# Can there be consistent countable FOL theories having no countable models?

$$\sf LS(\aleph_0)$$: Every model $$M$$ of a first order theory $$\sf T$$ with countable signature has an elementary submodel $$N$$ which is at most countable.

Now this theorem is equivalent over axioms of $$\sf ZF$$ to the axiom of dependet choice $$\sf DC"$$.

Now does this mean that in $$\sf ZF + \neg DC$$, we can have a consistent theorey in a countable signature that doesn't have a countable model?

No. If a theory is well-orderable, then it has a well-orderable model satisfying all the usual properties in $$\sf ZFC$$.
To see why, note that you can code all the thing into a set of ordinals $$A$$, then in $$L[A]$$ your theory exists and it is a model of $$\sf ZFC$$, so the theory has a countable model. But this is upwards absolute to $$V$$.