The axiom of choice for finite sets

Abstract

If 3f is a family of nonempty sets, then by a choice function on 3f we mean a function f on 3f such that f(S) ES for every S Ez3. The axiom of choice for finite sets may be formulated as follows: (ACF) If 3f is any family of nonempty finite sets, then there exists a choice function on 3F. We shall denote the axiom of choice (for families of arbitrary nonempty sets) by (AC). It is easy to see that the ordering principle (which asserts that every set can be totally ordered) implies (ACF).' On the other hand, Mostowski [4] has shown that (relative to a suitable system of axiomatic set theory) the ordering principle is actually weaker than (AC) so that (ACF), while it is surely a consequence of (AC), is not equivalent to (AC) [4, Korollar II, p. 250]. The object of the present note is to obtain equivalent formulations of (ACF) of the Zorn's lemma type. In fact, in ?1 we show that (ACF) is equivalent to both (ZLF1) and (ZLF2) below; the latter are obtained by restricting two familiar maximal element forms of Zorn's lemma to a special class of partially ordered sets, namely, to those with finitary covers (see the definition below). It is noteworthy, however, that not every form of Zorn's lemma, when so restricted, is equivalent to (ACF). For example, in ?2 we observe that if the statement of either (ZLF1) or (ZLF2) is modified merely by replacing the hypothesis of a least upper bound by that of an upper bound, then the resulting formulation is equivalent, not to (ACF), but to (AC) itself. And furthermore, when one formulates the natural maximal chain analogue of (ZLF1) and (ZLF2), one finds that the result is again equivalent not to (ACF) but to the full axiom of choice (AC). The first named author is pleased to record his indebtedness to Professor Herman Rubin for a number of instructive conversations on the subject of this note.