Definition:Definable
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Definable Element
Let $\mathcal M$ be an $\mathcal L$-structure with universe $M$.
Let $A$ be a subset of $M$.
Let $\bar b$ be an ordered $n$-tuple of elements from $M$.
Let $\mathcal L_A$ be the language formed by adding constant symbols to $\mathcal L$ for each element of $A$.
$\bar b$ is definable over $A$ if there is an $\mathcal L_A$-formula $\phi \left({\bar x}\right)$ with $n$ free variables such that the set $\left\{{\bar m \in M^n : \mathcal M \models \phi \left({\bar m}\right)}\right\}$ contains $\bar b$ but nothing else.
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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 $\mathcal M$ if there is a formula $\phi \left({x}\right)$ such that $a \in A$ iff $\mathcal M \models \phi \left({a}\right)$.
Sources
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 4.15$