On the Consistency Problem for Quine's New Foundations, NF

18 June 2015
Peter Aczel

In 1937 Quine introduced an interesting, rather unusual, set theory called New Foundations - NF for short.  Since then the consistency of NF has been a problem that remains open today.  But there has been considerable progress in our understanding of the problem. In particular NF was shown, by Specker in 1962, to be equiconsistent with a certain theory, TST^+ of simple types. Moreover Randall Holmes, who has been a long-term investigator of the problem, claims to have  solved the problem by showing that TST^+ is indeed consistent.  But the working manuscripts available on his web page that describe his possible proofs are not easy to understand - at least not by me.

In my talk I will introduce TST^+ and its possible models and discuss some of the interesting ideas, that I have understood, that Holmes uses in one of his possible proofs.  If there is time in my talk I will also mention a more recent approach of Jamie Gabbay who is taking a nominal sets approach to the problem.