---
tag: YR6HCMABTE
kind: concept
name: Model complete theory
msc: [03C10]
sources:
  - url: https://en.wikipedia.org/wiki/Model_complete_theory
same_as:
  wikidata: Q6888319
  wikipedia: Model complete theory
---

## Summary

A theory is model complete when every embedding between two of its models is elementary.
Then a smaller model agrees with a larger one on all first-order statements about its own elements.
Abraham Robinson introduced the notion as an algebraic counterpart of [[quantifier elimination|7FDDXV3JK7]], which implies it.
Model completeness is equivalent to the property that every formula is equivalent to an existential formula.
The theory of real closed fields in the language of rings is a standard model complete theory without quantifier elimination.

## Explanations

### L1 human

Suppose one model of a theory sits inside another.
The theory is model complete when the inner model cannot tell the difference.
Any first-order statement about its elements is true in the inner model exactly when it is true in the outer one.
For example, the real algebraic numbers sit inside the real numbers, and both are real closed fields.
A polynomial equation with real algebraic coefficients has a solution in one field exactly when it has one in the other.