligned} F(s) = {\left\{ \begin{array}{ll} f(s) &{} \text{ if } s \in \textrm{dom}(f); \\ \text{ the } \text{ element } \text{ of } G(A_{s}^n\text{) } \text{ with } &{} \\ \quad \text{ the } \text{ same }\ n\text {-location as}\ s &{} \text{ otherwise. } \end{array}\right. } \end{aligned}$$

Then F is a permutation of $A_{0}^{m+1}$ which extends f and satisfies conditions (1), (2) and (3) of the lemma. Therefore the function $\phi$ defined by

$$\begin{aligned} \phi (s) = {\left\{ \begin{array}{ll} F(s) &{} \text{ if } s \in A_{0}^{m+1}; \\ s &{} \text{ if } s \in A \setminus A_{0}^{m+1}. \end{array}\right. } \end{aligned}$$

extends f and is in $G_n$. ($\phi$ is bounded because the support of $\phi$ is a subset of $A_{0}^{m+1}$.) $\square$

Theorem 9

If $(X,\le )$ is a well ordered set of non-empty well orderable sets in ${\mathcal {N}}9$ and $G_n$ is a support of $(X,\le )$ where $n > 0$, then for every $y \in X$, $G_{n+1}$ fixes y pointwise.

Proof

Assume that $(X,\le )$ is a well ordered set with support $G_n$ and that $y \in X$ and $x \in y$. We prove the theorem by arguing by contradiction that for all $\beta \in G_{n+1}$, $\beta (x) = x$. Assume $\beta \in G_{n+1}$ and $\beta (x) \ne x$. $G_n$ fixes y since $G_n$ fixes X pointwise and since y is well orderable in ${\mathcal {N}}9$ there is $k > n$ such that $G_k$ fixes y pointwise. Therefore for any $\rho \in G_n$, $\rho (x) \in y$, and hence for all $\alpha \in G_k$, $\alpha (\rho (x)) = \rho (x)$. This contradicts the following lemma.

Lemma 6

Assume $x \in {\mathcal {N}}9$, n is a positive integer, and there exists $\beta \in G_{n+1}$ such that $\beta (x) \ne x$. Then for all $k \ge n$, there are $\rho \in G_n$ and $\alpha \in G_k$ such that $\alpha (\rho (x)) \ne \rho (x)$.

Proof

Assuming the hypotheses, then for $k = n$ or for $k = n+1$ we can take $\rho = \epsilon$, the identity permutation on A and $\alpha = \beta$. We will prove the lemma for $k = n+2$. The lemma will then follow by mathematical induction.

Since $\beta \in G$ there is an integer j such that the support of $\beta$ is a subset of $A_{0}^{j}$. We have assumed that $\beta \in G_{n+1}$ so $\beta$ fixes $A_{0}^{n+1}$ pointwise. Therefore there is an atom $s \notin A_{0}^{n+1}$ moved by $\beta$. From this we conclude that $j > n+1$.

The plan of the proof is to get an element $\rho$ of $G_n$ which takes each $n+1$-block contained in $A_{0}^{j}$ to an $n+2$-block. We will also make sure that if s and $s'$ have the same $n+1$-location then $\rho (s)$ and $\rho (s')$ have the same $n+2$-location. Then $\alpha$ will be defined so that it acts on $n+2$ blocks contained in $A_{0}^{j+1}$ by mirroring the action of $\beta$ on $n+1$-blocks contained in $A_{0}^{j}$. That is, $\alpha$ will be $\rho \beta \rho ^{-1}$.

Fix a bijection $i \mapsto (u_1(i), u_2(i))$ from $\omega$ onto $\omega \times \omega$ so that

$$\begin{aligned} u_1(0) = u_2(0) = 0. \end{aligned}$$

(6)

Define $f: A_{0}^{j} \rightarrow A_{0}^{j+1}$ by

$$\begin{aligned}&f(s_0, s_1, \ldots , s_{n-1}, s_n, s_{n+1}, \ldots , s_{j-1}, 0, 0, \ldots ) \\&\quad =(s_0, s_1, \ldots , s_{n-1}, u_1(s_n), u_2(s_n), s_{n+1}, \ldots , s_{j-1}, 0, 0, \ldots ) \end{aligned}$$

That is,

$$\begin{aligned} (f(s))_i = {\left\{ \begin{array}{ll} s_i &{} \text{ if } 0 \le i \le n-1; \\ u_1(s_n) &{} \text{ if } i = n; \\ u_2(s_n) &{} \text{ if } i = n+1; \\ s_{i-1} &{} \text{ if } i > n+1. \end{array}\right. } \end{aligned}$$

(But the first form is easier to work with.)

