YES, because it doesn't matter. Bill's question is not well framed (besides the potential for ambiguity in the answer arising from "Am I wrong to say that"). Is the question is about intuitionistic logic/Heyting algebra itself, or its domain of discourse, ostensibly posetal Kripke structures? If the former, then the canonical models obtained by dualizing Heyting algebras are Stone(-Priestley(-Heyting)) spaces, whose topology is compact, the logic being finitary. If the latter, and if Stone topology is more bother than discrete topology (surely the case from a pedagogical standpoint), then the Alexandrov topology is preferable; this in general is not compact. But variations between topologies having the same specialization order describe only what points are approached in the (infinite) limit. Since intuitionistic logic is entirely finitary and does not concern itself with limiting processes, it doesn't matter what topology Kripke semantics is based on. Hence YES, Bill is wrong to imply that it does matter (other than for the secondary reasons discussed above). Best, Vaughan
YES, because, given a poset X, the category of sheaves over X equipped with the Alexandrov topology is equivalent to the category of *covariant* functors X --> Set. A very important, basic fact, which is at the foundation of a lot of things in topos theory and computer science.
Cheers, François
I have been re-reading the chapter on intuitionism in Goldblatt's book, specifically the section on Kripke semantics. Am I wrong to say that Kripke semantics is based on the Alexandrov topology generated by the underlying poset?
Regards, Bill