A Number-theoretic Conjecture and Its Implication for Set Theory

For any set S let ∣ ∣ seq1-1(S) ∣ ∣ denote the cardinality of the set of all finite one-to-one sequences that can be formed from S, and for positive integers a let ∣ ∣a ∣ ∣ denote the cardinality of all functions from S to a. Using a result from combinatorial number theory, Halbeisen and Shelah have shown that even in the absence of the axiom of choice, for infinite sets S one always has ∣ ∣ seq1-1(S) ∣ ∣ 6= ∣ ∣2 ∣ ∣ (but nothing more can be proved without the aid of the axiom of choice). Combining stronger number-theoretic results with the combinatorial proof for a = 2, it will be shown that for most positive integers a one can prove the inequality ∣ ∣ seq1-1(S) ∣ ∣ 6= ∣ ∣a ∣ ∣ without using any form of the axiom of choice. Moreover, it is shown that a very probable number-theoretic conjecture implies that this inequality holds for every positive integer a in any model of set theory.