No countable models

Question

I want an example of a theory T with finite models of arbitrarily large size but T has no countably infinite model. I know that T has to be uncountable, but couldn't come up with an example.

Thanks!!!

Answer 1

In the language there is a relation symbol $\unlhd$ and a constant for every real. Let $\{q_i : i<\omega\}$ enumerate the rationals (or any other countable dense subset of the reals). Consider the following axioms:

$\unlhd$ is a linear order;

$r\unlhd s$ for every pair of reals $r\le s$;

If there are at least $n$ elements then $q_i\lhd q_j$ for all $i,j<n$ such that $q_i<q_j$.

This theory has finite models of any size. But in any infinite model all constants have to be interpreted in distinct elements. Hence it has the size of the continuum.