Consequences of arithmetic for set theory

In this paper, we consider certain cardinals in ZF (set theory without AC, the axiom of choice). In ZFC (set theory with AC), given any cardinals C and D, either C ≤ D or D ≤ C. However, in ZF this is no longer so. For a given infinite set A consider$\operatorname{seq}^{1 - 1}(A)$, the set of all sequences of A without repetition. We compare$|\operatorname{seq}^{1 - 1}(A)|$, the cardinality of this set, to |P(A)|, the cardinality of the power set of A. What is provable about these two cardinals in ZF? The main result of this paper is that$ZF \vdash \forall A(|\operatorname{seq}^{1 - 1}(A)| \neq|\mathscr{P}(\mathscr{A})|)$, and we show that this is the best possible result. Furthermore, it is provable in ZF that if B is an infinite set, then $|\operatorname{fin}(B)|