Timeline for What structure has been found for functions with this relationship.
Current License: CC BY-SA 3.0
13 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Sep 2, 2014 at 8:44 | history | edited | Paul Taylor | CC BY-SA 3.0 |
added 262 characters in body
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| Sep 2, 2014 at 4:26 | comment | added | François G. Dorais | Just so I'm sure I remember correctly, Pataraia's theorem is that in a DCPO $P$, every $f:P \to P$ has a least fixed point? (PS: There is no such thing as "below" on MO unless you're talking about the later parts of your own post.) | |
| Sep 1, 2014 at 20:47 | history | edited | Paul Taylor | CC BY-SA 3.0 |
added Pataraia and Bekic
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| Nov 19, 2013 at 21:02 | comment | added | Todd Trimble | Sure. Just clarifying what I had meant. | |
| Nov 19, 2013 at 20:17 | comment | added | Paul Taylor | This stuff was swapped out of my skull some time ago, I don't particularly want to go through old postings and this is getting to be rather a long discussion in broken comments, about which you have previously complained. What you say is broadly historically correct. I don't think anything more needs to be said. | |
| Nov 19, 2013 at 19:40 | comment | added | Todd Trimble | (cont.) about defining ordinal iterates of the function [denoted] s and its values at the least element... | |
| Nov 19, 2013 at 19:40 | comment | added | Todd Trimble | Well, what I had in mind in my first sentence is what you wrote here: mta.ca/~cat-dist/catlist/1999/harvey-friedman (post dated 30 Jan 1998). If I may: "For several years I was trying to prove (in an elementary topos, in particular without excluded middle, or the axiom of collection, which seems to me to be set-theoretic hocus pocus): Let (X, <=) be a poset with least element and directed joins, and s:X->X a monotone (not necessarily Scott continuous) function. Then s has a least fixed point... Because of my set-theoretic indoctrination, much as I rebelled against it, I set (cont.) | |
| Nov 19, 2013 at 18:48 | comment | added | Paul Taylor | I'm not sure what your first sentence means, Todd, but the second one is correct. Also, it was a design principle of the book that everything that is done for categories is first done for posets in Chapter III. Curiously, some of it is more difficult, if not impossible, in particular my attempts to define well founded coalgebras for monotone functions. | |
| Nov 19, 2013 at 18:27 | comment | added | Todd Trimble | I think I've spotted you -- maybe on the categories mailing list? -- also decrying the (perhaps all too typical?) approaches to fixed point theorems based on transfinite iteration. Is it correct to say, however, that big chunks of theory that you worked out on developing transfinite induction and recursion constructively, and that appear in your book, might not have been developed at all had you not missed Pataraia's proof at the time? (Not sure I've phrased all this optimally.) | |
| Nov 19, 2013 at 16:56 | comment | added | Paul Taylor | I had exactly Pataraia's proof in mind when I gave my "warning", because I really kicked myself when I first heard it. I had thought of every single step of his argument myself beforehand, but had failed to put these steps together in the right order! | |
| Jul 13, 2013 at 20:35 | comment | added | Todd Trimble | Your warning reminds me that I first learned Pataraia's fixed-point theorem through your book, Paul, and a beautiful gem of mathematical reasoning it is. A mathematical analogue of a truly great poem! | |
| May 16, 2013 at 12:55 | history | answered | Paul Taylor | CC BY-SA 3.0 |