Cantor’s Principle of Finitism or the Actualization of Infinite Potentialities

Abstract

The paper sets forth a critique of Georg Cantor’s philosophical justifications for the introduction of transfinite numbers, and thus actual infinities, in his set theory. “Critique” here has a twofold meaning: 1) the investigation concerns the conditions of possibility of set theory and not its strictly mathematical content, and 2) it consists in a pars destruens of Cantor’s “dogmatic” defense of his opaque mathematical entities. It is based on the examination of the classical definition of set given by Cantor in his 1883 foundational work, and it is developed through the analysis of some crucial conceptual pairs: one and many, completeness and incompleteness, totality and variability, and especially actuality and potentiality.