---
tag: CY1TXHF6KQ
kind: concept
name: Definable set
msc: [03C07, 03C40]
sources:
  - url: https://en.wikipedia.org/wiki/Definable_set
same_as:
  wikidata: Q2895824
  wikipedia: Definable set
---

## Summary

A definable set in a [[structure|983GZBQYT4]] is a set of tuples of elements that is cut out by a first-order formula, possibly with parameters.
Definable sets are the main objects of study in modern model theory.
Their behaviour, such as in o-minimal or stable structures, describes how complicated a structure is.

## Explanations

### L1 human

A definable set is a set that you can describe by a first-order formula in the language of the structure.
In the field of real numbers, the set of nonnegative numbers is definable by the formula "$x$ is a square", that is, $\exists y \, (y \cdot y = x)$.
The unit circle is definable by $x \cdot x + y \cdot y = 1$.
In the ordered set $(\R, <)$ without parameters, no single number can be defined, because a shift $x \mapsto x + c$ preserves the order.