nLab
Fraenkel-Mostowski model

Contents

Contents

Idea

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

We can then look at the submodel of VV consisting of sets that are (hereditarily) closed under “most” permutations of VV (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 VV be a model of ZFAZFA, and let AA be its set of atoms. Then any permutation of AA induces a unique non-trivial automorphism of VV.

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

  • \mathcal{F} is non-empty.

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

  • If HH \in \mathcal{F} and KHK \geq H, then KK \in \mathcal{F}.

  • If HH \in \mathcal{F} and gGg \in G, then gHg 1gHg^{-1} \in \mathcal{F}.

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

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

Theorem

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

Not a proof

See Felgner, Chapter III.B.

Category-theoretic approach

From a modern, structural foundations point of view, we can construct these models by viewing GAut(A)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 HGH \le G (where HH is allowed to vary), and morphisms are functions f:XYf\colon X\to Y which are equivariant for an open subgroup LHKL \le H \cap K, where HH and KK act on XX and YY 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} iI\{X_i\}_{i \in I} (at least not in the obvious way), since given functions f i:X iAf_i: X_i \to A, we cannot guarantee that the combined map ( iX i)A(\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 GG-action, and morphisms are the functions that are GG-equivariant. This gives the topos of continuous G-sets.

References

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 GG sets:

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

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

Last revised on July 15, 2016 at 10:25:19. See the history of this page for a list of all contributions to it.