Definition:Universal Model

Definition

Let $T$ be an $\LL$-theory.

Let $\kappa$ be an infinite cardinal.

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

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

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

for all models $\NN \models T$ with cardinality $\card \NN < \kappa$, there is an elementary embedding $j: \NN \to \MM$.

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