Zermelo–Fraenkel set theory with atoms


ZFA is a variant of the material set theory ZF which allows for objects, called atoms or urelements, which may be members of sets, but which themselves have no members (and are not the empty set either). There is a version ZFCA (analogous to ZFC) which includes the axiom of choice (AC). ZFA featured in early independence proofs, notably Fraenkel–Mostowski permutation models, for example showing AC is independent of the rest of the axioms of ZFA.

Zermelo?’s original 1908 axiomatisation of set theory included atoms, but they were soon discarded as a foundational approach as they could be modeled inside of atomless set theory.

Category-theoretic approach

From a modern, structural foundations point of view, one can build models of ZFA starting from ZFC by taking a topological group GG, considering the Boolean topos of sets with an action by an open subgroup HGH \le G (where HH is allowed to vary), and 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 construction outlined at pure set can be varied to construct well-founded pure sets using atoms [DR:which I believe they are elements of the transitive -sets.]

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!).

Revised on November 29, 2012 07:09:00 by David Roberts (