On counting equivalence relations

Abstract

The continuum hypothesis of G. Cantor is a starting point for much current work in set theory and mathematical logic. Most elementary texts go no further than a statement of the hypothesis, with a brief indication of its significance. The aim of this paper is to show the details of its application at an elementary level. We consider two sets; E, the set of all possible equivalence relations on the set of natural numbers N, and R, the set of real numbers, and present two proofs that they are equinumerous (E=R). The first proof uses only the basic results of transfinite arithmetic (within easy reach of first‐year undergraduates) to show that E ≤ R and E >N, and then an appeal to the continuum hypothesis immediately yields E=R. The second proof avoids the continuum hypothesis by a direct demonstration that E ≥ R, and this only involves the Schroder‐Bernstein theorem and a certain degree of care in showing that a certain map from the real interval (0, 1) to a set of subsequences of N is in fact a bijection.