Remarks on infinite factorials and cardinal subtraction in ZF$\mathsf{ZF}$

Abstract

The factorial of a cardinal a$\mathfrak {a}$, denoted by fraktura!$\mathfrak {a}!$, is the cardinality of the set of all permutations of a set which is of cardinality a$\mathfrak {a}$. We give a condition that makes the cardinal equality fraktura!−fraktura=fraktura!$\mathfrak {a}!-\mathfrak {a}=\mathfrak {a}!$ provable without the axiom of choice. In fact, we prove in ZF$\mathsf {ZF}$ that, for all cardinals a$\mathfrak {a}$, if ℵ0⩽2a$\aleph _0\leqslant 2^\mathfrak {a}$ and there is a permutation without fixed points on a set which is of cardinality a$\mathfrak {a}$, then fraktura!−fraktura=fraktura!$\mathfrak {a}!-\mathfrak {a}=\mathfrak {a}!$.