Incomplete countable theory T

Question

There is this problem I am working on that has two parts : Let $T$ be an incomplete countable theory. Either prove or provide an example for the following.

1)If T has an uncountable model, then T has a countable model.

2)If T has finite models and a denumerable model, then T has arbitrarily large finite models.

Any ideas of how to show the (1)?

In (2), I picked a cardinal $k$. I considered the $L'$-theory, which is the full theory of our infinite model and the extra axioms $c_a \neq c_b$ for all $a\neq b$. Is that enough?

Thank you in advance.

Answer 1

It sounds like you've already solved part (1): as others have said, you need to apply downward Löwenheim-Skolem.

For (2), the result is false. By way of a hint, can you write a sentence (in some language) that has only infinite models? Can you think of another sentence whose models are all smaller than some finite size? What happens when you take the disjunction of these sentences?

Answer 2

For the second question , what you could try to do is choose an $n$, such that you don't want there to be finite models with $n$ or more elements.

Once you've done that you can try and write the following sentences in a first order language :

"If there are $n$ distinct objects, then there are $n+1$ distinct objects"

And for $k\geq 0$, "If there are $n+k$ distinct objects, then there are $n+k+1$". Call this sentence $F_k$.

Then what kind of models does $T=\{F_k\mid k\in \Bbb{N}\}$ have ?