Timeline for How hard is it to destroy a diamond? (with a real)
Current License: CC BY-SA 3.0
22 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 8, 2015 at 9:01 | comment | added | Asaf Karagila♦ | @Avshalom: Considering the fact that until two minutes ago I have never heard on that paper, the answer is a solid "no". It seems interesting, and I'll take a look later. Thanks! | |
| Jun 8, 2015 at 8:55 | comment | added | Avshalom | Have you considered the forcings in the paper by Todd Eisworth and Peter Nyikos? people.math.sc.edu/nyikos/antidi.pdf | |
| Jun 3, 2015 at 13:27 | comment | added | Asaf Karagila♦ | @saf: If you look at the edit history on Mohammad's answer, you'll see essentially the content of the now-deleted comment[s?], but in any case that is an interesting result! | |
| Jun 3, 2015 at 13:13 | comment | added | saf | I am puzzled by Moahammad's comment, as Abraham and Todorcevic proved that in any model of GCH, there is a cardinal-preserving notion of forcing that adds no new reals and entails SH. see here: cs.bgu.ac.il/~abraham/papers/math/PartitionWithCH.ps | |
| May 28, 2015 at 23:40 | comment | added | Asaf Karagila♦ | @Avshalom: Yes, exactly. But since I'm (1) more comfortable with diamond; and (2) "How hard is it to destroy a club" would be ambiguous and less amusing! :-) | |
| May 28, 2015 at 23:32 | comment | added | Avshalom | So it would be enough/necessary to preserve CH but force the failure of $\clubsuit$. | |
| May 28, 2015 at 12:37 | comment | added | Ashutosh | In hindsight, Goldstern's comment was pretty relevant here. | |
| May 28, 2015 at 8:58 | vote | accept | Asaf Karagila♦ | ||
| May 27, 2015 at 11:34 | comment | added | Asaf Karagila♦ | @Goldstern: That is true, but it blows up the continuum. And here we want to preserve $\sf CH$. | |
| May 27, 2015 at 11:26 | comment | added | Goldstern | Shelah and Woodin showed how CH can be violated by adding a real in JSL 49:4 (1984). They give several examples; in one of them all cardinals are preserved. I do not know if ◊ holds in one of their ground models. | |
| May 27, 2015 at 4:09 | comment | added | Mohammad Golshani | I meant we start with $L$ and add a new real | |
| May 27, 2015 at 4:08 | comment | added | Monroe Eskew | @MohammadGolshani If $V=L$, then $L[r] = V = L$. | |
| May 27, 2015 at 4:08 | comment | added | Mohammad Golshani | The answer to your question is no, if $V=L,$ as then for any real $r, L[r]\models \Diamond+CH.$ | |
| May 27, 2015 at 4:07 | comment | added | Asaf Karagila♦ | @Monroe: I do not, that's a good question, probably easier to answer too. Can't you force a new subset to $\omega_1$ that fails to reflect somehow? | |
| May 27, 2015 at 3:28 | answer | added | Mohammad Golshani | timeline score: 10 | |
| May 27, 2015 at 2:56 | comment | added | Monroe Eskew | Do you know the answer if we weaken $r \subseteq \omega$ to $r \subseteq \omega_1$? | |
| May 26, 2015 at 19:51 | comment | added | Asaf Karagila♦ | @Vidit: And not to mention those bloody diamonds, too! :-) | |
| May 26, 2015 at 19:47 | comment | added | Vidit Nanda | I've upvoted in order to partially offset the inevitable backlash from environmentalists who react negatively to "blow up the continuum and destroy all the Suslin trees." | |
| May 26, 2015 at 17:51 | comment | added | Yair Hayut | Jensen's forcing for $SH+CH$ forces the failure of diamond without collapsing cardinals or blowing up the continuum, but its generic is certainly not a real. | |
| May 26, 2015 at 17:14 | comment | added | Asaf Karagila♦ | Yes, a comment very worth mentioning! | |
| May 26, 2015 at 17:12 | comment | added | Will Brian | Possibly worth mentioning: if $\mathbb{P}$ is ccc and $|\mathbb{P}| \leq \aleph_1$, then forcing with $\mathbb{P}$ preserves $\diamondsuit$ and CH. This is Exercise IV.7.58 in the newer set theory book by Kunen. | |
| May 26, 2015 at 16:54 | history | asked | Asaf Karagila♦ | CC BY-SA 3.0 |