Aristotle meets Frege: from Potentialism to Frege Arithmetic

The purpose of this paper is to present a genuinely potentialist account of Frege arithmetic. The (cardinal) numbers are not generated from Hume’s Principle, but rather from more or less standard principles of potentialism. The relevant version of Hume’s Principle is a principle stating a condition for numbers to be identical with each other. Essentially, (HP) tells us what we are generating— cardinal numbers—but the generation does not go through (HP) itself. We also develop an Aristotelian, potentialist set theory—in effect, a theory of hereditarily finite sets—a theory that is definitionally equivalent to Dedekind-Peano arithmetic.