---
tag: Q1VYK7362X
kind: definition
name: Axiom of constructibility
msc: ["03E45", "03E35"]
defines: 4QKRPRCSXH
uses: [6TF2JBTPPJ, 0HE5WDQ9FJ, 0YSCSX5YZC]
sources:
  - url: https://en.wikipedia.org/wiki/Axiom_of_constructibility
---

## Definition

Work in [[ZF|0HE5WDQ9FJ]], and let $L = \bigcup_\alpha L_\alpha$ be the [[constructible universe|6TF2JBTPPJ]], where $\alpha$ runs over the [[ordinals|0YSCSX5YZC]].
The *axiom of constructibility* is the sentence
$$\forall x \, \exists \alpha \, (\alpha \text{ is an ordinal} \wedge x \in L_\alpha).$$
It is written $V = L$, where $V$ is the class of all sets.

## Remarks

The sentence holds inside $L$ itself: the construction of $L_\alpha$ is absolute for transitive models of ZF that contain $\alpha$, so the class $L$ computed inside $L$ is again $L$.
This gives the relative consistency of $V = L$ with ZF.
For a non-example, take a forcing extension $L[c]$ of $L$ by a Cohen real $c$; the real $c$ is not in $L$, so $V = L$ fails in $L[c]$.
Under $V = L$ there is a well-order of the reals that is $\Delta^1_2$ in the [[projective hierarchy|8DZTNVTGY2]].
Scott proved that $V = L$ implies that there is no [[measurable cardinal|AKBWT323RQ]].