Restriction to finite sets The usual statement of the axiom of choice does not specify whether the collection of nonempty sets is finite or infinite, and thus implies that every finite collection of nonempty sets has a choice function. However, that particular case is a theorem of the Zermelo–Fraenkel set theory without the axiom of choice (ZF); it is easily proved by the principle of finite induction.Tourlakis (2003), pp. 209–210, 215–216. In the even simpler case of a collection of one set, a choice function just corresponds to an element, so this instance of the axiom of choice says that every nonempty set has an element; this holds trivially. The axiom of choice can be seen as asserting the generalization of this property, already evident for finite collections, to arbitrary collections.
Restriction to finite sets The usual statement of the axiom of choice does not specify whether the collection of nonempty sets is finite or infinite and thus implies that every finite collection of nonempty sets has a choice function However that particular case is a theorem of the Zermelo–Fraenkel set theory without the axiom of choice (ZF); it is easily proved by the principle of finite inductionTourlakis (2003) pp 209–210 215–216 In the even simpler case of a collection of one set a choice function just corresponds to an element so this instance of the axiom of choice says that every nonempty set has an element; this holds trivially The axiom of choice can be seen as asserting the generalization of this property already evident for finite collections to arbitrary collections