---
tag: G78W36EGMB
kind: concept
name: Inner model
msc: ["03E45"]
sources:
  - url: https://en.wikipedia.org/wiki/Inner_model
same_as:
  wikidata: Q6035663
  wikipedia: Inner model
---

## Summary

An inner model is a transitive class of sets that contains all [[ordinals|0YSCSX5YZC]] and satisfies the axioms of [[ZF|0HE5WDQ9FJ]].
It is a smaller universe of sets that lives inside the universe $V$ and has the same ordinals.
Inner models give consistency proofs: a statement true in an inner model is consistent with ZF.
The [[constructible universe|6TF2JBTPPJ]] $L$ is the smallest inner model, and the [[core model|A6ZJ8MSCAY]] is a larger inner model built to hold large cardinals.

## Explanations

### L1 human

Think of the universe of all sets as a large building.
An inner model is a smaller building inside it that has the same height, because it has every ordinal, but fewer rooms on each floor.
Inside the smaller building all the usual rules of set theory still hold.
For example, the constructible sets form such a building, and in it the continuum hypothesis is true.
So the continuum hypothesis cannot be disproved from the usual axioms.