             Gödel’s proof of the relative consistency of AC  with the axioms of set theory (see the entry on   Kurt Gödel)  rests on an entirely different idea: that of definability. He introduced a new hierarchy of  sets—the constructible hierarchy—by analogy with the cumulative type hierarchy. We recall that the latter is defined by the following recursion on the ordinals, where \(\sP(X)\) is the power set of \(X\), \(\alpha\) is an ordinal, and \(\lambda\) is a limit ordinal::