Editing Axiom of constructibility (section)

== Implications ==
The axiom of constructibility implies the [[axiom of choice]] (AC), given [[Zermelo–Fraenkel set theory]] without the axiom of choice (ZF).  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 [[Continuum hypothesis#The generalized continuum hypothesis|generalized continuum hypothesis]], the negation of [[Suslin's hypothesis]], and the existence of an [[analytical hierarchy|analytical]] (in fact, <math>\Delta^1_2</math>) [[non-measurable]] set of [[real number]]s, 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 [[zero sharp|0<sup>#</sup>]], which includes some "relatively small" large cardinals. For example, no cardinal can be ω<sub>1</sub>-[[Erdős cardinal|Erdős]] in ''L''.  While ''L'' does contain the [[initial ordinal]]s 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. [[measurable cardinal|measures]]) that 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 [[Philosophy of mathematics#Mathematical realism|realist]] bent, who believe that the axiom of constructibility is either true or false, most believe that it is false.<ref>"Before Silver, many mathematicians believed that <math>V\neq L</math>, but after Silver they knew why." - from {{cite|author=P. Maddy|journal=The Journal of Symbolic Logic|title=Believing the Axioms. I|url=https://www.cs.umd.edu/~gasarch/BLOGPAPERS/belaxioms1.pdf|volume=53|year=1988}}, p. 506</ref> This is in part because it seems unnecessarily "restrictive", as it allows only certain subsets of a given set (for example, <math>0^\sharp\subseteq \omega</math> can't exist), 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 axiom]]s. This point of view is especially associated with the [[Cabal (set theory)|Cabal]], or the "California school" as [[Saharon Shelah]] would have it.