---
tag: 4QKRPRCSXH
kind: concept
name: Axiom of constructibility
msc: ["03E45", "03E35"]
sources:
  - url: https://en.wikipedia.org/wiki/Axiom_of_constructibility
same_as:
  wikidata: Q1151112
  wikipedia: Axiom of constructibility
---

## Summary

The axiom of constructibility, written $V = L$, says that every set is constructible, that is, every set lies in the [[constructible universe|6TF2JBTPPJ]] $L$.
It is consistent with [[ZF|0HE5WDQ9FJ]] when ZF is consistent, and it decides many questions that ZFC leaves open.
It implies the [[axiom of choice|0HTGB13FE2]] and the generalized [[continuum hypothesis|0MN2MER9WN]].
It is incompatible with large cardinals such as [[measurable cardinals|AKBWT323RQ]], so most set theorists do not accept it as true.

## Explanations

### L1 human

The constructible universe is the thinnest possible world of sets that can be built from definable pieces.
The axiom of constructibility says that there are no other sets at all.
It is like saying that every set has a description at some stage of one fixed construction.
In such a world many hard questions get a definite answer; for example, there are exactly $\aleph_1$ real numbers.