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.