= Constructible-universe obstruction to a uniform inner-model proof of failure of choice
Every <inner model> $N$ of <ZF> contains the ambient <constructible universe> $L$. By induction, $L_\alpha^N=L_\alpha$: at successor stages, satisfaction in the same <set> structure and definability with parameters are absolute; at limits both constructions take the same <union>. Thus, if the ambient universe satisfies $V=L$, its only inner model of <ZF> is itself, and it satisfies the <axiom of choice>. A procedure that must produce a choice-failing inner model from every model of <ZF> is therefore impossible. This does not assert that arbitrary inner models of arbitrary choice universes satisfy choice; the obstruction concerns a uniform construction.
Back to article page
