Timeline for Are there any mathematical objects that exist but have no concrete examples? [closed]
Current License: CC BY-SA 3.0
24 events
| when toggle format | what | by | license | comment | |
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| Sep 12, 2021 at 11:29 | history | made wiki | Post Made Community Wiki by Stefan Kohl♦ | ||
| Apr 23, 2011 at 15:48 | comment | added | Jon Paprocki | Sorry for the vagueness! I think I've realized that I don't understand the question well enough to make it more concrete (a little ironic, given the nature of the question), so I don't think I'll try to rewrite it until I have a better grasp of what I'm asking. | |
| Apr 23, 2011 at 4:22 | history | closed |
Ryan Budney Pete L. Clark Dan Petersen Dmitri Pavlov Andrés E. Caicedo |
too localized | |
| Apr 23, 2011 at 3:29 | comment | added | Gerhard Paseman | How "concrete" do you mean? Are you willing to accept any uncountably infinite set? If you know of one, where can I buy it? Gerhard "Ask Me About System Design" Paseman, 2011.04.22 | |
| Apr 23, 2011 at 1:42 | answer | added | Justin Hilburn | timeline score: 14 | |
| Apr 23, 2011 at 0:50 | answer | added | Mark | timeline score: 3 | |
| Apr 23, 2011 at 0:02 | answer | added | Igor Rivin | timeline score: 10 | |
| Apr 22, 2011 at 23:47 | answer | added | Olivier Bégassat | timeline score: 18 | |
| Apr 22, 2011 at 22:52 | answer | added | Michael Renardy | timeline score: 1 | |
| Apr 22, 2011 at 21:23 | comment | added | Daniel Mehkeri | @Jon - are you unsatisfied with the non-principal ultrafilter example because "non-principal" is basically cheating? So e.g. do you prefer KConrad's example of well-ordering of the reals but not the non-trivial non-archimedean absolute value on C ? | |
| Apr 22, 2011 at 21:14 | comment | added | Theo Johnson-Freyd | This question is, at time of writing, likely to be closed. Since the comments section is already long, please bring discussion to tea.mathoverflow.net/discussion/1019 . And please vote up this comment so that it appears "above the fold". | |
| Apr 22, 2011 at 19:32 | answer | added | zroslav | timeline score: 3 | |
| Apr 22, 2011 at 19:24 | answer | added | Taylor Dupuy | timeline score: 4 | |
| Apr 22, 2011 at 19:04 | comment | added | Andrés E. Caicedo | @Jon: You need to specify better what you mean by "concrete", or the question becomes too vague to be useful. | |
| Apr 22, 2011 at 19:03 | comment | added | Andrés E. Caicedo | @Qiaochu: Pretty sure the intention was that only the principal ultrafilters can be "explicitly" described. | |
| Apr 22, 2011 at 18:14 | answer | added | Qiaochu Yuan | timeline score: 12 | |
| Apr 22, 2011 at 17:46 | comment | added | Harald Hanche-Olsen | “No concrete examples” does not imply “non-constructive”. For example, in the game of hex the first player has a winning strategy, which can be constructed by marking the (finite) game graph. However, I don't think anyone has described a winning strategy for the game. en.wikipedia.org/wiki/Hex_%28board_game%29 | |
| Apr 22, 2011 at 17:27 | comment | added | mephisto | At various times, elliptic curves over Q with a given (high) rank have been proved to exist but none exhibited. I don't know whether that is the case at present. | |
| Apr 22, 2011 at 17:24 | comment | added | Qiaochu Yuan | What is the one example of an ultrafilter that can be written down? (I assume you mean non-principal ultrafilter.) | |
| Apr 22, 2011 at 17:13 | comment | added | SNd | Additive and yet discontinuous function. | |
| Apr 22, 2011 at 17:05 | comment | added | Suvrit | I guess then uncomputable creatures such as "Chaitin's constant" also fall under the transcendence blanket? | |
| Apr 22, 2011 at 16:55 | comment | added | godelian | In fact all those examples are ultimately due to the non constructive nature of the axiom of choice. A choice function is postulated but in general no concrete examples can be given. Thus every equivalent form of choice (or every weaker form which is still independent of ZF), is in itself an example. | |
| Apr 22, 2011 at 16:47 | comment | added | KConrad | Lots of things proved to exist by Zorn's lemma are non-constructive, like a basis for R as a Q-vector space, a transcendence basis for R as a field extension of Q, a well-ordering of R, a nontrivial non-archimedean absolute value on C, a field isomorphism between the algebraic closure of Q_p and C,... | |
| Apr 22, 2011 at 16:37 | history | asked | Jon Paprocki | CC BY-SA 3.0 |