Intuitionistic Lowenheim-Skolem? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T02:41:17Z http://mathoverflow.net/feeds/question/2976 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/2976/intuitionistic-lowenheim-skolem Intuitionistic Lowenheim-Skolem? Mike Shulman 2009-10-28T01:36:39Z 2011-02-04T15:12:19Z <p>Is there a version of the L&ouml;wenheim-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.</p> http://mathoverflow.net/questions/2976/intuitionistic-lowenheim-skolem/3378#3378 Answer by Kirill Levin for Intuitionistic Lowenheim-Skolem? Kirill Levin 2009-10-30T03:09:22Z 2009-10-30T03:09:22Z <p>It's been a while, but I think Ebbinghaus, Flum &amp; 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. </p> http://mathoverflow.net/questions/2976/intuitionistic-lowenheim-skolem/3515#3515 Answer by Kristal Cantwell for Intuitionistic Lowenheim-Skolem? Kristal Cantwell 2009-10-31T06:05:08Z 2009-10-31T06:05:08Z <p>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." </p> <p>Skolem's paradox and constructivism Journal of Philosophical Logic Springer Netherlands Issue Volume 16, Number 2 / May, 1987</p> <p><a href="http://www.springerlink.com/content/t28583t748301t04/" rel="nofollow">http://www.springerlink.com/content/t28583t748301t04/</a></p> <p>Also page 341 of A Companion to Metaphysics By Jaegwon Kim</p> <p>"...there is no intuitionistically acceptable analogue of the classical downward Löwenheim-Skolem theorem"</p> http://mathoverflow.net/questions/2976/intuitionistic-lowenheim-skolem/54319#54319 Answer by Emil Jeřábek for Intuitionistic Lowenheim-Skolem? Emil Jeřábek 2011-02-04T15:12:19Z 2011-02-04T15:12:19Z <p>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.</p>