Axiom of constructibility

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The axiom of constructibility is a possible axiom for set theory in mathematics that asserts that every set is constructible. The axiom is usually written as

V = L,

where V and L denote the von Neumann universe and the constructible universe, respectively.

[edit] Implications

Acceptance of the axiom of constructiblity settles many natural mathematical questions independent of ZFC, the standard axiomatization of set theory.

The axiom of constructibility implies the generalized continuum hypothesis, the axiom of choice, the negation of Suslin's hypothesis, and the existence of a simple (Δ¹₂) non-measurable set of real numbers.

The axiom of constructibility implies the non-existence of those large cardinals with consistency strength greater or equal to 0# which even includes some relatively small large cardinals. Thus, the ω₁-st Erdős cardinal, $\eta_{\omega_1}$ , cannot exist in L. While L does contain the initial ordinals of those large cardinals (when they exist in a supermodel of L), and they are still initial ordinals in L, it excludes the auxiliary structures (e.g. measures) which endow those cardinals with their large cardinal properties.

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" (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.