Mathematical logic and foundations Definition

Definable set

Definition

Let $L$ be a signature, let $\mathcal{M}$ be an $L$ -structure with domain $M$ , and let $A \subseteq M$ . A set $X \subseteq M^n$ is definable with parameters from $A$ when there are an $L$ -formula $\varphi(x_1, \dots, x_n, y_1, \dots, y_m)$ and $b_1, \dots, b_m \in A$ such that $X = \set{(a_1, \dots, a_n) \in M^n : \mathcal{M} \models \varphi(a_1, \dots, a_n, b_1, \dots, b_m)}.$ The set $X$ is definable when it is definable with parameters from $M$ . It is definable without parameters, or $\emptyset$ -definable, when it is definable with parameters from $\emptyset$ .

Remarks

Example: in the field $(\R, +, \cdot, 0, 1)$ , the set $[0, \infty)$ is definable without parameters by $\varphi(x) = \exists y \, (y \cdot y = x)$ . Indeed a real number is a square of a real number exactly when it is nonnegative.

Non-example: in $(\R, <)$ , the set $\set{0}$ is not definable without parameters. The map $\sigma(x) = x + 1$ is an automorphism of $(\R, <)$ , and an automorphism maps each $\emptyset$ -definable set onto itself; but $\sigma(\set{0}) = \set{1}$ . With the parameter $0$ , the formula $x = y_1$ defines $\set{0}$ .

The definable subsets of $M^n$ are closed under finite unions, finite intersections and complements, and projections of definable sets are definable.

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Structure concept
Well-formed formula concept
Definable set definition

Tag	`ZTTPPXGB30`
URI	`m2://ZTTPPXGB30`
Kind	definition
Writing	W0 stub
MSC	03C07, 03C40
Defines	Definable set
Uses	Structure, Well-formed formula
Changed	2 November 2026

Definable set

Definition

Remarks

Reading path

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