Weak extensional collapse and appropriate technology in NFU
by
Randall Holmes
Boise State University

Something which is often not clear in considering the difference between NF and NFU is the reason why it is not possible to convert a model of NFU to a model of NF by extensional collapse: it is very easy to get from a model of ZFA (ZF with atoms) to a model of ZF. The explanation is that it is generally hard to do any kind of induction on membership in NFU. However, it is possible to carry out a kind of weak extensional collapse in Quine-style set theory: Marcel Crabbé has shown how to convert a model of stratified comprehension without any extensionality at all to a model of NFU. It is possible to reduce many arbitrary failures of extensionality to a single failure of extensionality (for empty sets). Recently, the author has exploited the idea behind Crabbés technique of weak extensional collapse to get an account of forcing in NFU which is natural in terms of NFU; it was already known that forcing can be done in NFU, but the previously known technique involved a simulation of the usual techniques in an interpretation of a fragment of Zermelo set theory in NFU.

Date received: April 11, 2000