Fregean abstraction in Zermelo-Fraenkel set theory: a deflationary account

TL;DR

This paper demonstrates that Fregean abstraction principles can be explicitly defined within ZF set theory, providing a deflationary account that sidesteps Russell's paradox by reducing abstraction to set-theoretic assertions.

Contribution

It offers a novel set-theoretic interpretation of Fregean abstraction principles, showing they are definable and reducible within ZF, thus avoiding Russell's paradox.

Findings

Fregean abstraction operators are definable in ZF set theory.

The approach provides a deflationary account of Fregean abstraction.

It sidesteps Russell's paradox by reducing abstraction to set-theoretic assertions.

Abstract

The standard treatment of sets and definable classes in first-order Zermelo-Fraenkel set theory accords in many respects with the Fregean foundational framework, such as the distinction between objects and concepts. Nevertheless, in set theory we may define an explicit association of definable classes with set objects $F\mapsto\varepsilon F$ in such a way, I shall prove, to realize Frege's Basic Law V as a ZF theorem scheme, Russell notwithstanding. A similar analysis applies to the Cantor-Hume principle and to Fregean abstraction generally. Because these extension and abstraction operators are definable, they provide a deflationary account of Fregean abstraction, one expressible in and reducible to set theory -- every assertion in the language of set theory allowing the extension and abstraction operators $\varepsilon F$, $\# G$, $\alpha H$ is equivalent to an assertion not using them. The analysis thus sidesteps Russell's argument, which is revealed not as a refutation of Basic Law~V as such, but rather as a version of Tarski's theorem on the nondefinability of truth, showing that the proto-truth-predicate "$x$ falls under the concept of which $y$ is the extension" is not expressible.