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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Choices from finite sets and choices of finite subsets

Author: Martin M. Zuckerman
Journal: Proc. Amer. Math. Soc. 27 (1971), 133-138
MSC: Primary 02.60
MathSciNet review: 0270905
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Abstract: In set theory without the axiom of choice we prove a consistency result involving certain ``finite versions'' of the axiom of choice. Assume that it is possible to select a nonempty finite subset from each nonempty set. We determine sets $ Z$, of integers, which have the property that $ n \in Z$ is a necessary and sufficient condition for the possibility of choosing an element from every $ n$-element set. Given any nonempty set $ P$ of primes, the set $ {Z_p}$, consisting of integers which are not ``linear combinations'' of primes of $ P$, is such a set $ Z$.

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Keywords: Axiom of choice for $ n$-element sets, multiple choice axioms, Fraenkel-Mostowski models, relative consistency
Article copyright: © Copyright 1971 American Mathematical Society

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