Editing Axiom of constructibility (section)

== Statements true in ''L'' ==
{{Unreferenced section|date=November 2017}}
Here is a list of propositions that hold in the [[constructible universe]] (denoted by ''L''):

* The [[Continuum hypothesis#The generalized continuum hypothesis|generalized continuum hypothesis]] and as a consequence
** The [[axiom of choice]]
* [[Diamondsuit]]
** [[Clubsuit]]
* [[Global square]]
* The existence of [[Morass (set theory)|morasses]]
* The negation of the [[Suslin hypothesis]]
* The non-existence of [[zero sharp|0<sup>#</sup>]] and as a consequence
** The non existence of all [[large cardinals]] that imply the existence of a [[measurable cardinal]]
* The existence of a <math>\Delta_2^1</math> set of reals (in the [[analytical hierarchy]]) that is not [[Lebesgue measure|measurable]].
* The truth of [[Whitehead problem|Whitehead's conjecture]] that every [[abelian group]] ''A'' with [[Ext functor|Ext]]<sup>1</sup>(''A'', '''Z''') = 0 is a [[free abelian group]].
* The existence of a definable [[well-order]] of all sets (the formula for which can be given explicitly).  In particular, ''L'' satisfies [[ordinal definable set|V=HOD]].
* The existence of a primitive recursive class surjection <math>F:\textrm{Ord}\to\textrm{V}</math>, i.e. a class function from Ord whose range contains all sets. <ref>W. Richter, [[Peter Aczel|P. Aczel]], [https://www.duo.uio.no/bitstream/handle/10852/44063/1973-13.pdf Inductive Definitions and Reflecting Properties of Admissible Ordinals] (1974, p.23). Accessed 30 August 2022.</ref>

Accepting the axiom of constructibility (which asserts that every set is [[constructible universe|constructible]]) these propositions also hold in the [[von Neumann universe]], resolving many propositions in set theory and some interesting questions in [[mathematical analysis|analysis]].