Upward Löwenheim-Skolem Theorem

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Let $T$ be an $\LL$-theory with an infinite model.

Then for each infinite cardinal $\kappa \ge \card \LL$, there exists a model of $T$ with cardinality $\kappa$.


The idea is:

to extend the language by adding $\kappa$ many new constants


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 $\LL^*$ be the language formed by adding new constants $\set {c_\alpha: \alpha < \kappa}$ to $\LL$.

Let $T^*$ be the $\LL^*$-theory formed by adding the sentences $\set {c_\alpha \ne c_\beta: \alpha, \beta < \kappa, \ \alpha \ne \beta}$ 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 $\MM$ of cardinality at most $\card \LL + \aleph_0$.

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 $\MM$.

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 $\MM^*$ of exactly size $\kappa$.

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


Source of Name

This entry was named for Leopold Löwenheim and Thoralf Albert Skolem.

Also see