Timeline for Does Well-Ordered Interval Power Set "WOIPS" principle , prove $\sf AC$ in $\sf ZFA$?
Current License: CC BY-SA 4.0
32 events
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| Jun 3 at 9:15 | answer | added | Zuhair Al-Johar | timeline score: 1 | |
| Jun 1 at 22:38 | comment | added | Guozhen Shen | @AsafKaragila Thanks for pointing out Truss's result. I have found his paper. | |
| Jun 1 at 18:10 | comment | added | Asaf Karagila♦ | @Joel: Elliot has given what is the gist of Specker's argument. (In fact, a slightly better one!) | |
| Jun 1 at 14:54 | comment | added | Asaf Karagila♦ | @GuozhenShen: An old theorem of Truss says that if there is some $\alpha$ such that there are no chains of type $\alpha$ (of distinct cardinals) between $X$ and $\mathcal P(X)$, then the axiom of choice holds. | |
| Jun 1 at 1:01 | answer | added | Elliot Glazer | timeline score: 6 | |
| May 31 at 13:20 | comment | added | Joel David Hamkins | No, this is not necessary. My comment was a suggestion that in the future, you should not make huge changes to a question that would make a mess of current comments or answers. | |
| May 31 at 13:15 | comment | added | Zuhair Al-Johar | @JoelDavidHamkins. If I post the original question as a separate question would that be reasonable? | |
| May 31 at 12:36 | comment | added | Joel David Hamkins | @EmilJeřábek I agree with the first part of your comment, but that is not the situation here, since the question was certainly not trivial, just because it admits an answer based on observations concerning a related proof in the research literature. We could have had a nice account of the Specker argument (not just a link) and remarks noting how it does or does not extend to the urelement context. Instead, we now have this comment thread, which has become basically incoherent in regard to the revised question. | |
| May 31 at 6:06 | comment | added | Emil Jeřábek | @JoelDavidHamkins On the contrary, if a question seemingly has an obvious answer, it is perfectly correct and much more constructive to seek clarification from the OP in a comment, rather than immediately post it as an answer only to find out that e.g. the OP forgot and/or misstated some requirement that makes the trivial answer invalid, or that the answerer has simply misunderstood something. Questions that admit trivial answers are not appropriate for this site anyway. So Andrés and Asaf have done the right thing. | |
| May 30 at 13:52 | comment | added | Joel David Hamkins | Zuhair, in the future please do not change your questions entirely like this after comments or answers have been posted. It makes the comment thread incoherent. For example, the initial comment of Andrés was a fully correct answer to your original question, but with your change it now has a different meaning that may be incorrect for the modified question. The recommended policy, when one wants major changes, is simply to post a new question. And incidently, this problem could have been avoided if the commentators had posted the answer as an answer instead of answering in the comments. | |
| May 30 at 13:49 | comment | added | Guozhen Shen | I do not even know whether the statement that there is at most one cardinal between an infinite cardinal and its power implies the axiom of choice in ZF. | |
| May 30 at 13:37 | history | edited | Zuhair Al-Johar |
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| May 30 at 12:07 | comment | added | Zuhair Al-Johar | @EmilJeřábek, Thanks! | |
| May 30 at 11:54 | history | edited | Zuhair Al-Johar | CC BY-SA 4.0 |
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| May 30 at 11:51 | comment | added | Emil Jeřábek | That sounds reasonable to me. | |
| May 30 at 11:50 | comment | added | Zuhair Al-Johar | @EmilJeřábek, should it be well ordered interval power set? | |
| May 30 at 11:47 | comment | added | Emil Jeřábek | The name is misleading, as the principle does not just ask the interval to be linearly ordered, but well ordered. | |
| May 30 at 10:48 | comment | added | Zuhair Al-Johar | @AsafKaragila, Agreed about the terminology. Thanks for the suggestion. | |
| May 30 at 10:47 | history | edited | Zuhair Al-Johar | CC BY-SA 4.0 |
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| May 30 at 10:19 | comment | added | Asaf Karagila♦ | Now that the question had been changed entirely, it's an interesting question. I'd suggest to change the name from $\sf GCH^{WO}$ to something else. This is no longer a generalised continuum hypothesis in any sense of the word. Perhaps "Linear Interval Power Set", or $\sf LIPS$. | |
| May 30 at 8:31 | history | edited | Zuhair Al-Johar | CC BY-SA 4.0 |
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| May 30 at 8:25 | history | edited | Zuhair Al-Johar | CC BY-SA 4.0 |
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| May 30 at 8:24 | history | undeleted | Zuhair Al-Johar | ||
| May 30 at 7:00 | history | deleted | Zuhair Al-Johar | via Vote | |
| May 30 at 7:00 | history | undeleted | Zuhair Al-Johar | ||
| May 30 at 6:47 | history | deleted | Zuhair Al-Johar | via Vote | |
| May 30 at 1:21 | comment | added | Joel David Hamkins | Probably best for someone simply to post the argument as an answer... | |
| May 30 at 1:13 | comment | added | Elliot Glazer | It also goes through in Z (without foundation). | |
| May 29 at 23:16 | comment | added | Asaf Karagila♦ | @AndrésE.Caicedo is correct. The atoms have no bearing on Specker's proof. If you suspect that it doesn't, simply go through it and edit your question to point out where exactly the assumption that we work in $\sf ZF$ comes into the proof. | |
| May 29 at 22:15 | comment | added | Zuhair Al-Johar | @AndrésE.Caicedo I think those are made in $\sf ZF$, not in $\sf ZFA$. I suspected it won't work here, because we don't have control over urelements, I mean over the size of $A$. | |
| May 29 at 21:58 | comment | added | Andrés E. Caicedo | The usual proof by Specker (as in p. 419 of math.bu.edu/people/aki/7.pdf) shows that AC holds, doesn't it? | |
| May 29 at 21:16 | history | asked | Zuhair Al-Johar | CC BY-SA 4.0 |