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

up vote 7 down vote accepted

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".

  • 2
    This is the reference. It takes work to translate the arguments there (usually presented using hypothesis stronger than Woodiness) into the optimal result, but all the ingredients are there. A very high level outline of the key steps is at the beginning of Woodin's book on the nonstationary ideal. If you read that first, then you will find it easier to go through Shelah's chapter and see what proofs you need to concentrate and work on. I do not know of a detailed account available in print. – Andrés E. Caicedo Oct 3 '11 at 16:25

I have a writeup Shelah's proof of the consistency of "NS is saturated" relative to "there is a Woodin cardinal" on my web page:

Jech (in the Chapter "Stationary Sets", from the Handbook of Set Theory) lists the reference as

Saharon Shelah. Iterated forcing and normal ideals on $\omega_1$. Israel Journal of Mathematics, 60(3):345–380, 1987.

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    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

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