---
tag: MVCH6N9SNW
kind: concept
name: Model
msc: [03C07]
sources:
  - url: https://en.wikipedia.org/wiki/Structure_(mathematical_logic)
---

## Summary

A model of a [[theory|92SWXST1ZM]] is a [[structure|983GZBQYT4]] in which every sentence of the theory is true.
Models give meaning to axioms: the models of the group axioms are exactly the groups.
A theory with a model is consistent, and by [[Gödel's completeness theorem|WQ1ZE255SF]] every consistent first-order theory has a model.

## Explanations

### L1 human

A theory is a list of rules, and a model is a concrete world in which all the rules hold.
The rules "$<$ is a strict linear order with no least and no greatest element" have the integers with their usual order as a model.
The rational numbers with their usual order are another model of the same rules.
The set $\set{1, 2, 3}$ with its usual order is not a model, because $1$ is a least element.