---
tag: Q9176JRJ5M
kind: statement
role: theorem
name: Gödel's consistency of CH and AC
msc: ["03E35", "03E45", "03E50"]
uses: [0HE5WDQ9FJ, 0HTGB13FE2, 0MN2MER9WN, 6TF2JBTPPJ, 4QKRPRCSXH]
about: [0MN2MER9WN, 0HTGB13FE2, 6TF2JBTPPJ]
sources:
  - url: https://en.wikipedia.org/wiki/Constructible_universe
---

## Statement

Let $L$ be the [[constructible universe|6TF2JBTPPJ]].

1. The axioms of [[ZF|0HE5WDQ9FJ]] and the [[axiom of constructibility|4QKRPRCSXH]] $V = L$ hold in $L$.
2. ZF together with $V = L$ proves the [[axiom of choice|0HTGB13FE2]] and the generalized [[continuum hypothesis|0MN2MER9WN]] $2^{\aleph_\alpha} = \aleph_{\alpha + 1}$ for every ordinal $\alpha$.
3. Hence, if ZF is consistent, then ZF together with the axiom of choice and the generalized continuum hypothesis is consistent.

## Remarks

The construction of $L$ can be carried out inside any model of ZF, and it is absolute, so $L$ computed inside $L$ is $L$ again; this gives part 1.
The axiom of choice holds in $L$ because $L$ has a definable well-order.
A set enters at a least stage and is defined there by a formula with parameters, and these data can be ordered.
For the continuum hypothesis, assume $V = L$; the condensation lemma shows that every subset of $\omega_\alpha$ lies in $L_{\omega_{\alpha + 1}}$, a set of size $\aleph_{\alpha + 1}$.
The result is a relative consistency proof: by [[Gödel's second incompleteness theorem|X9WYNYE29C]], ZF cannot prove its own consistency, so the hypothesis "ZF is consistent" cannot be removed.
Together with [[Cohen's theorem|EMGBZ46GCD]], it shows that the continuum hypothesis is independent of ZFC.

## Explanations

### L1 human

The continuum hypothesis says that there is no size of infinity strictly between the natural numbers and the real numbers.
Gödel found a world of sets, the constructible universe, in which all the usual axioms hold and the continuum hypothesis is true as well.
So the usual axioms can never disprove the continuum hypothesis.
The same world also shows that the axiom of choice cannot be disproved from the other axioms.

### L2 human

The theorem is the first half of the independence of the continuum hypothesis; the second half is Cohen's forcing argument.
The method is the method of inner models: one finds a definable class in which the axioms hold together with the desired statement.
Inner models alone cannot show that the continuum hypothesis can fail: if $V = L$, then $L$ is the only inner model, and the continuum hypothesis holds in it.
To show that the continuum hypothesis can fail, one needs a way to enlarge the universe, and this is what forcing does.