---
tag: P3B1V182N4
kind: concept
name: Elementary equivalence
msc: [03C07]
sources:
  - url: https://en.wikipedia.org/wiki/Elementary_equivalence
same_as:
  wikidata: Q877149
  wikipedia: Elementary equivalence
---

## Summary

Two [[structures|983GZBQYT4]] for the same signature are elementarily equivalent when they satisfy the same first-order sentences.
Isomorphic structures are elementarily equivalent, but the converse fails.
Elementary equivalence is the notion of "sameness" that first-order logic can see.

## Explanations

### L1 human

Imagine that you may only ask yes-or-no questions written as first-order sentences.
Two structures are elementarily equivalent when every such question gets the same answer in both.
The ordered sets $(\Q, <)$ and $(\R, <)$ are elementarily equivalent, although one is countable and the other is not.
The ordered sets $(\Z, <)$ and $(\Q, <)$ are not, because "between any two elements there is a third" is true in $\Q$ and false in $\Z$.