Fregean abstraction in set theory—a deflationary account, Italian Philosophy of Mathematics, September 2022

This will be a talk for the conference Philosophy of Mathematics: Foundations, Definitions and Axioms, the Fourth International Conference of the Italian Network for the Philosophy of Mathematics, 29 September to 1 October 2022.

Abstract. The standard 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 $\varepsilon F$, in such a way that Basic Law V is fulfilled: $$\varepsilon F =\varepsilon G\iff\forall x\ (Fx\leftrightarrow Gx).$$ 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. The main result leads also to a proof of Tarski’s theorem on the nondefinability of truth as a corollary to Russell’s argument. 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.

3 thoughts on “Fregean abstraction in set theory—a deflationary account, Italian Philosophy of Mathematics, September 2022

  1. Pingback: The math tea argument—must there be numbers we cannot describe or define? Pavia Logic Seminar | Joel David Hamkins

  2. Visser has some very general results about the result of adding predicative comprehension to a theory T: It’s mutually intererpretable both with Q + Con(T) and to the result of adding a Tarskian theory of truth to T. Other than the bit about definability in ZF, I’m not seeing how these results would go beyond Visser’s. Indeed, Visser’s result looks stronger.

    I’m also a bit puzzled why it doesn’t already follow from the proof that first-order (or predicative) BLV is consistent that Russell’s argument doesn’t refute Law V. That, after all, was precisely the claim that Dummett made that got Parsons (and me) to think about these restricted theories. Those constructions start (as it were) just from logic, but I’d guess that it would be a simple matter to redo them so that they began from any first-order theory that has infinite models. Francesca Boccuni’s paper “Plural Frege Arithmetic” might be thought of as doing something like that for arithmetic.

    • Thank you for your comment! Indeed, the main new part of my result is that I achieve BLV in a definable manner. That is, the map from the definable ZF classes $F$ to their extension object $\varepsilon F$ is first-order definable (in various senses explained in the paper) in the language of set theory without any extension operator. It follows that the extension operators are eliminable — every formula in the language with extensions is equivalent to one in the base language. This is the sense of “deflationary” that I intend in the title. Furthermore, to my way of thinking, I find the definability of the extension assignments to be absolutely at the center of many issues, such as the Julius Caesar problem, although this seems to be strangely neglected in prior work. The results of Parsons 87, Bell 94, and Burgess 2005, for example, all establish the consistency of BLV over any infinite first-order model, but do not achieve the definability of the extension assignments.

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