Why believe infinite sets exist?

Andrei Mărăşoiu

pp. 447-460

in: Axiomathes 28 (4), 2018.

Abstract

The axiom of infinity states that infinite sets exist. I will argue that this axiom lacks justification. I start by showing that the axiom is not self-evident, so it needs separate justification. Following Maddy's (J Symb Log 53(2):481–511, 1988) distinction, I argue that the axiom of infinity lacks both intrinsic and extrinsic justification. Crucial to my project is Skolem's (in: van Heijnoort (ed) From Frege to Gödel: a source book in mathematical logic, 1879–1931, Cambridge, Harvard University Press, pp. 290–301, 1922) distinction between a theory of real sets, and a theory of objects that theory calls "sets". While Dedekind's (in: Essays on the theory of numbers, pp. 14–58, 1888. http://www.gutenberg.org/ebooks/21016) argument fails, his approach was correct: the axiom of infinity needs a justification it currently lacks. This epistemic situation is at variance with everyday mathematical practice. A dilemma ensues: should we relax epistemic standards or insist, in a skeptical vein, that a foundational problem has been ignored?

Cited authors

Frege Gottlob