Is there a version of the LöwenheimSkolem theorem in intuitionistic logic? I'm particularly interested in the "downward" form. The standard proof I know uses the TarskiVaught test for elementary substructures, which in turn relies on the fact that "forall" is equivalent to "not exists not", and that fails intuitionistically.

I have a reference that says the downward LöwenheimSkolem theorem does not occur in intuitionistic logic. In the words of the abstract "even a very powerful version of intuitionistic set theory does not yield any of the usual forms of a countable downward LöwenheimSkolem theorem." Skolem's paradox and constructivism Journal of Philosophical Logic Springer Netherlands Issue Volume 16, Number 2 / May, 1987 http://www.springerlink.com/content/t28583t748301t04/ Also page 341 of A Companion to Metaphysics By Jaegwon Kim "...there is no intuitionistically acceptable analogue of the classical downward LöwenheimSkolem theorem" 


It's been a while, but I think Ebbinghaus, Flum & Thomas, in the book Mathematical Logic, get the LöwenheimSkolem theorems as a byproduct of the completeness theorem, which they prove using the Henkin construction. And I think that is fully constructivist. 


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 twosorted classical model in such a way that satisfaction of any intuitionistic formula in the original model is firstorder definable in the representation, and then applying the classical Löwenheim–Skolem theorem. 

