Timeline for cofinality of $(P(\kappa)/NS,\subseteq)$
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 20, 2013 at 8:09 | vote | accept | drizzy | ||
| Aug 19, 2013 at 2:57 | comment | added | Andreas Blass | @AsafKaragila Turn it upside down everywhere except Jerusalem; leave it right-side up (and delete the top element instead of $0$) in Jerusalem. | |
| Aug 19, 2013 at 1:38 | answer | added | Monroe Eskew | timeline score: 9 | |
| Aug 18, 2013 at 22:13 | comment | added | Asaf Karagila♦ | @Andreas: Now when you say turning it upside down, do you mean so it would follow the Jerusalem notation or the Universal notation? (And we have weekly arguments on the topic in our students seminar in HUJI... :-)) | |
| Aug 18, 2013 at 21:53 | comment | added | Andreas Blass | Turning the Boolean algebra $P(\kappa)/NS$ upside down, we can reformulate the question as asking for the smallest cardinality of a dense subset of the forcing notion $P(\kappa)/NS-\{0\}$. I think a good deal is known about this for $\kappa=\aleph_1$, for example the large-cardinal strength of this cofinality's being $\aleph_1$. But I don't recall seeing similar results for larger $\kappa$ such as $\aleph_{\omega+1}$. (Wait till Andres Caicedo, Philip Welch, and Paul Larson see this question; you'll get plenty of information.) | |
| Aug 18, 2013 at 19:59 | comment | added | Joel David Hamkins | Your edit involving NS makes the question more interesting, and now it is probably fine here. (Your other formulation was trivial.) | |
| Aug 18, 2013 at 19:55 | comment | added | drizzy | ok, how do i move it there? | |
| Aug 18, 2013 at 19:53 | comment | added | drizzy | edited to include some stuff about nonstationary ideal. | |
| Aug 18, 2013 at 19:53 | comment | added | Joel David Hamkins | Then take all the co-atoms (sets missing just one element). These are maximal, and must be included in any cofinal set, and form a cofinal set. So in this case, you would have $\lambda=\kappa$. Probably math.stackexchange.com may be a better fit for your question. | |
| Aug 18, 2013 at 19:52 | history | edited | drizzy | CC BY-SA 3.0 |
added 76 characters in body; edited title
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| Aug 18, 2013 at 19:51 | review | First posts | |||
| Aug 18, 2013 at 19:52 | |||||
| Aug 18, 2013 at 19:47 | comment | added | drizzy | I was thinking of $P(\kappa)$ as just not having the largest element. Is it still trivial? | |
| Aug 18, 2013 at 19:39 | comment | added | Joel David Hamkins | The way you define it, we could take $X=\{\kappa\}$, and have $\lambda=1$, since every $A\subset\kappa$ has $A\subset\kappa\in X$. So I think you haven't said exactly what you mean. That is, since this poset has a largest element, every cofinal collection must include it, and any collection that includes it will be cofinal. | |
| Aug 18, 2013 at 19:35 | history | asked | drizzy | CC BY-SA 3.0 |