---
tag: RMTZ5J64FX
kind: concept
name: Saturated model
msc: [03C50]
sources:
  - url: https://en.wikipedia.org/wiki/Saturated_model
same_as:
  wikidata: Q1473532
  wikipedia: Saturated model
---

## Summary

A saturated model is a model that realizes as many [[types|9V71Q3SD1G]] as its size allows.
More precisely, $M$ is $\kappa$-saturated when every complete type over fewer than $\kappa$ parameters of $M$ is realized in $M$.
It is saturated when it is $\abs{M}$-saturated.
Saturated models of a complete theory are unique up to isomorphism in each cardinality.
Every model of the theory of at most the same size embeds elementarily into a saturated one.
They serve as large "universal" models in which most model-theoretic work is done.

## Explanations

### L1 human

A saturated model has no gaps that its own language can describe.
If a description of an element mentions only a few given elements and is consistent, such an element already exists in the model.
For example, the rational numbers with their order are saturated.
Given finitely many rationals, a consistent wish such as "between $1/3$ and $1/2$" or "larger than all of them" is met by some rational number.
The integers with their order are not: no integer is "larger than $0$ and with infinitely many integers between it and $0$".