Choices from Finite Sets and Choices of Finite Subsets

Martin M. Zuckerman
1971 Proceedings of the American Mathematical Society  
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 "linear
more » ... "linear combinations" of primes of P, is such a set Z. (1) <j\J {"IAC, FS,(Vnè 2)idn) ^ n E Z)} also consistent. Let Z be the set of sets Z for which (1) holds.
doi:10.2307/2037276 fatcat:44qrbawkyrghve7gzco2g4kdhi