Counter example to Ryll-Nardzewski

My definition of $$\omega-$$categorical: a theory $$T$$ is $$\omega-$$categorical if any two countable models of $$T$$ are isomorphic. In particular, if $$T$$ doesn't have a countable model, then $$T$$ is $$\omega-$$categorical.

I am solving some exercises from Hodges shorter model theory, and while solving Ex 6.3.1, I wanted to figure out the following:

I either want to find an example or prove that such example cannot exist, of the following:

Find an example of a language $$L$$ and an infinite $$L$$ structure $$M$$, such that there is a finite tuple $$\vec{m}\in M$$ such that precisely one of the following two theories is $$\omega-$$categorical: $$Th(M)$$ and $$Th(M,\vec{m})$$.

If $$L$$ is countable, then by Ryll-Nardzewski, using the fact that "$$S_n(Th(M))$$ is finite for all $$n$$ iff $$S_n(Th(M,\vec{m}))$$ is finite for all $$n$$", we can see that such an example cannot exist.

Thanks for any help!

• I'm confused about what exactly you're asking. It seems like you know that when $L$ is countable, and $\overline{m}$ is a finite tuple from an $L$-structure $M$, then $\mathrm{Th}(M)$ is $\omega$-categorical if and only if $\mathrm{Th}(M,\overline{m})$ is $\omega$-categorical. So are you asking about whether this can fail when $L$ is uncountable? Nov 17 at 16:26
• by modifying the example here I think it is not hard to find examples of theories with many non-isomorphic countable models, but such that adding a single constant can preclude the existence of countable models. more precisely, if $M$ is any uncountable model of the theory in his post, then there is some non-standard $m\in M$, and then the theory of $(M,m)$ will have no countable models for the reason he describes Nov 17 at 18:49
• so eg you can take a two-sorted theory, where the first sort is just the theory Andreas describes in the post, and the second sort is your favorite non-$\omega$-categorical theory, and there are no relation or function symbols between the two sorts. then taking $M$ to be any uncountable model will let you get an example where adding a constant makes the theory vacuously $\omega$-categorical (as it has no countable models) even though $Th(M)$ is not $\omega$-categorical Nov 17 at 18:51
• however this is not so satisfying as the only reason the expanded theory is $\omega$-categorical is that it has no countable models. I think a non-vacuous example would be much nicer Nov 17 at 18:53
• You can think of a two-sorted structure in the following way: two predicates $A,B$ partition the universe; for the theory that you want to be referring to the first sort, "relativize the quantifiers to $A$" the same way as in set theory; similarly for $B$. This should work generally to simulate any finite number of sorts in a single-sorted structure. Nov 18 at 4:10