---
tag: 1GYEDS227W
kind: definition
name: Model theory
msc: [03C07, 03C98]
defines: HFTBG5PMXC
uses: [92SWXST1ZM, MVCH6N9SNW, 983GZBQYT4, 086QKPF0A6]
sources:
  - url: https://en.wikipedia.org/wiki/Model_theory
---

## Definition

Model theory is the part of mathematical logic whose objects are the following.

1. Formal languages, most often the language of [[first-order logic|086QKPF0A6]] over a signature.
2. [[Theories|92SWXST1ZM]], that is, sets of sentences of such a language.
3. [[Structures|983GZBQYT4]] for the signature, and the relation of truth between structures and sentences, in particular the [[models|MVCH6N9SNW]] of theories.

It studies the classes of models of theories and the sets definable in structures.

## Remarks

Example: the statement that every field elementarily equivalent to $\C$ is algebraically closed of characteristic $0$ is a result of model theory.
It concerns the class of models of the complete theory $\mathrm{Th}(\C)$.

Non-example: the computation of the homology of a sphere is not a result of model theory, because it concerns no formal language and no relation of truth.

Classical model theory works with first-order logic; other parts study [[infinitary logic|PQG7VAZ1R2]], finite structures or [[second-order logic|G6X7SZSNC5]].

## Explanations

### L1 human

Model theory is the study of mathematical structures through the sentences that are true in them.
It sits between logic and algebra: its theorems are about formulas, and its applications are often about fields, groups and orders.