Full Idea
The Axiom of Choice is a pure existence statement, without defining conditions. It was necessary to provide a foundation for Cantor's theory of transfinite cardinals and ordinal numbers, but its nonconstructive character engendered heated controversy.
Gist of Idea
Cantor's theories needed the Axiom of Choice, but it has led to great controversy
Source
Feferman / Feferman (Alfred Tarski: life and logic [2004], Int I)
Book Reference
Feferman,S/Feferman,A.B.: 'Alfred Tarski: life and logic' [CUP 2008], p.43