Theorem 2.65 in Woodin's [book](http://www.ams.org/mathscinet-getitem?mr=2723878) shows that a saturated ideal on $\omega\_1$ exists after Levy-collapsing a Woodin cardinal $\delta$ to $\omega\_2$. I am confused about the part of the argument where he shows that ideal he defines is a proper ideal.


Claim (2.2) on page 50 says we can find a countable structure $X \prec H\_{\delta^+}$ and an $X$-generic condition $p$ such that for all inaccessible $\gamma \in X \cap \delta$ and $Col(\omega\_1,<\gamma)$-names $\tau \in X$ for a semi-proper subset of $\mathcal P(\omega\_1)/NS$, there is a $\sigma \in X$ such that $p \Vdash \sigma \in \tau$ and $p \Vdash X \cap \omega\_1 \in \sigma$. He says we construct this pair $(X,p)$ by an elementary chain.


Suppose $X\_0 \prec H\_{\delta^+}$ and $p\_0 \restriction \gamma \in X\_0$. By definition of semi-proper, if $\tau \in X\_0$ is as above, then $p\_0 \Vdash\_\gamma (\exists y \in \tau) Sk(X\_0[G\_\gamma] \cup \{ y \}) \cap \omega\_1 = X\_0[G\_\gamma] \cap \omega\_1$ and $X\_0[G\_\gamma] \cap \omega\_1 \in y$. The problem I have is: How do we choose a *name* for $y$ with a similar property? There is a name $\sigma$ such that $p\_0$ forces $\sigma^G$ witnesses the above property of $y$ in $V[G\_\gamma]$, but could it be that $Sk(X\_0 \cup \{ \sigma \}) \cap \omega\_1 \not= X\_0 \cap \omega\_1$?


Or perhaps there is a quite different strategy for building the elementary chain. Thanks for your help!



Details for Woodin's forcing argument for a saturated ideal from the Levy collapse
Woodin's argument is wrong. Let us state his definition of the ideal. Assume $G \subseteq \mathrm{Col}(\omega\_1,<\delta)$ is generic over $V$.



> 
> Let $I\_0 \in V[G]$ be the set of $A \subseteq \omega\_1$ such that for some $f : \omega\_1 \to \mathcal{P}(\omega\_1) \setminus NS$,
> 
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> (1.1) $A = \{ \beta < \omega\_1 : \beta \notin f(\alpha)$ for all $\alpha < \beta \}$
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> 
> (1.2) If $\mathcal A = \{ f(\alpha) : \alpha < \omega\_1 \}$, then for some $\gamma < \delta$, $\gamma$ is strongly inaccessible in $V$, $\mathcal A \in V[G \cap V\_\gamma]$ and $\mathcal A$ is semiproper in $V[G \cap V\_\gamma]$.
> 
> 
> Let $I$ be the normal ideal generated by $I\_0$.
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> 
> 


As stated earlier in the chapter, Foreman, Magidor, and Shelah [proved](http://www.ams.org/mathscinet-getitem?mr=924672) that whenever $\delta$ is supercompact and $G \subseteq \mathrm{Col}(\omega\_1,<\delta)$ is generic, then in $V[G]$, every maximal antichain in $\mathcal P(\omega\_1)/NS$ is semiproper. Actually only something like $\beth\_4(\delta)$-supercompactness is used.


Assume $\kappa < \delta$ are both Woodin and $\beth\_4(\cdot)$-supercompact. Let $G\_\delta \subseteq \mathrm{Col}(\omega\_1,<\delta)$ be generic and let $G\_\kappa = G\_\delta \cap V\_\kappa$. Let $I\_\kappa \in V[G\_\kappa]$ and $I\_\delta \in V[G\_\delta]$ be the ideals as above, and note that $I\_\kappa \subseteq I\_\delta$.


By a well-known forcing argument (see [here](https://mathoverflow.net/questions/125308/forcing-diamond)), $V[G\_\kappa]$ satisfies $\Diamond(S)$ for every stationary $S \subseteq \omega\_1$. This easily implies that $NS \restriction S$ is not $\omega\_2$-saturated for any stationary $S$. In $V[G\_\kappa]$, the set of stationary $S$ which are in $I\_\kappa$ is dense in $\mathcal{P}(\omega\_1)/NS$. This is because otherwise there would be some $S$ such that $I\_\kappa \restriction S = NS \restriction S$.


Thus there is a maximal antichain of stationary sets $\mathcal A \subseteq I\_\kappa$ in $V[G\_\kappa]$, and it is semiproper. In $V[G\_\delta]$ there is an enumeration $\{ S\_\alpha : \alpha < \omega\_1 \}$ of $\mathcal A$, and the diagonal union $\nabla S\_\alpha$ is put into the dual filter to $I\_\delta$. But since $I\_\kappa \subseteq I\_\delta$ and $I\_\delta$ is normal in $V[G\_\delta]$, $\nabla S\_\alpha \in I\_\delta$. Thus $I\_\delta$ is not a proper ideal.


