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Full Idea
The axiom of choice actually amounts to admitting an absolutely indeterminate infinite set whose existence is asserted albeit remaining linguistically indefinable. On the other hand, as a process, it is unconstructible.
Gist of Idea
The axiom of choice must accept an indeterminate, indefinable, unconstructible set
Source
Alain Badiou (Briefings on Existence [1998], 2)
Book Ref
Badiou,Alain: 'Briefings on Existence', ed/tr. Madarsz,Norman [SUNY 2006], p.52
A Reaction
If only constructible sets are admitted (see 'V = L') then there is a contradiction.