# Well-founded models of fragments of Collection

## Zach McKenzie, University of Chester

Let M be the weak set theory obtained from ZF by removing the Collection Scheme, restricting the Separation Scheme to bounded formulae, and adding an axiom asserting that every set is contained transitive set. In this talk I will consider two formulations of the set-theoretic Collection Scheme restricted formulae that are Π_{n} in the Levy hierarchy: Π_{n}-Collection and Strong Π_{n}-Collection. It is known that, over M, Strong Π_{n}-Collection is equivalent to Π_{n}-Collection plus Σ_{n+1}-Separation, and Strong Π_{n}-Collection proves the consistency of M + Π_{n}-Collection. In this talk I will show that for all n > 0, every well-founded model of M + Π_{n}-Collection satisfies Strong Π_{n}-Collection. In particular, for n > 0, M + Strong Π_{n}-Collection does not prove the existence of a transitive model of M + Π_{n}-Collection. And, for n > 0, the minimum model of M + Strong Π_{n}-Collection and M + Π_{n}-Collection coincide. If time permits, I will indicate how the assumption that that the model is well-founded can be replaced with the assumption that the model instead satisfies a fragment of Foundation. This reveals new equivalences between subsystems of ZF that include the Powerset axiom.