Definition:Universal Model

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Definition

Let $T$ be an $\mathcal L$-theory.

Let $\kappa$ be an infinite cardinal.


A model $\mathcal M$ of $T$ is $\kappa$-universal if and only if:

for every model $\mathcal N$ of $T$ whose universe has cardinality strictly less than $\kappa$, there is an elementary embedding of $\mathcal N$ into $\mathcal M$.

That is, $\mathcal M$ is $\kappa$-universal if and only if:

for all models $\mathcal N \models T$ with cardinality $\left\vert{\mathcal N}\right\vert < \kappa$, there is an elementary embedding $j: \mathcal N \to \mathcal M$.


We say $\mathcal M$ is universal if it is $\kappa^+$-universal where $\kappa$ is the cardinality of the universe of $\mathcal M$ and $\kappa^+$ is the successor cardinal of $\kappa$.