# Supplementary Talks

## Friday 21 March

10:00 - 11:00 Thomas Forster (Cambridge University)

"The Category of Sets According to Stratified Set Theories"

It is a celebrated result proved by our Distinguished Overseas Visitor to this workshop that, according to Quine's NF, the category of sets is not cartesian closed. Other - quite different - proofs of this fact are now known, and some of them, at least, trade on the fact that some unstratified instances of the separation scheme are actually refutable in NF, and cartesian-closedness is not a stratified condition. Typically in NF when a classical theorem fails because of stratification problems, one can prove a stratified modification that ''feels the same''. There is a stratified version of cartesian-closedness that the sets of NF do in fact obey, and this is probably an avenue worth pursuing, given the possibility that stratified set theories hold out of the existence of a category of all categories. This is a report on work in progress on a project that has really only been started in the last few months.

11:30 - 12:30 Verónica Borja (Benemérita Universidad Autónoma de Puebla)

"Extensions of intuitionistic and Cw logics"

Among non-classical logics, intermediate logics and paraconsistent logics play a very important roll. Both families of logics emerged in the first half of the last century but they have now evolved into a complex but interesting mathematical field, with a strong trend to Computer Science applications. Algebraic and Kripke semantics have become a powerful tool for studying their properties. However several important problems are still open, for example, little is known about completeness, about possible translations or relations between these families. Due to their particular importance in the field of semantics for logic programs we focus on some extensions of logic intuitionism and some of Cw. We study their different characterizations and the behavior of the systems in order to reveal the relations between constructive and paraconsistent logics.

14:00 - 15:30 Adam Epstein (University of Warwick)

"What does it take to prove Thurston's Rigidity Theorem?"

Around thirty years ago, William Thurston proved a fundamental theorem in holomorphic dynamics concerning the existence and uniqueness of PSL_{2}**C**** **conjugacy classes of postcritically finite rational endomorphisms of **P**^{1 }with specified combinatorics. The uniqueness result is a rigidity theorem in a certain deformation theory. As in Kodaira-Spencer's deformation theory for complex manifolds, the corresponding infinitesimal rigidity theorem is a vanishing theorem for the first cohomology of a sheaf of infinitesimal automorphisms. Infinitesimal Thurston Rigidity is an essentially algebraic theorem: given complexity bounds, it may be expressed as an assertion in the elementary theory of fields. It is striking that the only known proof of this theorem is inherently transcendental and, in particular, not Galois invariant. Moreover, via of the completeness of the theory of algebraically closed fields of characteristic 0, this theorem may be coded into a true arithmetic statement which is not currently known to be provable in Peano Arithmetic.

16:00 Colin McLarty (Case Western Reserve University)

Colloquium "Categorical Foundations Today"

Saunders Mac Lane stressed foundations not as a priori philosophical justifications for mathematics but as "proposals for the organization of mathematics." He urged Lawvere's categorical foundations in this role. This talk will look at the current state of these categorical foundations in theory and in textbook practice. The talk will relate these foundations to the related sense of inquiry to find the minimal requirements for particular theorems or branches of mathematics, to some other styles of categorical foundations such as Homotopy Type Theory, and to objections to categorical foundations.