DIVISIBILITY OF DEDEKIND FINITE SETS

Abstract

A Dedekind-finite set is said to be divisible by a natural number n if it can be partitioned into pieces of size n. We study several aspects of this notion, as well as the stronger notion of being partitionable into n pieces of equal size. Among our results are that the divisors of a Dedekind-finite set can consistently be any set of natural numbers (containing 1 but not 0), that a Dedekind-finite power of 2 cannot be divisible by 3, and that a Dedekind-finite set can be congruent modulo 3, to all of 0, 1, and 2 simultaneously. (In these results, 2 and 3 serve as typical examples; the full results are more general.)