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
Z
, of integers, which have the property that
n
∈
Z
n \in Z
is a necessary and sufficient condition for the possibility of choosing an element from every
n
n
-element set. Given any nonempty set
P
P
of primes, the set
Z
p
{Z_p}
, consisting of integers which are not “linear combinations” of primes of
P
P
, is such a set
Z
Z
.