A deflationary account of Fregean abstraction in set theory, with Basic Law V as a ZFC theorem, Paris PhilMath Intersem 2023

This will be a talk for the Axe Histoire et Philosophie des mathématiques, Séminaire PhilMath Intersem 2023,  a collaborative event sponsored by the University of Notre Dame and le laboratoire SPHERE, Paris. The Intersem runs several weeks, but my talk will be 9 June.

Abstract. The set-theoretic distinction between sets and classes instantiates in important respects the Fregean distinction between objects and concepts, for in set theory we commonly take the universe of sets as a realm of objects to be considered under the guise of diverse concepts, the definable classes, each serving as a predicate on that domain of individuals. Although it is commonly held that in a very general manner, there can be no association of classes with objects in a way that fulfills Frege’s Basic Law V, nevertheless, in the ZF framework, it turns out that we can provide a completely deflationary account of this and other Fregean abstraction principles. Namely, there is a mapping of classes to objects, definable in set theory in senses I shall explain (hence deflationary), associating every first-order parametrically definable class F with a set object εF, in such a way that Basic Law V is fulfilled:  

   εF=εG ⇔ ∀x (Fx ⇔ Gx)

Russell’s elementary refutation of the general comprehension axiom, therefore, is improperly described as a refutation of Basic Law V itself, but rather refutes Basic Law V only when augmented with powerful class comprehension principles going strictly beyond ZF, one amounting, I argue, to a truth predicate in Frege’s system. The main result therefore leads to a proof of Tarski’s theorem on the nondefinability of truth as a corollary to Russell’s argument, independently of Gödel. A central goal of the project is to highlight the issue of definability and deflationism for the extension assignment problem at the core of Fregean abstraction.

Set theory with abundant urelements, STUK 10, Oxford, June 2023

This will be a talk for the Set Theory in the UK, STUK 10, held in Oxford 14 June 2023, organized by my students Clara List, Emma Palmer, and Wojciech Wołoszyn.

Abstract. I shall speak on the surprising strength of the second-order reflection principle in the context of set theory with abundant urelements. The theory GBcU with the abundant urelement axiom and second-order reflection is bi-interpretable with a strengthening of KM with a supercompact cardinal. This is joint work with Bokai Yao.