Full Idea
In 1938 Gödel proved that the Axiom of Choice is consistent with the other axioms of set theory.
Gist of Idea
The Axiom of Choice is consistent with the other axioms of set theory
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.44
A Reaction
Hence people now standardly accept ZFC, rather than just ZF.