Cambridge, the weekend of 26th-27th May 2007. Meeting Room 4, the Centre for Mathematical Sciences, Wilberforce Road, Cambridge.

Definite attendees: Marcel Crabbe, Thierry Libert, Andre Petry, Randall Holmes, Ali Enayat, Thomas Forster, Sergei Tupailo, Dang Vu, Athanasios Tzouvaras, Olivier Esser.

The conference is being funded by the St. Luke's Institute. (St Luke's also funded (i) Buddhism In Logic and Analytic Philosophy: whose proceedings are being prepared for publication even as I write this; and (ii) Logic and Rhetoric whose proceedings will appear in a special number of Logique et Analyse. It is not clear at this stage what will happen to the proceedings of NF 2007)

Provisional Programme We will not start talks until saturday mid-morning.

Dang Vu

Title: S and HS

A very short introduction to the stratified constructible S hierarchy, with some comments on its similarities/differences with respect to the constructible universe L and the class HS of hereditarily symmetric sets.

Randall Holmes

"A Forster term model of TST"

A Forster term model of a set theory is a model in which all elements are parameter-free set abstracts. In the case of TST, we must also allow type 0 constants. Solovay has proved using standard techniques from the theory of constructible sets that there is a Forster term model of TST. I have realized that a structure that I described to Thomas Forster during a visit in 1998 actually seems to be a Forster term model (in fact, probably the same one Solovay described). I will outline my proof of this fact. Nothing in my talk should be attributed to Solovay, nor should anything I say be construed as denying Solovay priority for this result (in particular, while I did describe this structure in 1998, I had no reason to believe it was a Forster term model and I was in fact fairly certain it was not).

Athanassios Tzouvaras

``Combinatorics related to NF consistency''

Ali Enayat

What NFU knows about Cantorian objects

In what follows NFU is Jensen's modification of Quine's NF, NFU(+) is NFU augmented with the axiom of infinity, and choice and NFU(-) is NFU plus the negation of the axiom of infinity. We plan to discuss various results about extensions of NFU(+/-), in particular:

1. The `equivalence' of the following two theories:

(a) NFU(+) augmented with the axiom "every Cantorian set is strongly Cantorian" ;

(b) Godel-Bernays theory of classes plus the axiom "the class of ordinals is weakly compact".

2 . The `equivalence' of the following two theories:

(a) NFU(-) augmented with the axiom "every Cantorian set is strongly Cantorian";

(b) ACA_0 (the predicative extension of Peano ArithmetiAc)

Marcel Crabbe

Ambiguous Cardinals

A cardinal number k is termed ''ambiguous" if it is indiscernible from 2 to the power of k. In a more specific way, k is ambiguous if the natural typed structure over a set X of size k is elementarily equivalent to the natural structure over the power set of X. Thanks to this notion, different results stemming from Specker's disproof of the axiom of choice will be extracted from the NF milieu and recast in the framework of the politically correct set theory

Sergei Tupailo