Full Idea
The usual accounts of ZF are not restricted to subsets that we can describe, and that is what justifies the axiom of choice.
Gist of Idea
The Axiom of Choice relies on reference to sets that we are unable to describe
Source
David Bostock (Philosophy of Mathematics [2009], 8.4 n36)
Book Reference
Bostock,David: 'Philosophy of Mathematics: An Introduction' [Wiley-Blackwell 2009], p.256
A Reaction
This contrasts interestingly with predicativism, which says we can only discuss things which we can describe or define. Something like verificationism hovers in the background.
Related Idea
Idea 18136 If we can only think of what we can describe, predicativism may be implied [Bostock]