---
tag: PNGY60EBDJ
kind: concept
name: Strongly minimal theory
msc: [03C45]
specializes: [T7JW5QY6KR]
sources:
  - url: https://en.wikipedia.org/wiki/Strongly_minimal_theory
same_as:
  wikidata: Q7624689
  wikipedia: Strongly minimal theory
---

## Summary

A strongly minimal theory is a complete first-order [[theory|92SWXST1ZM]] with infinite models.
In each of its models, every definable subset of the line is finite or cofinite.
These theories are the simplest infinite structures from the point of view of definability.
In a strongly minimal theory, algebraic closure behaves like linear span, so each model has a well-defined dimension.
They are basic building blocks of [[stable theories|T7JW5QY6KR]] and are central to the study of [[categorical theories|1JJCGNX240]].

## Explanations

### L1 human

Think of a structure where a formula in one variable can only cut out a few points, or everything except a few points.
Nothing in between, like a half-line or an infinite set with an infinite complement, can be described.
The theory of an infinite set with no structure at all is the simplest case: a formula can only pick out finitely many named elements or their complement.
Another case is the theory of [[algebraically closed fields|MT8TDVPBXB]]: a nonzero polynomial in one variable has finitely many roots.
So in $\C$, every definable set of numbers is finite or cofinite.