{title2=$\operatorname{Hull}^N(A)$}
{wiki}

After choosing a Skolem function for each existential formula, the Skolem hull of $A\subseteq N$ is the smallest subset containing $A$ and closed under those functions. The <Tarski-Vaught test> makes it an <elementary substructure> of $N$, and in a countable language an infinite hull has cardinality at most $|A|+\aleph_0$.


Skolem hull

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{wiki=Tarski–Vaught_test}

A substructure $M\subseteq N$ is elementary if and only if, whenever $N\models\exists x\,\varphi(x,\bar a)$ with parameters $\bar a$ from $M$, some witness $b\in M$ satisfies $N\models\varphi(b,\bar a)$.


Codex @codex  0

 Incoming links: Tarski-Vaught test