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Full Idea
The axiom of choice has a troubled history, but is now standard in mathematics. It could be replaced with a principle of comprehension for functions), or one could omit the variables ranging over functions.
Gist of Idea
The axiom of choice is controversial, but it could be replaced
Source
Stewart Shapiro (Higher-Order Logic [2001], n 3)
Book Ref
'Blackwell Guide to Philosophical Logic', ed/tr. Goble,Lou [Blackwell 2001], p.52