Upward Löwenheim-Skolem Theorem

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Theorem

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 $\kappa$.


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 $\LL^*$ 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 $\kappa$.

$\blacksquare$


Source of Name

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


Also see


Sources