Paul Taylor wrote:
In my work on ASD, particularly its application to real analysis, I have wanted to refer to classical sources as a background, but on none of the relevant topics have I considered the Wikipedia article to be anywhere near satisfactory. All spaces are Hausdorff, and Excluded Middle is a Fact. I have thought about rewriting the articles on Dedekind cuts, locally compact spaces and some other things, but am afraid that my contributions will just be "reverted".
In my understanding of Heyting algebras/lattices/semilattices, excluded middle fails for the algebras themselves but not for my understanding of them, where the partial order x <= y in a Heyting algebra is either true or false with no middle ground allowed. I have had little luck absorbing the logic of Heyting algebras into my own mathematical thinking. I furthermore worry that if ever I were to succeed my insights might become even less penetrating than they already are. On a related note, a careful reading of Max Kelly's "Basic Concepts of Enriched Category Theory" reveals that it is thoroughly grounded in Set, as I pointed out in August 2006 in my initial Wikipedia article on Max. I gave some thought to how one might eliminate Set from the treatment, without much success, and concluded that Max's judgment there was spot on. My feeling about these recommended Brouwerian modes of thoughts is that they are something like locker room accounts of social and other conquests: great stories about things that never actually happened, but which with sufficient repetition convince one that they must surely have occurred. The self-evident is merely an hypothesis that is so convenient, and that has been assumed for so long, that we can no longer imagine it false. This is just as true for Excluded Middle itself as for its negation. I happen to find Excluded Middle more convenient than its negation, but that's just me and perhaps others have had the opposite experience. Then there are those who accept neither Excluded Middle nor its negation, which takes us into the Hall of Mirrors that I always find myself in when I go down this particular rabbit-hole. Vaughan Pratt