Using the definition, we see that f has the following properties.

• 1. f is a bijection from $A_{0}^{j}$ onto $A_{0}^{j+1}$ (because the function $i \mapsto (u_1(i), u_2(i))$ is a bijection from $\omega$ onto $\omega \times \omega$).

• 2. f fixes $A_{0}^{n}$ pointwise (using Eq. 6).

• 3. If s and t are in A and $\textrm{bc}^n(s) = \textrm{bc}^n(t)$ then $\textrm{bc}^n(f(s)) = \textrm{bc}^n(f(t))$.

• 4. For $s \in \textrm{dom}(f) = A_{0}^{j}$, the n-location of s is

  $$\begin{aligned} (s_0, s_1, \ldots , s_{n-1}) \end{aligned}$$

  which is the same as the n-location of f(s).

• 5. For $s \in \textrm{dom}(f)$, $\textrm{bc}^{n+2}(f(s)) = (s_{n+1}, s_{n+2}, \ldots , s_{j-1}, 0, 0, \ldots ) = \textrm{bc}^{n+1}(s)$.

• 6. For $s \in \textrm{dom}(f)$, the $n+2$-location of f(s) is

  $$\begin{aligned} (s_0, s_1, \ldots , s_{n-1}, u_1(s_n), u_2(s_n)). \end{aligned}$$

• 7. $\textrm{bc}^{n+2}(f(A_{0}^{n+1})) = \textrm{bc}^{n+1}(A_{0}^{n+1}) = (0, 0, 0, \ldots )$ (by Eq. (6).

Using items (2), (3) and (4) above we see that f satisfies conditions (1), (2) and (3) of the hypotheses of Lemma 5. Further, condition (4) is satisfied since $\textrm{dom}(f) \cup \textrm{ran}(f) \subseteq A_{0}^{j+1}$. Applying the lemma we obtain a $\rho \in G_n$ that extends f.

Let $\alpha = \rho \beta \rho ^{-1}$. To complete the proof, we need to argue that $\alpha (\rho (x)) \ne \rho (x)$ and that $\alpha \in G_{n+2}$. For the first of these we note that $\alpha (\rho (x)) = \rho (\beta (\rho ^{-1}(\rho (x)))) = \rho (\beta (x))$. If this is equal to $\rho (x)$ we conclude that $\beta (x) =x$ which contradicts our assumptions that $\beta (x) \ne x$.

For the proof that $\alpha \in G_{n+2}$ we will need the following sublemma.

Sublemma 1

Assume $s \in A$. Then,

• 1. If $A_{s}^{n+1} \subseteq A_{0}^{j}$ then $f(A_{s}^{n+1}) = A_{s}^{n+2}$.

• 2. If $s \not \in A_{0}^{j+1}$ then $\alpha (s) = s$.

Proof

For part (1) assuming that $A_{s}^{n+1} \subseteq A_{0}^{j}$. It follows from (5) in the list of properties of f that $f(A_{s}^{n+1}) \subseteq A_{f(s)}^{n+2}$ From this we conclude that $A_{f(s)}^{n+2} \cap A_{0}^{j+1} \ne \emptyset$ (since $\textrm{ran}(f) = A_{0}^{j+1}$). Since $n+2 < j+1$ we apply item (3) in the list following Definition 4 to conclude that $A_{f(s)}^{n+2} \subseteq A_{0}^{j+1} = \textrm{ran}(f)$. To show that every element of $A_{f(s)}^{n+2}$ is in $f(A_{s}^{n+1})$ assume $t \in A_{f(s)}^{n+2}$. By the previous remark, $t \in \textrm{ran}(f)$ so $t = f(s')$ for some $s' \in A_{0}^{j}$. Since t and f(s) are in the same $n+2$-block, $\textrm{bc}^{n+2}(f(s')) = \textrm{bc}^{n+2}(t) = \textrm{bc}^{n+2}(f(s))$. By item (5) in the list of properties of f, we have

$$\begin{aligned} \textrm{bc}^{n+1}(s') = \textrm{bc}^{n+2}(f(s')) = bc^{n+2}(f(s)) = bc^{n+1}(s) \end{aligned}$$

so $s'$ and s are in the same $n+1$-block, namely $A_{s}^{n+1}$. Hence, $t = f(s') \in f(A_{s}^{n+1})$.

