---
tag: T7JW5QY6KR
kind: concept
name: Stable theory
msc: [03C45]
sources:
  - url: https://en.wikipedia.org/wiki/Stable_theory
same_as:
  wikidata: Q7595779
  wikipedia: Stable theory
---

## Summary

A complete theory is stable when its models have few [[types|9V71Q3SD1G]].
For some infinite cardinal $\lambda$, every parameter set of size $\lambda$ has at most $\lambda$ complete types over it.
Equivalently, no formula can define an infinite linear order on tuples, a condition called the order property.
Stable theories, introduced and developed by Saharon Shelah, admit a good theory of independence and dimension.
Algebraically closed fields, vector spaces and modules give stable theories, while dense linear orders and real closed fields do not.

## Explanations

### L1 human

Stability measures how complicated the possible behaviours of elements in a theory are.
In a stable theory, fixing a set of parameters leaves relatively few ways for a new element to sit relative to them.
In the rational numbers with their order, every cut of a countable set gives a different behaviour, so there are too many, and the theory is unstable.
In algebraically closed fields, a new element is either algebraic over the parameters, in finitely many ways for each polynomial, or transcendental, and this keeps the count small.