The Upwards Löwenheim-Skolem Theorem

If a finite order theory has an infinite model, then has an uncountable model.

Proof

Add uncountably many new constants to the language.
Consider

By assumption has an infinite model, which is a model of every finite subset of .
By Compactness Theorem has a model ,
i.e. a model of together with an injection

Note

For any set , can take Hartogs’ Lemma
Then the proof yields a model of that cannot inject into .