For part (2) we assume $s \not \in A_{0}^{j+1}$. Since, $\textrm{ran}(f) =A_{0}^{j+1}$ and $\rho$ extends f (and is a permutation of A), $\rho ^{-1}(s) \not \in A_{0}^{j}$, and hence $\beta (\rho ^{-1}(s)) =\rho ^{-1}(s)$. Therefore

$$\begin{aligned} \alpha (s) = \rho (\beta (\rho ^{-1}(s))) = \rho (\rho ^{-1}(s)) = s. \end{aligned}$$

This completes the proof of the sublemma. $\square$

To prove $\alpha \in G_{n+2}$, we argue that conditions (1), (2) and (3) in the definition of $G_n$ are true (with n replaced by $n+2$).

• Condition (1) is the requirement that $\alpha$ fixes $A_{0}^{n+2}$ pointwise. If $s \in A_{0}^{n+2}$ then $s \in A_{0}^{j+1}$ so $s \in \textrm{ran}(f)$. Therefore $\rho ^{-1}(s) = f^{-1}(s) \in A_{0}^{n+1}$ (using the sublemma, item (1) ). Therefore, since $\beta \in G_{n+1}$, $\beta (\rho ^{-1}(s)) = \rho ^{-1}(s)$. We conclude that $\alpha (s) = \rho (\beta (\rho ^{-1}(s))) = s$.

• For condition (2) we must show that for any $n+2$-block $B = A_{s}^{n+2}$, $\alpha (B)$ is an $n+2$-block. Since $j + 1 > n + 2$ (see the remark in the second paragraph of the proof of the lemma.), every $n+2$-block is either contained in $A \setminus A_{0}^{j+1}$ or contained in $A_{0}^{j+1}$. In the first case part (2) of the sublemma gives us $\alpha (B) = B$. In the second case $B \subseteq \textrm{ran}(f)$ so $\rho ^{-1}(B) = f^{-1}(B)$ which by the sublemma part (1) is an $n+1$-block contained in $A_{0}^{j}$. Since the support of $\beta$ is a subset of $A_{0}^{j}$ and $\beta \in G_{n+1}$, $\beta (\rho ^{-1}(B))$ is an $n+1$ block contained in $A_{0}^{j}$. Applying part (1) of the sublemma again we conclude that $f(\beta (\rho ^{-1}(B)))$ is an $n+2$-block. Therefore, since $\rho$ extends f, $\rho (\beta (\rho ^{-1}(B)))$ is an $n+2$-block. So $\alpha (B)$ is an $n+2$ block.

• To prove condition (3) we assume $t \in A$ and argue that the $n+2$-location of $\alpha (t)$ is the same as the $n+2$-location of t. If $t \not \in A_{0}^{j+1}$ then the conclusion follows from part (2) of the sublemma. If $t \in A_{0}^{j+1}$ then $t \in \textrm{ran}(f)$ so $\rho ^{-1}(t) = f^{-1}(t) = s$ for some $s \in A_{0}^{j} = \textrm{dom}(f)$. By item (6) in the list of properties of f, the $n+2$-location of t is $(s_0, s_1, \ldots , s_{n-1}, u_1(s_n),u_2(s_n))$ and the $n+1$-location of $\rho ^{-1}(t) = f^{-1}(t)$ is $(s_0, s_1, \ldots , s_{n})$. Since $\beta$ fixes $n+1$-locations, the $n+1$ location of $\beta (\rho ^{-1}(t))$ is $(s_0, s_1, \ldots , s_{n})$. By item (6) in the list of properties of f, the $n+2$-location of $f(\beta (\rho ^{-1}(t))) = \rho (\beta (\rho ^{-1}(t))) = \alpha (t)$ is $(s_0, s_1, \ldots , u_1(s_n), u_2(s_n))$.

This completes the proof of the lemma. $\square$

The lemma gives a contradiction and therefore the proof of the theorem is complete. $\square$

8.3 Versions of ${\textsf{AC}}$ in ${\mathcal {N}}9$

Theorem 10

In ${\mathcal {N}}9$, the union of a well-ordered collection of well orderable sets can be well ordered and ${{\textsf{AC}}}^{{{\textsf{WO}}}}_{{{\textsf{WO}}}}$, ${{\textsf{LW}}}$ and $\forall {\mathfrak {m}}, 2{\mathfrak {m}} = {\mathfrak {m}}$ are true, but the axiom of choice for families of two-element sets and ${{\textsf{MC}}}^{\aleph _{0}}$ are false.

