A Generalization of Cantor’s Theorem

Abstract

One of the most important results in basic set theory is without doubt Cantor’s Theorem which states that the power set of any set X is strictly bigger than X itself. Specker once stated, without providing a proof, that a generalization is possible: for any natural exponent m, there is a natural number N for which if X has at least N distinct elements, then the power set of X is strictly bigger than X^m. The aim of this paper is to formalize and prove Specker’s claim and to provide a way to compute the values of N for which the theorem holds.