Timeline for preserving saturated ideals
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Jul 7, 2015 at 20:16 | comment | added | Monroe Eskew | It is proved here: ams.org/mathscinet-getitem?mr=2322360. This is the simplest forcing notion to achieve this, but I find Kunen's (very different) argument from a huge cardinal easier, which is described in Foreman's Handbook chapter. Diamond also holds in Kunen's model because a countably closed collapse is used in the second and final stage. | |
| Jul 7, 2015 at 20:12 | comment | added | Ashutosh | Can you tell me what the saturated ideal is? A reference will suffice, thanks! | |
| Jul 7, 2015 at 19:57 | comment | added | Monroe Eskew | If you Levy collapse a Woodin cardinal to be $\omega_2$ then there is a saturated ideal in the extension. Also diamond holds because it is always forced by this countably closed collapse. | |
| Jul 7, 2015 at 19:49 | comment | added | Ashutosh | Can you tell me the simplest way to get a model of diamond plus there is a saturated ideal on $\omega_1$? | |
| Jul 7, 2015 at 19:31 | comment | added | Monroe Eskew | If $Col(\omega,\omega_1)$ completely embeds into $\mathcal P(\omega_1)/I$, then we can use a variant of Foreman's Duality Theorem to compute the generated ideal and show that it is not saturated. However in this situation there is another saturated ideal in the extension. Actually I don't know of a model of CH + saturated ideal + the quotient algebra doesn't have this factor. | |
| Jul 7, 2015 at 19:26 | comment | added | Ashutosh | I was considering the possibility that the ideal generated by the old saturated ideal stays saturated. Is this ever possible after adding a Cohen subset of $\omega_1$? I will stop blabbering now. | |
| Jul 7, 2015 at 19:20 | comment | added | Ashutosh | Monroe, this query is out of my ignorance but can you show that adding one Cohen subset of $\omega_1$ preserves this? since this forces diamond, it must possibly kill old (say e.g., the non stationary ideal) saturated ideals. | |
| Jul 1, 2015 at 0:17 | history | asked | Monroe Eskew | CC BY-SA 3.0 |