Intuitionistic Lowenheim-Skolem? 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.
 A: 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.
A: I have a reference that says the downward Löwenheim-Skolem 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 Löwenheim-Skolem theorem."
Charles McCarty & Neil Tennant, Skolem's paradox and constructivism.
Journal of Philosophical Logic.
Springer Netherlands.
Issue   Volume 16, Number 2 / May, 1987.
https://doi.org/10.1007/BF00257838
Also
page 341 of
A Companion to Metaphysics
By Jaegwon Kim
"...there is no intuitionistically acceptable analogue
of the classical downward Löwenheim-Skolem theorem"
A: It's been a while, but I think Ebbinghaus, Flum & Thomas, in the book Mathematical Logic, get the Löwenheim-Skolem theorems as a byproduct of the completeness theorem, which they prove using the Henkin construction. And I think that is fully constructivist. 
