Timeline for Forcing out of L[U] when we have a precipitous ideal in V
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 27 at 21:45 | vote | accept | Toby Meadows | ||
| Mar 27 at 21:45 | |||||
| Mar 24, 2022 at 12:55 | answer | added | Farmer S | timeline score: 8 | |
| Mar 24, 2022 at 0:36 | comment | added | Farmer S | @JasonZeshengChen In fact the filter $U$ is uniquely determined by the critical point, assuming we take it to be normal in $L[U]$; anyway if we allow it to be non-normal, we still get $U'$ such that $L[U']=L[U]$. | |
| Mar 23, 2022 at 21:11 | comment | added | Jason Zesheng Chen | (continued) From a cursory look at the proof in Jech. I think the proof of (2) carries over to the case of forcing with Silver collapse. But of course, I could be wrong. | |
| Mar 23, 2022 at 21:08 | comment | added | Jason Zesheng Chen | Asaf, I think for some ground model and $U$, it's possible for there to be no generic filter for the Levy collapse in this situation. For example, start with $V_0=L[U]$ with a measurable $\kappa$ and normal measure $U$. Forcing to collapse it to $\omega_1$ with Silver collapse, then in $V=V_0[G]$, $\kappa$ is $\omega_1$ and has a precipitous ideal. If we happen to pick the $U$ we started with, then $L[U]$ here is just the ground model. But since $V$ is a Silver collapse extension, there is no $V_0$-generic filter for Levy collapse (see here). | |
| Mar 23, 2022 at 10:34 | comment | added | Asaf Karagila♦ | @Farmer: Oh, I should write a book titled "On the Dangers of Posting Comments Immediately After Waking Up". Regardless, since $\omega_1$ is the measurable of $L[U]$, there's a filter for the point wise collapse of all the ordinals below it. I guess the only missing ingredient is to combine them together? | |
| Mar 23, 2022 at 10:26 | comment | added | Farmer S | @AsafKaragila Why is there a Levy collapse filter in $V$? | |
| Mar 23, 2022 at 6:43 | comment | added | Asaf Karagila♦ | Yes, just do the usual Levy collapse. The tricky part is if you want the original ideal to be generic over L[U], which I guess could be impossible. | |
| Mar 22, 2022 at 23:20 | history | asked | Toby Meadows | CC BY-SA 4.0 |