Upward Löwenheim-Skolem Theorem

Theorem
Let $T$ be an $\mathcal L$-theory with an infinite models.

Then for each infinite cardinal $\kappa \ge \left|{\mathcal L}\right|$, there exists a model of $T$ with cardinality $\kappa$.

Proof
The idea is:
 * to extend the language by adding $\kappa$ many new constants

and:
 * to extend the theory by adding sentences asserting that these constants are distinct.

It is shown that this new theory is finitely satisfiable using an infinite model of $T$.

Compactness then implies that the new theory has a model.

Some care needs to be taken to ensure that we construct a model of exactly size $\kappa$.

Let $\mathcal L^*$ be the language formed by adding new constants $\left\{ {c_\alpha: \alpha < \kappa}\right\}$ to $\mathcal L$.

Let $T^*$ be the $\mathcal L^*$-theory formed by adding the sentences $\left\{ {c_\alpha \ne c_\beta: \alpha, \beta < \kappa, \ \alpha \ne \beta}\right\}$ to $T$.

We show that $T^*$ is finitely satisfiable:

Let $\Delta$ be a finite subset of $T^*$.

Then $\Delta$ contains:
 * finitely many sentences from $T$

along with:
 * finitely many sentences of the form $c_\alpha \ne c_\beta$ for the new constant symbols.

Since $T$ has an infinite model, it must have a model $\mathcal M$ of cardinality at most $\left|{\mathcal L}\right| + \aleph_0$.

This model already satisfies everything in $T$.

So, since we can find arbitrarily many distinct elements in it, it can also be used as a model of $\Delta$ by interpreting the finitely many new constant symbols in $\Delta$ as distinct elements of $\mathcal M$.

Since $T^*$ is finitely satisfiable, it follows by the Compactness Theorem that $T^*$ itself is satisfiable.

Since $T^*$ ensures the existence of $\kappa$ many distinct elements, this means it has models of size at least $\kappa$.

It can be proved separately or observed from the ultraproduct proof of the compactness theorem that $T^*$ then has a model $\mathcal M^*$ of exactly size $\kappa$.

Since $T^*$ contains $T$, $\mathcal M^*$ is a model of $T$ of size $\kappa$.

Also see

 * Overflow Theorem