Abstract: Abstract 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 and , either ≤ or ≤ . However, in ZF this is no longer so. For a given infinite set A consider seq 1-1 ( A ), the set of all sequences of A without repetition. We compare |seq 1-1 ( A )|, the cardinality of this set, to | |, 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 ⊢ ∀ A (| seq 1-1 ( 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 | fin(B)| < | (B*) | even though the existence for some infinite set B * of a function ƒ from fin( B *) onto ( B *) is consistent with ZF.