Proof

In [4], $\forall {\mathfrak {m}}, 2{\mathfrak {m}} = {\mathfrak {m}}$ was shown to be true in ${\mathcal {N}}9$ and the axiom of choice for families of two-elements sets was shown to be false.

The fact that the union of a well-ordered collection of well orderable sets can be well ordered follows from Theorem 9 and ${\textsf{AC}}^{{\textsf{WO}}}_{{\textsf{WO}}}$ follows from this. (In [4], ${\textsf{AC}}^{{\textsf{WO}}}_{{\textsf{fin}}}$ was shown to be true in ${\mathcal {N}}9$.)

Claim

${\textsf{LW}}$ is true in ${\mathcal {N}}9$.

Proof

We will show that ${\mathcal {N}}9$ satisfies condition (*) of Theorem 1. To this end, let $x\in {\mathcal {N}}9$ and also let $n\in \omega$ such that $G_{n}$ does not support x. (Recall that $\{G_{n}:n\in \omega \}$ is a filter base for the filter J used to construct ${\mathcal {N}}9$, see Definition 4.) Then there exists $\eta \in G_n$ such that $\eta (x) \ne x$.

Since $\eta \in G_n$, the set $\eta ' = \{ (A_{s}^{n}, A_{\eta (s)}^{n}) : s \in A \}$ is a permutation of the set of n-blocks. (See item 2 in the definition of $G_n$.) Since $\eta$ also fixes n-locations, for any $s \in A$,

$$\begin{aligned} \eta (s) = (s \upharpoonright n)^{\frown }\textrm{bc}(\eta '(A_{s}^{n})). \end{aligned}$$

By Lemma 1, there is a permutation $\tau '$ of the set of n-blocks such that

• 1. $\{ B : \tau '(B) \ne B \} \subseteq \{ B : \eta '(B) \ne B \}$,

• 2. $(\tau ')^2 = \epsilon$, and

• 3. $(\eta ' \tau ')^2 = \epsilon$.

$\tau '$ determines a permutation $\tau$ of A defined by

$$\begin{aligned} \tau (s) = (s \upharpoonright n)^{\frown }\textrm{bc}(\tau '(A_{s}^{n})). \end{aligned}$$

Then $\tau$ has the following properties:

• 1. $\tau \in G_n$

• 2. $\tau ^2 = \epsilon$ and

• 3. $(\eta \tau )^2 = \epsilon$

Since $\eta (x)\ne x$, we have that either $\tau (x) \ne x$ or $\eta \tau (x) \ne x$, and since both $\tau$ and $\eta \tau$ are in $G_{n}$ and have finite order, we conclude that (*) is satisfied. Hence, by Theorem 1, ${\textsf{LW}}$ is true in ${\mathcal {N}}9$.

Claim

${\textsf{MC}}^{\aleph _{0}}$ is false in ${\mathcal {N}}9$.

Proof

For each $n \in \omega$, let ${\mathcal {C}}_n$ be the set of n-blocks and let $W_n = \{ \phi ({\mathcal {C}}_n) : \phi \in G \}$. Then each $W_n$ is supported by G and therefore ${\mathcal {W}} = \{ W_n : n \in \omega \}$ is a countably infinite set in ${\mathcal {N}}9$. We will show by contradiction that ${\mathcal {W}}$ has no multiple choice function in ${\mathcal {N}}9$.

Therefore assume that f is such a function which is in ${\mathcal {N}}9$ with support $G_n$. By our assumptions $f(W_{n+1})$ is a finite non-empty subset of $W_{n+1}$ with support $G_n$. Choose $x \in f(W_{n+1})$ then $x = \gamma ({\mathcal {C}}_{n+1})$ for some $\gamma \in G$. Since the support of $\gamma$ is bounded, there is a $k \in \omega$ such that the support of $\gamma$, $\{ a \in A : \gamma (a) \ne a \}$, is a subset of $A^{k}_{0}$ and we assume without loss of generality that $k \ge n+1$. It follows that for any $n+1$-block B, either B is disjoint from $A^{k}_{0}$ or $B \subseteq A^{k}_{0}$. Therefore ${\mathcal {C}}_{n+1}$ is the disjoint union

$$\begin{aligned} {\mathcal {C}}_{n+1} = \{ B \in {\mathcal {C}}_{n+1} : B \cap A^{k}_{0} = \emptyset \} \cup \{ B \in {\mathcal {C}}_{n+1} : B \subseteq A^{k}_{0} \}. \end{aligned}$$

