Arbitrariness and Frege Arithmetic

Abstract

This paper focuses on a less-known version of Abstractionism, that we’ll call Arbitrary Frege Arithmetic. This system aims to restore Fregean Logicist project (of a logical derivation of second-order Peano Arithmetic) by adopting, as much as possible, its original tools, namely a logical second-order system augmented with a (consistent) version of Basic Law V. The major differences from Frege’s Grundgesetze will be a recasted presentation of the abstractionist vocabulary and a weakening of first-order classical logic in a negative free logic one. Both these choices are supported by philosophical and formal motivations that allow us to consider them completely compatible with the abstractionist spirit.