basic Fraenkel model


The basic Fraenkel model is a model of the set theory ZFA that doesn’t satisfy the axiom of choice. It was one of the first examples of a permutation model of set theory. In the basic Fraenkel model, we have a countable set of atoms AA, and we take the full permutation group GG of AA. The normal filter of subgroups is generated by the stabilizers of finite subsets of AA.

The second Fraenkel model is similar, but uses the countable group (/2) (\mathbb{Z}/2\mathbb{Z})^\mathbb{N}.


Material set theory

In the language of material set theory, this is just a Fraenkel-Mostowski model given by a countable set of atoms AA with the full permutation group GG acting on it. The normal filter of subgroups is generated by the stabilizers of finite subsets of AA, ie. groups of the form

Stab(K)={gG:xK,gx=x}, Stab (K) = \{ g \in G: \forall x \in K, gx = x\},

where KK is a finite subset of AA.

Structural set theory

We can take the same group GG as above, with topology generated by the stabilizers Stab(K)Stab (K). We can then form the topos of continuous G-sets, whose objects are sets with a continuous action of GG (with the set given the discrete topology), and morphisms are the GG-equivariant maps. This topos is also known as the Schanuel topos.

The internal logic of this topos can be identified with the standard logic of the material model of ZFA constructed above, if we interpret the universal quantifiers in ZFA suitably (eg. via stack semantics, or other semantics as mentioned in Fraenkel-Mostowski models)


It is clear that the set of atoms in the Fraenkel-Mostowski model cannot be linearly ordered. So the ordering principle (that every set can be linearly ordered) fails. Consequently, the axiom of choice, boolean prime ideal theorem etc. all fail. In fact, even countable choice fails.

We can consider the object 𝒜\mathcal{A} consisting of the set of atoms with the obvious action of GG. It can be shown that this does not biject with any finite set (ie. a set that is a finite coproduct of the terminal object), but any injection 𝒜𝒜\mathcal{A} \to \mathcal{A} must be a surjection (this is true both as an external statement or as a statement in the internal logic). So this is an example of a Dedekind-finite but not finite set.


We can consider the case where we have more atoms, or allow stabilizers of larger subsets. In different cases, weak versions of Choice may hold. For example, if we have 1\aleph_1 many atoms, and take the normal filter generated by stabilizers of countable subsets, then the axiom of choice holds for any well-ordered family of sets. Details can be found in the book Consequences of the Axiom of Choice.


The model is given by 𝒩1\mathcal{N}1 in the book

  • Paul Howard, Jean E. Rubin, Consequences of the Axiom of Choice, AMS Mathematical Surveys and Monographs 59

The various properties of the model are listed in detail. Variations of the model are listed as 𝒩12\mathcal{N}12 and 𝒩16\mathcal{N}16.

Revised on September 28, 2016 12:20:12 by Dexter Chua (