A Note on Superamorphous Sets and Dual Dedekind-Infinity

In the absence of the axiom of choice there are several possible nonequivalent ways of translating the intuitive idea of “infinity” into a mathematical definition. In [10], Tarski investigated some natural infinity notions notions, and his research was continued by Levy [8], Truss [12], Spǐsiak and Vojtas [9], Howard and Yorke [4] and others. The most prominent definitions of finiteness are the true finiteness (equipotent to a bounded set of natural numbers), which is equivalent to Tarskifiniteness (every family of subsets has a maximal element), and the much weaker Dedekind-finiteness (not equipotent to any proper subset). In a recent survey paper [1], Degen recalled the notions of “weak Dedekind infinity” and “dual Dedekind infinity” (see below for definitions) and asked if they were in fact equivalent. A forcing construction of a weakly Dedekind set which is not dually Dedekind is given by Truss in [12], and also announced by Diel [2]. We give here an alternate and more elementary construction of such a set, which does not use the methods of forcing but instead relies on (the consistency of) the existence of a superamorphous set. Answering another question posed in [1], we also show that “inexhaustibility” is not a notion of infinity unless the axiom of choice is assumed.