---
tag: 8WZF9SBP39
kind: concept
name: Elementary substructure
msc: [03C07]
specializes: [676H7EGTKJ]
sources:
  - url: https://en.wikipedia.org/wiki/Elementary_equivalence
---

## Summary

An elementary substructure of a [[structure|983GZBQYT4]] is a [[substructure|676H7EGTKJ]] that agrees with the big structure on first-order formulas.
Each formula with parameters from the substructure has the same truth value in both.
The big structure is then an elementary extension of the small one.
Elementary substructures are the substructures that first-order logic cannot distinguish from the whole, even when it may name their elements.

## Explanations

### L1 human

A substructure can look different from the whole structure even when it is isomorphic to it.
The positive integers sit inside the natural numbers with the usual order.
In the positive integers the number $1$ is the least element, but in the natural numbers it is not, since $0 < 1$.
An elementary substructure forbids such differences: every first-order property of its elements is the same in the small and in the big structure.
The rational numbers are an elementary substructure of the real numbers, as ordered sets.