Big Implies Saturated

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Theorem

Let $\mathcal M$ be an $\mathcal L$-structure.

Let $\kappa$ be a cardinal.


If $\mathcal M$ is $\kappa$-big, then it is $\kappa$-saturated.


Proof

The idea of the proof is:

to go to some elementary equivalent structure where a type is realized

and:

to interpret a new relation symbol as the singleton containing some realization of the type.

This lets us write sentences about the extension saying that:

there must exist an element satisfying the relation

and:

every element satisfying the relation satisfies all of the formulas in the type.

Then using bigness, these sentences carry over into $\mathcal M$.


Let $p$ be a complete $n$-type over $A$, where $A$ is a subset of the universe of $\mathcal M$ with cardinality strictly less than $\kappa$.


We are to show that $p$ is realized in $\mathcal M$.


Let $\mathcal L_A$ be the language obtained from $\mathcal L$ by adding constant symbols for each element of $A$.

Viewing $\mathcal M$ as an $\mathcal L_A$-structure in the natural way, let $\operatorname {Th}_A (\mathcal M)$ be the collection of $\mathcal L_A$-sentences satisfied by $\mathcal M$.


By definition of type, $p \cup \operatorname {Th}_A \left({\mathcal M}\right)$ is satisfiable by some $\mathcal L_A$-structure $\mathcal N$.


Since $\mathcal N$ satisfies $p$, some $n$-ordered tuple $\bar b$ in it realizes $p$.


Now, let $\mathcal L_A^*$ be the language obtained from $\mathcal L_A$ by adding a new relation symbol $R$.

We can extend $\mathcal N$ to be an $\mathcal L_A^*$-structure by interpreting the symbol $R$ as the singleton $R^\mathcal N = \left\{{\bar b}\right\}$.


Since $\mathcal N \models \operatorname {Th}_A \left({\mathcal M}\right)$, we have that $\mathcal M$ and $\mathcal N$ are elementary equivalent as $\mathcal L_A$-structures.


We have that $\mathcal M$ is $\kappa$-big by assumption.

Hence there is some relation $R^\mathcal M$ on the universe of $\mathcal M$ such that:

$\left({\mathcal M, R^\mathcal M}\right)$ is elementary equivalent to $\left({\mathcal N, R^\mathcal N}\right)$ as $\mathcal L_A^*$-structures.


But $\left({\mathcal N, R^\mathcal N}\right)$ satisfies the $\mathcal L_A^*$-sentences:

$\exists \bar x R \left({\bar x}\right)$

and

$\forall \bar x \left({R \left({\bar x}\right) \to \phi \left({\bar x}\right)}\right)$

for each $\phi \left({\bar x}\right)$ in $p$.


Hence $\left({\mathcal M, R^\mathcal M}\right)$ also satisfies these sentences, and so there must be some $\bar d \in R^\mathcal M$ realizing $p$.

So $\mathcal M$ realizes $p$.

$\blacksquare$