Timeline for Is it possible to define higher cardinal arithmetics
Current License: CC BY-SA 3.0
34 events
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| Apr 13, 2017 at 12:19 | history | edited | CommunityBot |
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| S Aug 25, 2016 at 6:08 | history | bounty ended | CommunityBot | ||
| S Aug 25, 2016 at 6:08 | history | notice removed | CommunityBot | ||
| Aug 20, 2016 at 14:02 | comment | added | მამუკა ჯიბლაძე | In fact I now realized that I know one widely recognized example of this, but for ordinals rather than cardinals: $\omega\uparrow\omega=\varepsilon_0$. | |
| Aug 18, 2016 at 18:25 | answer | added | მამუკა ჯიბლაძე | timeline score: 2 | |
| Aug 18, 2016 at 15:30 | answer | added | Will Brian | timeline score: 29 | |
| Aug 17, 2016 at 5:12 | answer | added | Alex Mennen | timeline score: 6 | |
| S Aug 17, 2016 at 4:40 | history | bounty started | Mohammad Golshani | ||
| S Aug 17, 2016 at 4:40 | history | notice added | Mohammad Golshani | Authoritative reference needed | |
| Dec 2, 2014 at 16:55 | comment | added | Gottfried Helms | I tried to initiate an additional discussion with possibly further expertise at math.eretrandre.org/tetrationforum/showthread.php?tid=938 However, I'm no expert to judge whether this shall come out to be interesting at all. | |
| Nov 25, 2014 at 4:27 | history | edited | Mohammad Golshani | CC BY-SA 3.0 |
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| Nov 24, 2014 at 19:47 | answer | added | Noah Schweber | timeline score: 13 | |
| Nov 24, 2014 at 19:02 | comment | added | GH from MO | @NoahS: Thanks for your interesting comment! | |
| Nov 24, 2014 at 18:59 | comment | added | Noah Schweber | @GHfromMO, there's certainly an interesting argument about what "logic" ought to mean. For example, Quine has said that second-order logic "set theory in sheep's clothing," which can be read as drawing a line between 'legitimate' forms of logic and whatever sort of thing set theory is; to some extent, depending on what day it is, I agree with this. However, as understood in practice, "logic" as a subfield tends to be defined to include: set theory, proof theory, computability theory, model theory, and various related subjects (nonclassical logics, sometimes parts of category theory, etc.). | |
| Nov 24, 2014 at 18:48 | comment | added | GH from MO | @EmilJeřábek: Yes, I realized later that the set-theory tag is of lower level here (or on the arXiv) than the lo.logic tag. In fact, I have not realized before that the top level tags here are identical to those of the arXiv. I am no set theorist, nor a logician, but I studied both subjects. The way I see it, neither is part of the other, although both are important for the foundation of mathematics. This might be a naive point of view and I don't want to argue about it. If experts see a certain tag appropriate, then it should be there. | |
| Nov 24, 2014 at 18:19 | comment | added | Emil Jeřábek | @GHfromMO: It’s perfectly standard to classify set theory as a subfield of logic, and specifically, foundations of mathematics. You can see that it is included in the tag wiki for lo.logic, and anyway, there is no other top-level arXiv tag where it would fit. | |
| Nov 24, 2014 at 18:11 | comment | added | GH from MO | Let me emphasize, as a number theorist, that the number theory tag was inappropriate. Please do not put it back. | |
| Nov 24, 2014 at 18:08 | history | edited | GH from MO |
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| Nov 24, 2014 at 18:06 | comment | added | GH from MO | @NoahS: I see your point, but I don't fully agree. In some sense logic is also part of set theory, e.g. it is hard to do model theory without set theory. But even if a theory is part of the other, it does not deserve a tag automatically. For example, number theory is also part of set theory (as every mathematical theory is), but still we don't assign the set theory tag to every number theory problem. Anyways, I will put back the logic tag. Ah, I just see you put it back. Again, I don't fully agree. | |
| Nov 24, 2014 at 18:05 | comment | added | GH from MO | @FrançoisG.Dorais: I see your point, but I don't fully agree. Anyways, I will put back the combinatorics tag. | |
| Nov 24, 2014 at 18:04 | history | edited | Noah Schweber |
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| Nov 24, 2014 at 18:03 | comment | added | Noah Schweber | @GHfromMO, set theory is part of logic . . . | |
| Nov 24, 2014 at 17:42 | comment | added | François G. Dorais | @GHfromMO: I think this is a combinatorics question. Cardinal number operations are defined combinatorially, so the relevant question is: Is there a combinatorial interpretation of tetration? | |
| Nov 24, 2014 at 17:12 | comment | added | GH from MO | @AsafKaragila: I don't see why the logic tag would be appropriate here. The way I see it, the question is purely set-theoretical. But feel free to put back the logic tag. | |
| Nov 24, 2014 at 17:05 | comment | added | Asaf Karagila♦ | @GHfromMO: Generally, the logic tag is appropriate for anything that would be appropriate on the logic section of arXiv. This includes this question. | |
| Nov 24, 2014 at 16:57 | comment | added | Wojowu | Isn't standard notation for tetration $\uparrow\uparrow$? Other than that, we don't even have any combinatorial sort of definition for tetration, unlike exponentiation, say, so I find it quite unlikely for such satisfactory definition to exist. | |
| Nov 24, 2014 at 16:21 | comment | added | GH from MO | I removed the number theory, combinatorics, and logic tags. This question is really about cardinal arithmetic, hence belongs to set theory. | |
| Nov 24, 2014 at 16:21 | history | edited | GH from MO | CC BY-SA 3.0 |
removed the number theory, combinatorics, and logic tags. This question is really about cardinal arithmetic, hence belongs to set theory.
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| Nov 24, 2014 at 9:40 | comment | added | Asaf Karagila♦ | The problem with cardinal arithmetic is that it generally insists on being discontinuous. | |
| Nov 24, 2014 at 8:06 | comment | added | მამუკა ჯიბლაძე | In fact already $2{\uparrow}\aleph_0$ becomes problematic, so maybe the first question must be how to interpret this. | |
| Nov 24, 2014 at 6:08 | comment | added | Mohammad Golshani | Yes, for ordinal arithmetic, we can easily define hyperoperations, but it seems in the case of cardinal arithmetic, the situation is not known (at least I have not seen anything)!! | |
| Nov 24, 2014 at 5:54 | comment | added | bof | "Hyperoperations" on ordinal numbers have been treated e.g. by Doner & Tarski, but of course you are aware of that. | |
| Nov 24, 2014 at 5:43 | history | edited | Mohammad Golshani | CC BY-SA 3.0 |
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| Nov 24, 2014 at 5:27 | history | asked | Mohammad Golshani | CC BY-SA 3.0 |