A number-theoretic conjecture and its implication for set theory

For any set S let |seq(S)| denote the cardinalityof the set of all finite one-to-one sequences that can be formed from S, and for positive integers a let |a^S| 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 any infinite set S, the cardinality |seq(S)| is never equal to the cardinality |2^S| (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, without using any form of the axiom of choice, that the cardinalities |seq(S)| and |a^S| are different. Moreover, it is shown that a very probable number-theoretic conjecture implies that this holds for every positive integer a in any model of set theory.

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