Definition:Definable

Definition

Definable Element

Let $\MM$ be an $\LL$-structure with universe $M$.

Let $A$ be a subset of $M$.

Let $\bar b$ be an ordered $n$-tuple of elements from $M$.

Let $\LL_A$ be the language formed by adding constant symbols to $\LL$ for each element of $A$.

$\bar b$ is definable over $A$ if there is an $\LL_A$-formula $\map \phi {\bar x}$ with $n$ free variables such that the set $\set {\bar m \in M^n :\MM \models \map \phi {\bar m} }$ contains $\bar b$ but nothing else.

Definable Set

$A$ is a definable set in $\MM$ if and only if there exists a formula $\map \phi x$ such that:

$a \in A \iff \MM \models \map \phi a$

Sources

• 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 4.15$