The axiom of constructibility implies the axiom of choice, given Zermelo–Fraenkel set theory without the axiom of choice. It also settles many natural mathematical questions that are independent of Zermelo–Fraenkel set theory with the axiom of choice ; for example, the axiom of constructibility implies the generalized continuum hypothesis, the negation of Suslin's hypothesis, and the existence of an analyticalnon-measurableset of real numbers, all of which are independent of ZFC. The axiom of constructibility implies the non-existence of those large cardinals with consistency strengthgreater or equal to 0#, which includes some "relatively small" large cardinals. Thus, no cardinal can be ω1-Erdős in L. While L does contain the initial ordinals of those large cardinals, and they are still initial ordinals in L, it excludes the auxiliary structures which endow those cardinals with their large cardinal properties. Although the axiom of constructibility does resolve many set-theoretic questions, it is not typically accepted as an axiom for set theory in the same way as the ZFC axioms. Among set theorists of a realist bent, who believe that the axiom of constructibility is either true or false, most believe that it is false. This is in part because it seems unnecessarily "restrictive", as it allows only certain subsets of a given set, with no clear reason to believe that these are all of them. In part it is because the axiom is contradicted by sufficiently strong large cardinal axioms. This point of view is especially associated with the Cabal, or the "California school" as Saharon Shelah would have it.
Significance
The major significance of the axiom of constructibility is in Kurt Gödel's proof of the relative consistency of the axiom of choice and the generalized continuum hypothesis to Von Neumann–Bernays–Gödel set theory. Namely Gödel proved that is relatively consistent, and that in thereby establishing that AC and GCH are also relatively consistent. Gödel's proof was complemented in later years by Paul Cohen's result that both AC and GCH are independent, i.e. that the negations of these axioms are also relatively consistent to ZF set theory.
Statements true in ''L''
Here is a list of propositions that hold in the constructible universe :
The generalized continuum hypothesis and as a consequence
The existence of a definable well-order of all sets. In particular, L satisfies V=HOD.
Accepting the axiom of constructibility these propositions also hold in the von Neumann universe, resolving many propositions in set theory and some interesting questions in analysis.
