---
tag: A6ZJ8MSCAY
kind: concept
name: Core model
msc: ["03E45", "03E55"]
specializes: [G78W36EGMB]
sources:
  - url: https://en.wikipedia.org/wiki/Core_model
same_as:
  wikidata: Q5170200
  wikipedia: Core model
---

## Summary

A core model is an [[inner model|G78W36EGMB]], usually written $K$, built from fine-structural pieces like the [[constructible universe|6TF2JBTPPJ]].
Unlike $L$, it is large enough to contain some large cardinal structure.
It is defined under an assumption that rules out large cardinals beyond a certain level.
An example of such an assumption is that there is no inner model with a [[Woodin cardinal|CMC2C1D53W]].
Under such an assumption the core model is close to the universe $V$, in the sense of covering properties like [[Jensen's covering theorem|RJE1W5NRE3]] for $L$.
The theory of the core model links large cardinals with the strength of combinatorial statements.

## Explanations

### L1 human

The constructible universe is a very thin and well understood world of sets.
If [[zero sharp|HCTNBZZQ0W]] does not exist, the constructible universe is close to the full universe, and $L$ plays the role of the core model.
If larger objects exist, one needs a bigger inner model that still has the good structure of $L$; this is the core model.
It is the largest inner model of its kind that can be built from below, and it approximates the full universe well.
Its exact definition depends on which large cardinals are assumed not to exist.