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**1 - 1**of**1**### A NUMBER-THEORETIC CONJECTURE AND ITS IMPLICATION FOR SET THEORY

"... Abstract. For any set S let � � seq 1-1 (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 S � � denote the cardinality of all functions from S to a. Using a result from combinatorial number theory, Halbeise ..."

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Abstract. For any set S let � � seq 1-1 (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 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 infinite sets S one always has � � seq 1-1 (S) � � � = � �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 the inequality � � seq 1-1 (S) � � � = � �a S � � 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. 1.