The cardinality of the partitions of a set in the absence of the Axiom of Choice

Abstract

Abstract In the Zermelo–Fraenkel set theory (ZF), $|\textrm {fin}(A)|<2^{|A|}\leq |\textrm {Part}(A)|$ for any infinite set $A$, where $\textrm {fin}(A)$ is the set of finite subsets of $A$, $2^{|A|}$ is the cardinality of the power set of $A$ and $\textrm {Part}(A)$ is the set of partitions of $A$. In this paper, we show in ZF that $|\textrm {fin}(A)|<|\textrm {Part}_{\textrm {fin}}(A)|$ for any set $A$ with $|A|\geq 5$, where $\textrm {Part}_{\textrm {fin}}(A)$ is the set of partitions of $A$ whose members are finite. We also show that, without the Axiom of Choice, any relationship between $|\textrm {Part}_{\textrm {fin}}(A)|$ and $2^{|A|}$ for an arbitrary infinite set $A$ cannot be concluded.