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."

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"

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"

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."

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 Löwenheim-Skolem theorem"

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."

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"