---
tag: HFTBG5PMXC
kind: concept
name: Model theory
msc: [03C07, 03C98]
sources:
  - url: https://en.wikipedia.org/wiki/Model_theory
same_as:
  wikidata: Q467606
  wikipedia: Model theory
---

## Summary

Model theory is the branch of mathematical logic that studies formal [[theories|92SWXST1ZM]] together with their [[models|MVCH6N9SNW]].
It asks which classes of [[structures|983GZBQYT4]] a set of axioms describes, and which subsets of a structure a formal language can define.
Its tools, such as the [[compactness theorem|1NR3HAWG9D]], have applications in algebra, number theory and geometry.

## Explanations

### L1 human

Model theory looks at mathematics from two sides at once: the sentences you write and the objects that make them true.
For example, the field axioms are sentences, and the fields $\Q$, $\R$ and $\C$ are objects that satisfy them.
A typical question is: if two structures satisfy the same sentences, how alike must they be?
Another is: which sets of points can a formula describe inside a given structure?