It also settles many natural mathematical questions that are independent of Zermelo–Fraenkel set theory with the axiom of choice (ZFC); for example, the axiom of constructibility implies the generalized continuum hypothesis, the negation of Suslin's hypothesis, and the existence of an analytical (in fact,
) non-measurable set of real numbers, all of which are independent of ZFC.
The axiom of constructibility implies the non-existence of those large cardinals with consistency strength greater or equal to 0#, which includes some "relatively small" large cardinals.
This point of view is especially associated with the Cabal, or the "California school" as Saharon Shelah would have it.
Especially from the 1950s to the 1970s, there have been some investigations into formulating an analogue of the axiom of constructibility for subsystems of second-order arithmetic.
A few results stand out in the study of such analogues: 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.)
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 (
Here is a list of propositions that hold in the constructible universe (denoted by L): Accepting the axiom of constructibility (which asserts that every set is constructible) these propositions also hold in the von Neumann universe, resolving many propositions in set theory and some interesting questions in analysis.