A Note on Superamorphous Sets and Dual Dedekind-Infinity

Abstract

We give a simple example of a set that is weakly Dedekind infinite (= can be mapped onto omega) but dually Dedekind finite (=cannot be mapped noninjectively onto itself), namely, the power set of a superamorphous set. (A infinite set is superamorphous if all finitary relations on it are definable in the language of equality.) We also show that the property of "inexhaustibility" is not closed under supersets unless the full axiom of choice holds.