Philosophy of Mathematics seminar

Joel David Hamkins, Oxford

Fregean abstraction in Zermelo-Fraenkel set theory  

The standard treatment of sets and classes in Zermelo-Fraenkel set theory instantiates in many respects the Fregean foundational distinction between objects and concepts, for in set theory we commonly take the sets as 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 often asserted that there can be no association of classes with objects in a way that fulfills Frege’s Basic Law V, nevertheless, in the ZF framework I have described, it turns out that Basic Law V does hold, and provably so, along with other various Fregean abstraction principles. These principles are consequences of Zermelo-Fraenkel ZF set theory in the context of all its definable classes. Namely, there is an injective mapping from classes to objects, definable in senses I shall explain, 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. The main result leads also to a proof of Tarski’s theorem on the nondefinability of truth as a corollary to Russell’s argument.

Commentary can be made on the speaker’s blog at http://jdh.hamkins.org/fregean-abstraction-in-zf-set-theory-oxford-november-2021/