Timeline for singularize the least inaccessible?
Current License: CC BY-SA 3.0
6 events
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| Jan 28, 2015 at 15:56 | comment | added | Yair Hayut | @MonroeEskew: Maybe the paper of Gitik "Changing cofinalities and the non-stationary ideal" will be useful for this question. Theorem II in this paper states that it's consistent relative to supercompact that for some inaccessible $\kappa$ there is stationary $S\subseteq S^\kappa_\omega$ such that $NS_\kappa \restriction S$ is saturated, so forcing a generic ultrafilter will force $\text{cf }\kappa = \omega$, won't add bounded subsets and will be $\kappa$-c.c. I think that it's possible to arrange that $\kappa$ is the first inaccessible in this model. | |
| Jan 28, 2015 at 15:16 | comment | added | Asaf Karagila♦ | Yair, what about homogeneous forcing? :-) | |
| Jan 28, 2015 at 15:15 | comment | added | Yair Hayut | Yes, you're right. Since $\mathbb{P}\ast \tilde{Col}$ is not $\kappa^+$-c.c., while $\mathbb{NM}\ast Col$ is $\kappa^+$-c.c. (even of size $\kappa$), the quotient forcing $\mathbb{R}$ is not $\kappa^+$-c.c. I don't know if it is possible to singularize the first inaccessible using a $\kappa^+$-c.c. forcing. | |
| Jan 28, 2015 at 15:01 | comment | added | Monroe Eskew | Very interesting. So if I understand the definitions correctly it looks like $\mathbb P * \tilde{Col}$ is not $\kappa^+$-c.c. (because of $\tilde{Col}$). I wonder if we can singularize the least inaccessible $\kappa$ with a $\kappa^+$-c.c. forcing as in the ordinary Prikry forcing. | |
| Jan 28, 2015 at 14:57 | vote | accept | Monroe Eskew | ||
| Jan 28, 2015 at 14:20 | history | answered | Yair Hayut | CC BY-SA 3.0 |