Definition:Big Model

Definition

Let $\MM$ be an $\LL$-structure with universe $M$.

Let $\kappa$ be a cardinal.

$\MM$ is $\kappa$-big if for every subset $A \subset M$ with cardinality $\card A < \kappa$, the following holds:

if $\LL_A$ is the language obtained from $\LL$ by adding new constant symbols for each $a \in A$, then

for every language $\LL_A^*$ obtained by adding a new relation symbol $R$ to $\LL_A$, and

for every $\LL_A^*$-structure $\NN$ such that $\MM$ and $\NN$ are elementary equivalent as $\LL_A$-structures,

there is a relation $R^\MM$ on $M$ such that $\struct {\MM, R^\MM}$ is elementary equivalent to $\NN$ as an $\LL_A^*$-structure.

Note

Note that any function symbol or constant symbol can be replaced by a relation symbol along with suitable sentences mentioning only that symbol. So, the focus on relation symbols in the definition is just for convenience.

Informally, being $\kappa$-big means that $\MM$ already has all of the structural features that are consistent with the behavior of $\MM$ and the parameters in $A$.