Definition:Definable

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.

Also defined as
The following definition is sometimes used. It is not equivalent in general. However, these definitions are equivalent when working in a saturated model. This is proved in Definable iff Singleton Orbit.

$\bar b$ is definable over $A$ if every $A$-automorphism is an $A, b$-automorphism.

Definable Set
We say that the subset $A$ is a definable set in $\MM$ if there is a formula $\map \phi x$ such that $a \in A$ iff $\MM \models \map \phi a$.