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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 nS.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 Zp, consisting of integers which are not "lineardoi:10.2307/2037276 fatcat:44qrbawkyrghve7gzco2g4kdhi