CANTOR’S THEOREM MAY FAIL FOR FINITARY PARTITIONS

Abstract

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.