# A result of Shelah about the nonstationary ideal

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".

Do you know where I can find a proof of this result?

Thank you

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Try also Chapter XVI of "Proper and Improper Forcing" (entitled "Large ideals on $\aleph_1$ 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".
Saharon Shelah. Iterated forcing and normal ideals on $\omega_1$. Israel Journal of Mathematics, 60(3):345–380, 1987.
I read through the introduction to this paper; he mentions that things were improved to the use of $\kappa$ satisfying "$Pr_b$" (these are Woodin cardinals; Shelah is using old notation) but that the result wasn't written up yet. – Todd Eisworth Oct 3 '11 at 15:12