The Potential in Frege's Theorem

TL;DR

This paper investigates whether Frege's Theorem can be reconciled with a potential infinity perspective, showing that first-order but not second-order arithmetic can be interpreted in this setting, impacting the logicist's stance on actual infinities.

Contribution

It demonstrates that in a potential infinity framework, first-order Peano arithmetic is interpretable, but second-order arithmetic is not, challenging previous assumptions about the scope of Frege's Theorem.

Findings

First-order Peano arithmetic can be interpreted in the potential infinity setting.

Second-order Peano arithmetic cannot be interpreted in this setting.

Weakening the metaphysical claim of actual infinities affects the mathematics recovered.

Abstract

Is a logicist bound to the claim that as a matter of analytic truth there is an actual infinity of objects? If Hume's Principle is analytic then in the standard setting the answer appears to be yes. Hodes's work pointed to a way out by offering a modal picture in which only a potential infinity was posited. However, this project was abandoned due to apparent failures of cross-world predication. We re-explore this idea and discover that in the setting of the potential infinite one can interpret first-order Peano arithmetic, but not second-order Peano arithmetic. We conclude that in order for the logicist to weaken the metaphysically loaded claim of necessary actual infinities, they must also weaken the mathematics they recover.