TITLE: Intuitionistic Lowenheim-Skolem?
QUESTION [13 upvotes]: Is there a version of the Löwenheim-Skolem theorem in intuitionistic logic?  I'm particularly interested in the "downward" form.  The standard proof I know uses the Tarski-Vaught test for elementary substructures, which in turn relies on the fact that "forall" is equivalent to "not exists not", and that fails intuitionistically.

REPLY [6 votes]: Is the question about the Löwenheim–Skolem theorem for classical models in intuitionistic metatheory, or about the Löwenheim–Skolem theorem for intuitionistic (Kripke) models in classical metatheory? The latter certainly holds. One can prove it easily by realizing that a Kripke model can be represented by a suitable two-sorted classical model in such a way that satisfaction of any intuitionistic formula in the original model is first-order definable in the representation, and then applying the classical Löwenheim–Skolem theorem.