A result of Shelah about the nonstationary ideal - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T07:30:27Z http://mathoverflow.net/feeds/question/77043 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/77043/a-result-of-shelah-about-the-nonstationary-ideal A result of Shelah about the nonstationary ideal Stefan Hoffelner 2011-10-03T13:58:14Z 2011-10-03T14:38:00Z <p>Suppose that $\kappa$ is a regular cardinal and let $NS$ be the ideal of its nonstationary subsets. One can consider the Boolean algebra $P(\kappa) /NS$ and say that (if $\lambda$ is another cardinal) $NS$ is $\lambda$ saturated iff there are no antichains in $P(\kappa) / NS$ of length $\lambda$. It is an elegant result of Gitik and Shelah that $NS$ cannot be $\kappa^+$ saturated for every regular $\kappa > \aleph_1$, on the other hand Foreman Magidor and Shelah could show that assuming a supercompact cardinal it is consistent that $NS$ of $\omega_1$ is $\aleph_2$ saturated. These results are all well known and one can find them for example in Jech's book. However it is stated there that Shelah eventually found that even a Woodin cardinal suffices to obtain the consistency of the statement "$NS$ on $\omega_1$ is $\aleph_2$ saturated". </p> <p>Do you know where I can find a proof of this result? </p> <p>Thank you</p> http://mathoverflow.net/questions/77043/a-result-of-shelah-about-the-nonstationary-ideal/77045#77045 Answer by Michael Blackmon for A result of Shelah about the nonstationary ideal Michael Blackmon 2011-10-03T14:12:15Z 2011-10-03T14:12:15Z <p>Jech (in the Chapter "Stationary Sets", from the Handbook of Set Theory) lists the reference as</p> <blockquote> <p>Saharon Shelah. Iterated forcing and normal ideals on $\omega_1$. Israel Journal of Mathematics, 60(3):345–380, 1987.</p> </blockquote> http://mathoverflow.net/questions/77043/a-result-of-shelah-about-the-nonstationary-ideal/77049#77049 Answer by Todd Eisworth for A result of Shelah about the nonstationary ideal Todd Eisworth 2011-10-03T14:31:10Z 2011-10-03T14:38:00Z <p>Try also Chapter XVI of "Proper and Improper Forcing" (entitled "Large ideals on <code>$\aleph_1$</code> from smaller cardinals"). It's hard to tell exactly what's in there, but he does say in the chapter he will "keep old promises from 84-85 mentioned in [Sh:253]", where [Sh:253] is the paper Michael mentions, and he does claim to be replacing certain hypotheses used earlier by the assumption "$\lambda$ is a Woodin cardinal".</p>