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Jun 29, 2021 at 14:09 comment added მამუკა ჯიბლაძე If one understands completeness of the self-hosting semantics for IPC as the claim that every Heyting algebra is the algebra of all truth values in some elementary topos, then this has been proven by Pataraia but still unpublished (for which, I must admit, I am responsible).
Feb 24, 2021 at 1:30 vote accept ttbo
Feb 23, 2021 at 15:09 comment added Mike Shulman Yes, you got it right, except that as Andrej says, the set of intuitionistic truth values is not just $\{0,1\}$.
Feb 23, 2021 at 7:51 comment added ttbo I tried to semi-formalize the first part of your answer, as far as I understood it, and updated my question with the results. Can you let me know if I said anything wrong, or if I didn't get your meaning properly? Thanks.
Feb 19, 2021 at 17:36 history answered Mike Shulman CC BY-SA 4.0