# Definition:Saturated Model

## Definition

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

Let $\kappa$ be an infinite cardinal.

A model $\mathcal{M}$ of $T$ is $\kappa$-saturated if for every subset $A$ of the universe of $\mathcal{M}$ of cardinality strictly less than $\kappa$, and for every $n\in\mathbb N$, every complete $n$-type $p$ over $A$ is realized in $\mathcal{M}$.

That is, $\mathcal{M}$ is $\kappa$-saturated if for all $A \subseteq \mathcal{M}$ with $|A|<\kappa$, and for all $n\in\mathbb N$, each $p \in S_{n}^{\mathcal{M}} (A)$ is realized in $\mathcal{M}$.

We say $\mathcal{M}$ is saturated if it is $\kappa$-saturated where $\kappa$ is the cardinality of the universe of $\mathcal{M}$.