Choices from finite sets and choices of finite subsets

Zuckerman, Martin M.

doi:10.1090/S0002-9939-1971-0270905-5

Choices from finite sets and choices of finite subsets
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by Martin M. Zuckerman PDF

Proc. Amer. Math. Soc. 27 (1971), 133-138 Request permission

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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Über die Unabhängigkeit des Wohlordnungssätzes vom Ordnungsprinzip

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