Mathematical Logic Seminar Abstracts Autumn 2006

Elementary Patterns of Resemblance are finite sequences of nested trees whihc satisfy certain simple conditions (explained during the talk). They were introduced by T. J. Carlson as a new approach to ordinal notation systems and have a complex combinatorial structure, despite their simple definition which essentially involves teh notion of elementary substructure. We will illustrate the development of ordinal arithmetical tools based on the method of Skolem hulling. This method will be motivated and explained up to the ordinal of ID_

The classical version of Quine's set theory NF is known to be strong but not known to be consistent. The strength emerges in some highly bizarre proofs (Of the axiom of infinity and the negation of the axiom of choice) and it is not clear what their constructive content is. Naturally one wonders about the status of the constructive version of the theory. Is there perhaps an easy proof of its consistency, perhaps using realizability ideas arising from Specker's equiconsistency for NF and version of typed set theory?

Nothing of any significance has been published on constructive NF, and this paper is the result of my attempts to prepare - in collaboration with Randal Holmes - a background survey paper which would be useful to people thinking of working on this topic.