Timeline for A principle of mathematical induction for partially ordered sets with infima?
Current License: CC BY-SA 4.0
28 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 5, 2019 at 23:50 | comment | added | Gerhard Paseman | @Lspice: If memory serves, Arturo studied under Bergman and I considered Arturo to be more familiar with a reference for Bergman's 245 notes than I. (This was also when MathOverflow was smaller; of the people who occupied Evans Hall around the same time I did, Joel Hamkins, Greg Kuperberg, and Arturo Magidin were the three names I recognized best that were also regulars on this forum. Those were cosier times on MathOverflow.) Gerhard "Working On Coquetteishness And Charm" Paseman, 2019.03.05. | |
| Mar 5, 2019 at 22:15 | comment | added | LSpice | I'm not sure why the coyness in @GerhardPaseman's reference, but, Arturo Magidin apparently not having shown up, I think these are the referenced notes, with the relevant discussion being in §5.3, beginning on p. 124. | |
| Feb 19, 2019 at 7:45 | history | edited | Zach Teitler | CC BY-SA 4.0 |
link to Kalantari paper
|
| Sep 12, 2010 at 0:03 | history | edited | Pete L. Clark |
edited tags
|
|
| Sep 11, 2010 at 20:41 | comment | added | Joel David Hamkins | May I suggest a set-theory tag? | |
| Sep 11, 2010 at 19:04 | vote | accept | Pete L. Clark | ||
| Sep 10, 2010 at 12:25 | comment | added | Joel David Hamkins | Pete, it is a very nice question! | |
| Sep 10, 2010 at 11:20 | answer | added | Joel David Hamkins | timeline score: 15 | |
| Sep 10, 2010 at 2:15 | answer | added | Pete L. Clark | timeline score: 3 | |
| Sep 10, 2010 at 2:14 | comment | added | Gerhard Paseman | Hopefully Arturo Magidin will show up and give a reference to George Bergman's exposition of induction over partial orders. If he doesn't, search for Bergman's Math 245 notes on universal algebra. Gerhard "Ask Me About System Design" Paseman, 2010.09.09 | |
| Sep 10, 2010 at 1:37 | vote | accept | Pete L. Clark | ||
| Sep 11, 2010 at 19:04 | |||||
| Sep 10, 2010 at 1:09 | comment | added | Cam McLeman | @Pete: Your VIGRE talk was based on the Kalantari paper? Or is there a better reference? I'm intrigued. | |
| Sep 10, 2010 at 1:01 | answer | added | François G. Dorais | timeline score: 13 | |
| Sep 10, 2010 at 0:38 | history | edited | Pete L. Clark | CC BY-SA 2.5 |
added 250 characters in body; added 4 characters in body; edited title
|
| Sep 10, 2010 at 0:34 | comment | added | Pete L. Clark | @Francois: good point. I should have realized this when I discussed the need to add a minimum element in the well-founded case. @Gerald: that's a very interesting remark. Do you have any references? | |
| Sep 10, 2010 at 0:32 | comment | added | Peter LeFanu Lumsdaine | For a version that doesn't mention the minimal element, remove (POI1), and where (POI3) mentions $[0,x)$, replace it with $\{ y\ |\ y < x\}$. If there is a minimal element, then this is the same as before: the new (POI3') is equivalent to the old (POI3) for $x \neq 0$, and in the case $x = 0$, it gives back the old (POI1). Of course, an order with no minimal element can never satisfy these, since its empty subset will be inductive. But for generalisation to posets with possibly multiple minimal elements, this reformulation could be helpful. | |
| Sep 10, 2010 at 0:28 | answer | added | Harry Gindi | timeline score: 2 | |
| Sep 10, 2010 at 0:26 | comment | added | Gerald Edgar | If you look way back at the original work, you find that the Heine-Borel theorem was thought of as a kind of "continuous induction" for intervals [a,b] in the real line. | |
| Sep 10, 2010 at 0:13 | comment | added | François G. Dorais | Note that well-founded posets don't necessarily have infima: every nonempty set has a minimal element but it is not necessarily unique. So the generalization you seek will not fully capture well-founded induction. | |
| Sep 9, 2010 at 23:51 | history | edited | Pete L. Clark | CC BY-SA 2.5 |
edited body; added 15 characters in body
|
| Sep 9, 2010 at 23:46 | comment | added | Willie Wong | @Pete: the references in the later paragraphs to (i),(ii), (iii) should be to (POI1), (POI2), and (POI3), for clarity. | |
| Sep 9, 2010 at 23:45 | comment | added | Steve Kass | Small point: I think you've misstated POI3. You wrote: For all x∈S, if [0,x)⊂S, then x∈S. Do you mean: For all x∈**X**,... ? As you stated it, POI3 is vacuously true. | |
| Sep 9, 2010 at 23:38 | comment | added | Willie Wong | As a side remark: induction over totally ordered sets with minimum element is rather familiar to analysts, especially to PDE people. In fact it is so universal in evolutionary PDEs, I think every paper I've read in the subject uses it somewhere or another. | |
| Sep 9, 2010 at 23:34 | history | edited | Pete L. Clark | CC BY-SA 2.5 |
added 69 characters in body
|
| Sep 9, 2010 at 23:34 | comment | added | Pete L. Clark | @Dave: Yes, I think you're right. I tend to think of a Noetherian space as one which satisfies ACC for open subsets, but I guess I'm just making trouble for myself: better to think of it as satisfying DCC for closed subsets! | |
| Sep 9, 2010 at 23:28 | comment | added | Dave Anderson | About the terminology: I always thought "noetherian" induction came from the fact that Spec of a noetherian ring has the DCC property you mention (due to the order-reversing nature of the functor)... whereas Spec of an artinian ring tends to be rather uninteresting, set-theoretically. | |
| Sep 9, 2010 at 23:26 | comment | added | user5810 | Not answering your question, but I would tend to split POI2 into "S is closed downward" and "For all x in S, if x is maximal in S, then x is maximal in X". | |
| Sep 9, 2010 at 22:51 | history | asked | Pete L. Clark | CC BY-SA 2.5 |