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 .