For B in the first of these two sets $\gamma (B) = B$. So $x= \gamma ({\mathcal {C}}_{n+1})$ is the disjoint union

$$\begin{aligned} x&= \{ B \in {\mathcal {C}}_{n+1} : B \cap A^{k}_{0} = \emptyset \} \end{aligned}$$

(7)

$$\begin{aligned}&\quad \cup \{ \gamma (B) : B \in {\mathcal {C}}_{n+1} \text{ and } B \subseteq A^{k}_{0} \}. \end{aligned}$$

(8)

The first of the two sets above ($\{ B \in {\mathcal {C}}_{n+1} : B \cap A^{k}_{0} = \emptyset \}$) is infinite so we choose a countably infinite subset $B_i$, $i \in \omega$ where $B_i \ne B_j$ if $i \ne j$. (It is possible to choose $\{ B_i : i \in \omega \}$ so that this set is countable in the ${\mathcal {N}}9$ but for our purposes this is not required.) We now choose for each $i \in \omega$ an n-block $D_i$ which is a subset of the $n+1$-block $B_{i}$. For each $i \in \omega$ with $i \ge 1$, we let $\psi _i$ be the element of $G_n$ which interchanges the two n-blocks $D_0$ and $D_i$ and fixes all other atoms. (That is, $\psi _i$ is the product of transpositions $\prod _{s \in D_0} (s,s_i)$ where for each $s \in D_0$, $s_i$ is the element of $D_i$ with the same n-location as s.)

We note two things about $\psi _i$:

• Since $f(W_{n+1})$ is supported by $G_n$, all of the sets $\psi _i(x)$ ($i \ge 1$) are in $f(W_{n+1})$.

• $\psi _i$ fixes $A^{k}_{0}$ pointwise and also fixes every element of $\{B \in {\mathcal {C}}_{n+1} : B \cap A^{k}_{0} = \emptyset \} \setminus \{B_0, B_i \}$ pointwise. Therefore, using Eq. (7), $\psi _i(x)$ is the disjoint union

  $$\begin{aligned} \psi _i(x)&= \{B :B \in {\mathcal {C}}_{n+1} \text{ and } B \cap A^{k}_{0} \} \setminus \{B_0, B_i \} \\&\cup \{ \psi _i(B_0), \psi _i(B_i) \} \\ {}&\cup \{ \gamma (B) : B \in {\mathcal {C}}_{n+1} \text{ and } B \subseteq A^{k}_{0} \}. \end{aligned}$$

$\psi _i(B_0)$ is the $n+1$-block $B_0$ with the sub-n-block $D_0$ replaced by the n-block $D_i$ and $\psi _i(B_i)$ is the $n+1$-block $B_i$ with the sub-n-block $D_i$ replaced by the n-block $D_0$. Therefore neither $B_0$ nor $B_i$ are in $\psi _i(x)$.

Assume that $k, j \in \omega$ are both greater than zero and that $k \ne j$. Then (among other differences) $B_k \in \psi _j(x) \setminus \psi _k(x)$ so that $\psi _k(x) \ne \psi _j(x)$. Since all of the sets $\psi _i(x)$ ($i > 0$) are in $f(W_{n+1})$ and $f(W_{n+1})$ is finite, we have a contradiction. This completes the proof of the claim. $\square$

The above arguments complete the proof of the theorem. $\square$

9 Summary

Table 2 summarizes what is known (and unknown) about our six models.

Table 2 Forms of AC in our models

10 Open questions

• 1. Does ${\textsf{LW}}+ \forall {\mathfrak {m}},2{\mathfrak {m}}={\mathfrak {m}}$ imply ${\textsf{MC}}^{\aleph _{0}}_{\aleph _{0}}$ in ${\textsf{ZFA}}$?

• 2. Does $\forall {\mathfrak {m}},2{\mathfrak {m}}={\mathfrak {m}}$ imply ${\textsf{AC}}^{{\textsf{WO}}}_{{\textsf{fin}}}$? (Recall that $\forall {\mathfrak {m}},2{\mathfrak {m}}={\mathfrak {m}}$ $\Rightarrow$ ${\textsf{DF}}= {\textsf{F}}$ $\Rightarrow$ ${\textsf{AC}}^{\aleph _{0}}_{{\textsf{fin}}}$, where ${\textsf{AC}}^{\aleph _{0}}_{{\textsf{fin}}}$ is the axiom of choice for countably infinite families of non-empty finite sets.)

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 $y \in {\mathcal {M}}$. $\square$