Contents

foundations

# Contents

## Idea

It can be shown via $\epsilon$-induction that any model of ZF has no non-trivial automorphisms. However, if we have a model $V$ of ZFA with a set of atoms $A$, then any permutation of the atoms in $A$ gives rise to a non-trivial automorphism of $V$.

We can then look at the submodel of $V$ consisting of sets that are (hereditarily) closed under “most” permutations of $V$ (in a sense made precise by a normal filter), and it turns out this gives a transitive model? of ZFA. In some sense, imposing the symmetry condition above causes the atoms to be “indistinguishable”, and thus often causes the axiom of choice to fail, since there is no uniform way of picking out atoms.

By tweaking the choice of permutations, we can produce models in which some weak version of choice holds (such as the boolean prime ideal theorem), while full (or even countable) choice fails, and thus proving their independence (in ZFA).

Two such examples are the basic Fraenkel model and the second Fraenkel model which gave the first examples of the independence of the axiom of choice from a set theory (I think!).

## Definition

Let $V$ be a model of $ZFA$, and let $A$ be its set of atoms. Then any permutation of $A$ induces a unique non-trivial automorphism of $V$.

We pick a subgroup $G \leq Aut(A)$ of permutations of the atoms, and let $\mathcal{F}$ be a normal filter of subgroups of $G$, ie. a collection of subgroups such that

• $\mathcal{F}$ is non-empty.

• If $H, K \in \mathcal{F}$, then $H \cap K \in \mathcal{F}$.

• If $H \in \mathcal{F}$ and $K \geq H$, then $K \in \mathcal{F}$.

• If $H \in \mathcal{F}$ and $g \in G$, then $gHg^{-1} \in \mathcal{F}$.

We say $x$ is ($\mathcal{F}$-)symmetric if its stabilizer $stab(x) \in \mathcal{F}$. We let $\tilde{V}$ be the class of hereditarily symmetric sets.

We usually pick $\mathcal{F}$ so that for each atom $a \in A$, the singleton $\{a\}$ is symmetric. Otherwise, the non-symmetric elements would never occur in the class $\tilde{V}$.

###### Theorem

The class $\tilde{V}$ is a model of ZFA.

## Category-theoretic approach

From a modern, structural foundations point of view, we can construct these models by viewing $G \leq Aut(A)$ as a topological group, where the topology is generated by the normal filter of subgroups $\mathcal{F}$. We then construct the category whose objects are sets with an action by an open subgroup $H \le G$ (where $H$ is allowed to vary), and morphisms are functions $f\colon X\to Y$ which are equivariant for an open subgroup $L \le H \cap K$, where $H$ and $K$ act on $X$ and $Y$ respectively. The resulting category is in fact a Boolean topos.

The construction outlined at pure set can be varied to construct well-founded pure sets using atoms [:which I believe they are elements of the -sets.]

While this approach more-or-less translates the construction of the permutation model directly, the resulting topos in general fails to be (externally) complete. In particular, we cannot construct the coproduct of an infinite family $\{X_i\}_{i \in I}$ (at least not in the obvious way), since given functions $f_i: X_i \to A$, we cannot guarantee that the combined map $(\coprod_i X_i) \to A$ has an open stabilizer, as the infinite intersection of open sets need not be open.

Instead, we can consider the logical subtopos consisting of sets with a full $G$-action, and morphisms are the functions that are $G$-equivariant. This gives the topos of continuous G-sets.

Fraenkel-Mostowski Models are discussed in

• Ulrich Felgner, Models of ZF-Set Theory, Lecture Notes in Mathematics 223 (1971)

• Thomas J. Jech, The Axiom of Choice (1973)

The following papers discuss how we can deduce properties of the permutation model from properties of the corresponding categories of $G$ sets:

• Michael Fourman, Sheaf models for set theory (pdf)

• Andreas Blass, Andre Scedrov, Complete topoi representing models of set theory (pdf)