Programme:

13:30-14:15 Catrin Campbell-Moore (Bristol): Supervaluationism and omega-consistency.

14:20-15:05 Pablo Dopico Fernandez (KCL): The semantic indeterminacy of the concept of set.

15:05-15:25 Coffee Break

15:25-16:10 Carlo Nicolai (KCL): Some prolegomena to non-well-founded instantiation.

16:20-17:05 Beatrice Buonaguidi (KCL): Non-well-founded classes in hyperuniverses

17:15-18:00 Kentaro Fujimoto (Bristol): Uniform reflection principles and Feferman's schematic reflective closures.

 

Abstracts:

Beatrice Buonaguidi: Non-well-founded classes in hyperuniverses

Barwise and Moss (1996) introduce a theory of non-well-founded classes which is meant to extend Aczel's ZFA, with the aim of providing an axiomatisation of a so-called 'arbitrary' notion of class, and of offering a principled solution to the set-theoretic and class-theoretic paradoxes in a non-well-founded setting. Their theory, however, presents some mathematical and philosophical shortcomings. This talk gestures towards addressing some of these. In particular, we suggest to consider the hyperuniverses introduced by Forti and Hinnion (1989) as a semantic framework to model a theory of strongly extensional classes addressing some of the shortcomings in Barwise and Moss. This is based on joint work with Carlo Nicolai.

 

Carlo Nicolai: Some prolegomena to non-well-founded instantiation

The talk concerns the broader project extracting adequate property theories from anti-foundation principles. I focus on the task of recovering full ZF in such (non-extensional, non-wellfounded systems). I adapt and modify some known results to achieve this.

 -

Pablo Dopico: The semantic indeterminacy of the concept of set.

Let’s say that a predicate is extensionally deficient when there are borderline cases for its applicability, such as the predicate ‘being bald’. Drawing on a classification of extensionally deficient predicates proposed by Michael Dummett, in this paper we argue that the standard conception of set possessed by set theorists is likely to be indefinitely precisifiable; that is, regardless of how we precisify the predicate ‘being a set’, there is always a way to precisify it further that settles more borderline cases. For our claim, we rely on the development of the large cardinal axiom program from the early days of set theory until today. This is the first part of joint work with Davide Sutto (University of Oslo)

-

Kentaro Fujimoto: Uniform reflection principles and Feferman's schematic reflective closures

Feferman formulated a variety of formal systems in an attempt to vindicate his life-long conviction that the Feferman-Schuette ordinal is the predicative limit. Among them is the system of the schematic reflective closure of Peano Arithmetic PA. According to him, the engines for systematically generating larger and larger systems whose acceptability is implicit in acceptance of the starting theory are what have come to be called reflection principles, and one of the main purposes of his schematic reflective closures is “to give a more perspicuous generation procedure for predicativity" without use of progression of those reflection principles. In this talk, I will examine Feferman's concept of schematic reflective closure, consider a theory for predicativity directly in terms of the reflection principles, and then investigate how these two kinds of theories are related.

 -

Catrin Campbell-Moore: Supervaluationism and omega-consistency

Supervaluational Kripkean theories of truth result in triviality when we restrict to omega-consistent precisifications. This talk presents an alternative finitary supervaluational jump which is non-trivial for omega-consistent precisifications. The restriction to omega-consistent precisifications can be motivated when we consider alternative properties which are brought about as definite than just looking at when the precisifications agree on what is definitely true. We result in a picture of indefinite truth where truth is definitely omega-consistent, although definite-truth is omega-inconsistent.