Suppose a first order set theory $S$ has a countable model. Does it follow that there is a countable ordinal $\alpha$ so that $L\_\alpha$ in the constructible hierarchy is a model of $S$?



Are countable models constructible?
For the question in the text, the answer is no. Let $S$ consist of (1) a sufficient finite number of ZF axioms to provide an absolute (i.e., $\Delta\_1$ in the Lévy hierarchy) definition of the constructible hierarchy and (2) the sentence formalizing $V\neq L$.


For the question in the title, asking only for a constructible model and not necessarily one of the form $L\_\alpha$, the answer is yes if the theory itself is constructible, but not in general.


