Timeline for Can there be a tree of height $\omega_2$ having all levels countable, with no cofinal branch?
Current License: CC BY-SA 3.0
17 events
| when toggle format | what | by | license | comment | |
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| May 31, 2015 at 6:59 | comment | added | Avshalom | Following on, there is an attractive proposition that says if $I$ is a $\kappa$-complete ideal on $\kappa$ and is $\lambda$-saturated for some $\lambda < \kappa$, then $\kappa$ has the tree property. It's 16.4 in Kanamori's The Higher Infinite. | |
| May 30, 2015 at 23:42 | comment | added | Andrés E. Caicedo | For a particular case, see here. | |
| May 30, 2015 at 21:26 | comment | added | Joel David Hamkins | It turns out that the tree under consideration was not actually as initially advertised; it was not actually narrow, but rather had some relation to small sets at each level, in a way that prevented Kurepa's argument from applying directly. | |
| May 30, 2015 at 12:02 | comment | added | Monroe Eskew | I see. Was it actually covered by Kurepa's theorem? | |
| May 30, 2015 at 12:02 | comment | added | Joel David Hamkins | Here is a link to Avshalom's post: mathoverflow.net/a/181440/1946 | |
| May 30, 2015 at 12:01 | comment | added | Joel David Hamkins | @MonroeEskew A set theorist whom I respect had a particular tall narrow tree and wanted to show it had a cofinal branch. I said I thought this was immediate, since all tall narrow trees had cofinal branches, but this implication was doubted and instead a more complicated large cardinal construction was undertaken to get a branch. | |
| May 30, 2015 at 11:04 | comment | added | Avshalom | The question was answered in the following post too: On a weak tree property for inaccessible cardinals. | |
| May 30, 2015 at 6:59 | comment | added | Monroe Eskew | I would be interested to know what data supported the opposite intuition. | |
| May 30, 2015 at 5:06 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
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| May 30, 2015 at 4:14 | vote | accept | Joel David Hamkins | ||
| May 30, 2015 at 4:08 | answer | added | Mohammad Golshani | timeline score: 12 | |
| May 30, 2015 at 4:05 | comment | added | Joel David Hamkins | Mohammad, kindly post that as an answer! | |
| May 30, 2015 at 4:05 | comment | added | Joel David Hamkins | See here: books.google.com/… | |
| May 30, 2015 at 4:02 | comment | added | Mohammad Golshani | Kanamori's book "the higher infinite", proposition 7.9, page 78 | |
| May 30, 2015 at 4:01 | comment | added | Joel David Hamkins | Mohammad, that would fulfill my intuition! But I was somehow convinced to abandon that intuition in the conversation this evening. Can you post a reference? | |
| May 30, 2015 at 3:58 | comment | added | Mohammad Golshani | I think it is a theorem of Kurepa that if $T$ has height $\kappa$ and all levels have size $<\lambda,$ for some $\lambda<\kappa,$ then $T$ has a cofinal branch | |
| May 30, 2015 at 3:51 | history | asked | Joel David Hamkins | CC BY-SA 3.0 |