Definition:Definable/Element

Definition

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

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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.

Sources

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