A Note on Weakly Dedekind Finite Sets

Abstract

A set $A$ is Dedekind infinite if there is a one-to-one function from $\omega$ into $A$. A set $A$ is weakly Dedekind infinite if there is a function from $A$ onto $\omega$; otherwise $A$ is weakly Dedekind finite. For a set $M$, let ${dfin}^{\ast }\left(M\right)$ denote the set of all weakly Dedekind finite subsets of $M$. In this paper, we prove, in Zermelo–Fraenkel (ZF) set theory, that $\left|{dfin}^{\ast }\right(M\left)\right|<\left|\mathcal{P}\right(M\left)\right|$ if ${dfin}^{\ast }\left(M\right)$ is Dedekind infinite, whereas $\left|{dfin}^{\ast }\right(M\left)\right|<\left|\mathcal{P}\right(M\left)\right|$ cannot be proved from ZF for an arbitrary $M$.