Full Idea
If the Axiom of Choice says we can choose one member from each of a set of non-empty sets and put the chosen elements together in a set, this licenses the constructor to do an infinite amount of choosing.
Gist of Idea
The Axiom of Choice seems to license an infinite amount of choosing
Source
Stewart Shapiro (Philosophy of Mathematics [1997], 6.3)
Book Reference
Shapiro,Stewart: 'Philosophy of Mathematics:structure and ontology' [OUP 1997], p.188
A Reaction
This is one reason why the Axiom was originally controversial, and still is for many philosophers.