Extension Realizing All Types

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

Let $M$ be its universe.

There is an elementary extension $\mathcal N$ of $\mathcal M$ such that every type over $M$ (relative to $\mathcal M$) is realized in $\mathcal N$.

Proof
Let $S \left({\mathcal M}\right)$ denote the set containing all complete types over $M$ of every number of free variables.

Let $\kappa = \left\vert{S \left({\mathcal M}\right)}\right\vert$.

Use a bijection between $\kappa$ and $S \left({\mathcal M}\right)$ to write the elements of $S \left({\mathcal M}\right)$ as $p_\alpha$ for $\alpha < \kappa$.

For each $\alpha < \kappa$, let $N_\alpha$ be an elementary extension of $\mathcal M$ which has an ordered tuple $\bar a_\alpha$ realizing $p_\alpha$.

Such extensions and tuples exist by Type is Realized in some Elementary Extension.

We will construct the extension claimed by the theorem as the union over a chain of elementary extensions defined using Transfinite Induction.

At each step, we use the Elementary Amalgamation Theorem to add on $N_\alpha$.

Base case $\alpha = 0$:

Let $\mathcal B_0 = \mathcal M$.

Note that $\mathcal B_0$ is an elementary extension of $\mathcal M$ by choice of $\mathcal N_0$.

Limit ordinals $\alpha \le \kappa$:

Let $\displaystyle \mathcal B_\alpha = \bigcup_{\beta \mathop < \alpha} \mathcal B_\beta$

$\mathcal B_\alpha$ is an elementary extension of $\mathcal M$ by Union of Elementary Chain is Elementary Extension.

Successor ordinals $\alpha + 1$ for $\alpha < \kappa$:

We have that $\mathcal B_\alpha$ and $\mathcal N_\alpha$ are both elementary extensions of $\mathcal M$.

By the Elementary Amalgamation Theorem, there is:
 * an elementary extension $\mathcal B_{\alpha+1}$ of $\mathcal B_\alpha$

and:
 * an elementary embedding $g_\alpha: \mathcal N_\alpha \to \mathcal B_{\alpha + 1}$ which is the identity on $M$ viewed as a subset of $\mathcal N_\alpha$.

Note that $g_\alpha \left({\bar a_\alpha}\right)$ realizes $p_\alpha$ in $\mathcal B_{\alpha + 1}$ since $g_\alpha$ is elementary.

Thus $\mathcal B_{\alpha+1}$ is an elementary extension of $\mathcal M$ which contains $g_\alpha (\bar a_\alpha)$ realizing $p_\alpha$.

Now, let $\displaystyle \mathcal B = \bigcup_{\alpha < \kappa} \mathcal B_\alpha$.

$\mathcal B$ is an elementary extension of each $\mathcal B_\alpha$ by Union of Elementary Chain is an Elementary Extension.

In particular, $\mathcal B$ is an elementary extension of $\mathcal B_0 = \mathcal M$

We have that:
 * each $p_\alpha$ is realized in $\mathcal B_{\alpha + 1}$ by the corresponding $g_\alpha \left({\bar a_\alpha}\right)$

and:
 * $\mathcal B$ is an elementary extension of each $\mathcal B_{\alpha + 1}$

Thus we have that $g_\alpha \left({\bar a_\alpha}\right)$ realizes $p_\alpha$ in $\mathcal B$.