abstract
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A number-theoretic conjecture and its implication for set theory

Lorenz Halbeisen

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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.