Editing Axiom of constructibility (section)

== Significance ==
The major significance of the axiom of constructibility is in [[Kurt Gödel]]'s 1938 proof of the relative [[consistency]] of the [[axiom of choice]] and the [[generalized continuum hypothesis]] to [[Von Neumann–Bernays–Gödel set theory]].  (The proof carries over to [[Zermelo–Fraenkel set theory]], which has become more prevalent in recent years.)

Namely Gödel proved that <math>V=L</math> is relatively consistent (i.e. if <math>ZFC + (V=L)</math> can prove a contradiction, then so can <math>ZF</math>), and that in <math>ZF</math>

:<math>V=L\implies AC\land GCH,</math>

thereby establishing that AC and GCH are also relatively consistent.

Gödel's proof was complemented in 1962 by [[Paul Cohen]]'s result that both AC and GCH are ''independent'', i.e. that the negations of these axioms (<math>\lnot AC</math> and <math>\lnot GCH</math>) are also relatively consistent to ZF set theory.