Cantor's theorem may fail for finitary partitions

Abstract

A partition is finitary if all its members are finite. For a set $A$, $\mathscr{B}(A)$ denotes the set of all finitary partitions of $A$. It is shown consistent with $\mathsf{ZF}$ (without the axiom of choice) that there exist an infinite set $A$ and a surjection from $A$ onto $\mathscr{B}(A)$. On the other hand, we prove in $\mathsf{ZF}$ some theorems concerning $\mathscr{B}(A)$ for infinite sets $A$, among which are the following: (1) If there is a finitary partition of $A$ without singleton blocks, then there are no surjections from $A$ onto $\mathscr{B}(A)$ and no finite-to-one functions from $\mathscr{B}(A)$ to $A$. (2) For all $n\in\omega$, $|A^n|<|\mathscr{B}(A)|$. (3) $|\mathscr{B}(A)|\neq|\mathrm{seq}(A)|$, where $\mathrm{seq}(A)$ is the set of all finite sequences of elements of $A$.