Timeline for stationary tower forcing
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Feb 12, 2015 at 9:55 | comment | added | Yair Hayut | @MonroeEskew: I spoke too soon. The indestructibility argument that I had only works for $\kappa$-distributive forcing from the ground model. Maybe one can extend it to a large class of forcings. The idea is that for a forcing $\mathbb{P}$ of cardinality $\mu < \lambda$ as above, the supercompact Prikry forcing for $P_\kappa\mu$ will add a $V$-generic for $\mathbb{P}$. The quotient forcing $\mathbb{R}_{\mu^\prime}/\mathbb{P}$, for $\mu^\prime > \mu$ is still $\mu^\prime^+$.c.c. We need to deal we the mutual generity of the Prikry sequences and this can be done by adding a Cohen real. | |
| Feb 12, 2015 at 8:16 | comment | added | Monroe Eskew | @YairHayut: This looks very useful. Can you sketch the indestructibility you claim? | |
| Feb 12, 2015 at 7:59 | comment | added | Mohammad Golshani | My next project is to extend the above construction to build a model in which all uncountable cardinals are inaccessible in $HOD$, but it seems much more complicated. | |
| Feb 12, 2015 at 7:57 | comment | added | Mohammad Golshani | @YairHayut Thanks for extra information. What is the most interesting thing for me is that $\aleph_{\omega+1}$ of $M$ is inaccessible in $HOD^M$, because $HOD^M \subseteq N.$ I have not written all the details of this fact, but this is the main motivation of my work on this project. Do you know if it is known it is consistent that $\aleph_{\omega+1}$ is inaccessible in $HOD$? | |
| Feb 12, 2015 at 7:43 | comment | added | Yair Hayut | cont: In particular - in $N$, the fact that for unboundedly many cardinals $\mu$ it is possible collapse cardinals below them and force $\mu$ to be $\aleph_{\omega + 1}$ in a $\mu$.c.c. forcing is indestructible by any forcing of size $<\lambda$ that doesn't change the $\aleph_n$-s. | |
| Feb 12, 2015 at 7:41 | comment | added | Yair Hayut | @MohammadGolshani: You results is stronger: Let $N = V[G]$, where $G$ is a generic for the standard Prikry forcing on $\kappa$ with collapses (the normal measure on $\kappa$ projected from the measure on $P_\kappa \lambda$). In this model, for every $\mu$, let $\mathbb{R}$ be the quotient between the supercompact Prikry forcing with collapses using the measure on $P_\kappa \mu^+$ (from the ground model), and $G$. This forcing is $\mu$-centered if $\mu > \kappa$. | |
| Feb 12, 2015 at 6:18 | comment | added | Mohammad Golshani | By defining over the ground model a Prirky forcing with suitable collapses, and then defining a weak projection from supercompact prikry forcing into it (to guarantee that the $\omega-$sequence added by them and the collapses are the same, so that the cardinal structure below $\aleph_\omega$ is the same in both $M$ and $N$). | |
| Feb 12, 2015 at 6:14 | comment | added | Monroe Eskew | How do you show the model $N$ exists? | |
| Feb 11, 2015 at 18:52 | comment | added | Mohammad Golshani | @AsafKaragila Your welcome, in fact I learnt such an idea from Prof. Magidor; your supervisor!. | |
| Feb 11, 2015 at 17:09 | comment | added | Asaf Karagila♦ | When I asked if it's even consistent, this was the sort of idea that I had in mind. Thanks for reinforcing my intuition! | |
| Feb 11, 2015 at 12:35 | history | answered | Mohammad Golshani | CC BY-SA 3.0 |