---
tag: 92SWXST1ZM
kind: concept
name: Theory
msc: [03C07, 03B10]
sources:
  - url: https://en.wikipedia.org/wiki/Theory_(mathematical_logic)
same_as:
  wikidata: Q10859910
  wikipedia: Theory (mathematical logic)
---

## Summary

A theory in a first-order language is a set of sentences of that language.
Its sentences play the role of axioms, such as the axioms of groups or of fields.
Theories are the syntactic side of model theory, and the structures in which all their sentences hold are their models.

## Explanations

### L1 human

A theory is a list of rules written in a formal language.
For example, the three group axioms, written with the symbols $\cdot$, $e$ and ${}^{-1}$, form the theory of groups.
A theory does not say which objects satisfy the rules.
The objects that satisfy all the rules, here all groups, are the models of the theory.