Seeking Structure for the Collection of Rieger-Bernays Permutation models

Given any structure for the language of set theory and a permutation of its domain one obtains a new structure for the language of set theory. These are Rieger-Bernays models. Thus we have a surjection from the full symmetric group of the domain to a family of structures. Both groups and families-of-structures have natural topologies and it is natural to wonder what sensible things can be said about the way in which the topologies on this group and on this family interact. The techniques of Rieger-Bernays models first came to attention in an NF context but this material can also be applied in a ZF context too, and quite a lot of what follows below will carry over naturally. This paper was begun in 2003/4 when Olivier Esser was in Cambridge on a postdoctoral fellowship working with Thomas Forster on the problem mentioned in the above abstract. It is designed in the first instance as a briefing paper for the second author’s Ph.D. students. It had been hoped that it would fairly swiftly turn into something that the authors could publish, to justify the postdoc funding for 2003/4 and justify further appeals for funds in 2004/5, but the problems turned out to be far too intractable. 1 Problems to look at 1. Given a model of NF, form a topology from the ∼n equivalence classes as in GPG. What happens? 2. Show that if φ is a ∀∗∃∗-formula that is consistent with NF then NF ` φ; 3. Show that If Φ is a consistent ∃∗∀∗ sentence then {σ : V σ |= Φ} contains a permutation of finite support; 4. Obtain a sensible classification of unstratified formulæ so we can prove theorems with the flavour: if φ of syntactic nastiness > n then {σ : V σ |= σ}∩ C(Jn) = ∅. 5. Complete J0 wrt the various topologies?