@BillyRubina For every cardinal number, there exists a larger cardinal number. Thus every first-order theory with an infinite model has a larger infinite model.

@BillyRubina Besides the answer @user76284 gave, a simpler answer is that there are at least two infinite cardinalities, that of ℕ and P(ℕ), so clearly LS implies that the existence of an infinite model implies the existence of another model with different domain size.

@MaliceVidrine right?

but damn haha lumsdaine, shulman, bauer and co are really clearing up that area

@BillyRubina - These are sort of two different questions. For you you get models of given cardinalities, the most common way is to add a-many or b-many new constants to the language, along with axioms saying that no two of these constants refer to the same thing. Since the hypothesis is that T has infinite models, you can satisfy every finite subset of this new theory, and compactness gets you a model.

Which may give you a model bigger than the actual cardinality you're aiming for, but you can whittle it down to exactly the desired cardinality.

As for how different they can be, that really depends on the theory. Some theories are really tame, others have a wide range of weird models.

It should also be mentioned that a lot of theories have very wild models even within the same cardinality. Even restricting to countable models, models of PA can get kind of strange looking.