Timeline for A Shelah group in ZFC?
Current License: CC BY-SA 4.0
25 events
| when toggle format | what | by | license | comment | |
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| Jun 11, 2023 at 20:08 | vote | accept | Taras Banakh | ||
| May 31, 2023 at 15:51 | answer | added | Taras Banakh | timeline score: 6 | |
| Dec 6, 2022 at 5:13 | history | edited | Taras Banakh | CC BY-SA 4.0 |
Corrected 6641 to 6640.
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| Dec 6, 2022 at 4:49 | answer | added | Taras Banakh | timeline score: 4 | |
| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
Commonmark migration
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| Oct 30, 2018 at 9:23 | history | edited | Taras Banakh | CC BY-SA 4.0 |
Added a reference to a result of Protasov
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| Oct 30, 2018 at 6:29 | history | edited | Taras Banakh | CC BY-SA 4.0 |
Added a link to a MO-problem in which it is justified why the number $n_0$ in Shalh's result should be 6640
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| Oct 24, 2018 at 0:26 | comment | added | François G. Dorais | Easton's Theorem only applies to regular cardinals. To have $2^\lambda>\lambda^+$ everywhere, including at singular $\lambda$, is much more difficult and requires some very large cardinal hypotheses: jstor.org/stable/2944324 | |
| Oct 23, 2018 at 23:17 | comment | added | Taras Banakh | @FrançoisG.Dorais No, Shelah construction seems to hold for any infinite cardinal $\lambda$ with $\lambda^+=2^\lambda$. I mentioned the Easton's Theorem to argume that ZFC does not imply the existence of a cardinal $\lambda$ with $\lambda^+=2^\lambda$. Or such a cardinal always exists? | |
| Oct 23, 2018 at 21:59 | comment | added | François G. Dorais | Taras, does Shelah's construction rely on regular $\lambda$, or did you not mean Easton's Theorem in your previous comment? | |
| Oct 23, 2018 at 13:29 | comment | added | user35370 | Ah okay I see. I do seem to remember that n-Shelah $\aleph_1$ groups exist was done in the paper too, but I could be misremembering (I guess I should look in the paper | |
| Oct 23, 2018 at 13:21 | comment | added | Taras Banakh | @PaulPlummer It is the construction: for every cardinal $\lambda$ with $\lambda^+=2^\lambda$ Shelah constructs a 6643-Shelah group of cardinality $\lambda^+$. But by the Easton's Theorem cardinals $\lambda$ with $\lambda^+=2^\lambda$ need not exist in ZFC. On the other hand, such a cardinal $\lambda$ (namely $\lambda=\aleph_0$ exists under CH. | |
| Oct 23, 2018 at 13:17 | comment | added | user35370 | I could be misremembering but what made the examples CH(I don't recall anything relying on CH)? | |
| Oct 23, 2018 at 13:08 | answer | added | YCor | timeline score: 14 | |
| Oct 23, 2018 at 13:04 | comment | added | Taras Banakh | @AsafKaragila This is also a good question: is there an almost Shelah group, which is not Shelah? | |
| Oct 23, 2018 at 12:58 | comment | added | Asaf Karagila♦ | I think that history will tell us, but there is nothing which is "almost Shelah". You're either Shelah or not at all Shelah. :-) | |
| Oct 23, 2018 at 8:05 | comment | added | YCor | In general the use of multiple names makes definition not intuitive, and I thank our ancestors to have found many useful intuitive names to many definitions, and coined words that makes them both flexible and intuitive (such as "uniform", etc). | |
| Oct 23, 2018 at 7:28 | history | edited | Taras Banakh | CC BY-SA 4.0 |
Added some problems
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| Oct 23, 2018 at 6:44 | comment | added | Taras Banakh | @Jan_Ch. Thank you for the comment. In order to fix the problem I have changed "weak" to "almost" | |
| Oct 23, 2018 at 6:42 | history | edited | Taras Banakh | CC BY-SA 4.0 |
introduced Kurosh groups
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| Oct 23, 2018 at 5:49 | comment | added | YCor | @Jan_Ch. Refrain from using "weak" in lieu of "weakly" would already fix the issue. | |
| Oct 23, 2018 at 5:44 | comment | added | Jan_Ch. | When creating names for new mathematical objects I would propose to refrain from using weak + name :-) | |
| Oct 23, 2018 at 0:09 | comment | added | Taras Banakh | @NateEldredge Thank you for the comment. I have corrected the title. | |
| Oct 23, 2018 at 0:07 | history | edited | Taras Banakh | CC BY-SA 4.0 |
Corrected 651 to 6643
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| Oct 22, 2018 at 23:46 | history | asked | Taras Banakh | CC BY-SA 4.0 |