Timeline for The Rise and Fall of Dictators & How it Depends on Our Choice
Current License: CC BY-SA 4.0
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
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| Jun 23, 2018 at 13:29 | history | reopened |
Morteza Azad R.P. Jan-Christoph Schlage-Puchta Pace Nielsen Guntram |
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| Jun 20, 2018 at 19:16 | review | Reopen votes | |||
| Jun 23, 2018 at 13:29 | |||||
| Jun 9, 2018 at 17:58 | comment | added | Morteza Azad | On another forum, a colleague asked me whether I am aware of any similar result for the infinite case of Gibbard–Satterthwaite theorem, a counterpart of Arrow's Impossibility Theorem. Actually yes, there is a paper of Ilan Nehama along these lines. It is basically using the same techniques (i.e. non-principal ultrafilters) that the papers mentioned in the original post are based on. I left a link for the readers who might be interested in this topic. | |
| Jun 9, 2018 at 17:23 | comment | added | Morteza Azad | @AndreasBlass (+1) Hmmm... Yes! Good point, Andreas! By the way, does anybody here know what other works of Shelah had been around questions of economics origin such as his work on Arrow's Impossibility theorem?! It has been fairly surprising to me to find a work of him along these lines. | |
| Jun 9, 2018 at 14:58 | history | closed |
Andrés E. Caicedo user6976 Stefan Kohl♦ Chris Godsil Morteza Azad |
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| Jun 9, 2018 at 13:07 | comment | added | Andreas Blass | Nitpicking about your statement that "There are uncountable sets for which AD allows the existence of free ultrafilters": This is true but it's a vast understatement. AD implies the existence of free ultrafilters on $\aleph_1$ (and on the set of Turing degrees). | |
| Jun 9, 2018 at 10:29 | comment | added | Morteza Azad | @AsafKaragila (+1) Advice taken! I understand your point, Asaf! Admittedly, I would say that this post is not possibly the most optimal one when it comes to the Math Complexity/Length ratio and the main point of the question would have been much easier to grasp without some of the additional remarks. Actually, I have been searching for a quite different fact on determinacy but suddenly got politically distracted by the news coming from Korean Peninsula! Accidentally my two separate mathematical and political searches merged into one and led me to learn Arrow's theorem and post here! :-) | |
| Jun 9, 2018 at 9:18 | comment | added | Asaf Karagila♦ | Let me add some criticism, that when you formulate your questions in these long and elaborate prosaic ways, you end up obfuscate the mathematics. Which would have been much simpler to answer otherwise, as remarked by Monroe. | |
| Jun 9, 2018 at 8:54 | comment | added | Morteza Azad | @MonroeEskew (+1) I see! Thanks for the explanation, Monroe! | |
| Jun 9, 2018 at 8:47 | comment | added | Monroe Eskew | Large cardinals are not relevant. It is consistent relative to ZF that ZF holds and there are no nonprincipal ultrafilters on any set. The proof of Arrow’s theorem says that the outcome of the election is determined by what happens on ultrafilter-many ballots. So in the model with no n.p.u.f.’s there is always a dictator. | |
| Jun 9, 2018 at 8:27 | comment | added | Morteza Azad | @MonroeEskew Somehow yes (though, the details need checking)! So I think it would be possible to weave an answer for the above questions using the known set-theoretic theorems. Any suggestion for the missing large cardinal assumptions in the question? | |
| Jun 9, 2018 at 8:08 | comment | added | Monroe Eskew | The technical content of your question boils down to the existence of nonprincipal ultrafilters. See mathoverflow.net/questions/59157/… | |
| Jun 9, 2018 at 8:02 | history | edited | Morteza Azad | CC BY-SA 4.0 |
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| Jun 9, 2018 at 7:14 | history | edited | Morteza Azad | CC BY-SA 4.0 |
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| Jun 9, 2018 at 3:22 | review | Close votes | |||
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| Jun 9, 2018 at 3:07 | history | edited | Morteza Azad | CC BY-SA 4.0 |
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| Jun 9, 2018 at 2:35 | history | asked | Morteza Azad | CC BY-SA 4.0 |