Type is Realized in some Elementary Extension

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Theorem

Let $\MM$ be an $\LL$-structure.

Let $A$ be a subset of the universe of $\MM$.

Let $p$ be an $n$-type over $A$.

There exists an elementary extension of $\MM$ which realizes $p$.

Proof

The idea is to work in a language with constant symbols for all elements of $\MM$ and show that the union of $p$ and the elementary diagram of $\MM$ is satisfiable.

Since $\MM$ naturally embeds into any model of such a theory, this will prove the theorem.

Let $\LL_\MM$ be the language obtained by adding to $\LL$ constant symbols for each element of $\MM$.

Denote by $\map {\operatorname {Diag}_{\mathrm {el} } } \MM$ the elementary diagram of $\MM$.

Let $T$ be $p \cup \map {\operatorname {Diag}_{\mathrm{el} } } \MM$.

We will show that $T$ is finitely satisfiable.

It will follow by the Compactness Theorem that $T$ is satisfiable.

To this end, let $\Delta$ be a finite subset of $T$.

We have that $\Delta$ is finite.

Thus it consists of:

finitely many $\LL_A$-sentences $\phi_0, \dotsc, \phi_n$ from $p$ (which are $\LL_\MM$ sentences since $A \subseteq \MM$)

along with:

finitely many $\LL_\MM$-sentences $\psi_0,\dots,\psi_k$ from $\map {\operatorname{Diag}_{\mathrm{el} } } \MM$.

By definition, $p$ is satisfiable by some $\LL_A$-structure $\NN$ such that:

$\NN \models p \cup \map {\operatorname{Th}_A} \MM$

Thus, since $\phi_0, \dotsc, \phi_n \in p$:

$\NN$ satisfies $\phi_0, \dotsc, \phi_n$.

We will show that the same $\NN$ also satisfies $\psi_0, \dotsc, \psi_k$.

The obstacle to overcome is that the $\psi_i$ are $\LL_\MM$-formulas, and we only know $\NN$ as an $\LL_A$-structure which satisfies sentences with parameters from $A$.

The $\psi_i$ may have parameters from $\MM$ outside of $A$.

The idea is to quantify away the excess parameters and appropriately select the interpretation of new symbols so that $\NN$ is a good $\LL_\MM$-structure.

Explicitly:

Let $\psi$ be the conjunction $\psi_0 \wedge \cdots \wedge \psi_k$.

Note that since $\psi$ is an $\LL_\MM$-sentence, it can be written as an $\LL_A$-formula $\map \psi {\bar b}$, where $\bar b$ is a tuple of parameters from $\MM$ not in $A$.

By existentially quantifying away the tuple $\bar b$, we obtain an $\LL_A$-sentence $\exists \bar x \map \psi {\bar x}$.

Now, since $\MM \models \map \psi {\bar b}$, we have:

$\MM \models \exists \bar x: \map \psi {\bar x}$

Hence $\exists \bar x: \map \psi {\bar x}$ is in $\map {\operatorname{Th}_A} \MM$.

By choice of $\NN$, it follows that:

$\NN \models \exists \bar x: \map \psi {\bar x}$

and thus there must be some tuple $\bar c$ of elements from $\NN$ such that:

$\NN \models \map \psi {\bar c}$

Now, by interpreting the $\LL_\MM$-symbols $\bar b$ as the elements $\bar c$, we can view $\NN$ as an $\LL_\MM$-structure which satisfies:

$\phi_0 \wedge \cdots \wedge \phi_n \wedge \psi_0 \wedge \cdots \wedge \psi_k$.

Thus $\NN$ satisfies all of $\Delta$.

This demonstrates that $T$ is finitely satisfiable and hence satisfiable by the Compactness Theorem.

This means that there is an $\LL_\MM$-structure $\MM^*$ which satisfies:

$p \cup \map {\operatorname{Diag}_{\mathrm {el} } } \MM$

Since $\MM^*$ interprets a symbol for each element of $\MM$, there is an obvious embedding of $\MM$ into $\MM^*$.

This embedding is elementary since $\MM^*$ satisfies the elementary diagram of $\MM$.

Thus $\MM^*$ is an elementary extension of $\MM$.

Finally, since $\MM^*$ satisfies $p$, there must be a tuple of elements $\bar d$ such that $\MM^* \models \map \phi d$ for each $\map \phi {\bar v} \in p$.

Thus $\MM^*$ realizes $p$.

$\blacksquare$