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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 $f : \omega_1 \to \mathcal{P}(\omega_1) \setminus NS$,

 

(1.1) $A = \{ \beta < \omega_1 : \beta \notin f(\alpha)$ for all $\alpha < \beta \}$

 

(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 $I_0$.

As stated earlier in the chapter, Foreman, Magidor, and Shelah proved 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), $V[G_\kappa]$ 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.

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 $f : \omega_1 \to \mathcal{P}(\omega_1) \setminus NS$,

(1.1) $A = \{ \beta < \omega_1 : \beta \notin f(\alpha)$ for all $\alpha < \beta \}$

(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 $I_0$.

As stated earlier in the chapter, Foreman, Magidor, and Shelah proved 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), $V[G_\kappa]$ 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.

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 $f : \omega_1 \to \mathcal{P}(\omega_1) \setminus NS$,

(1.1) $A = \{ \beta < \omega_1 : \beta \notin f(\alpha)$ for all $\alpha < \beta \}$

(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 $I_0$.

As stated earlier in the chapter, Foreman, Magidor, and Shelah proved 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), $V[G_\kappa]$ 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.

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 $f : \omega_1 \to \mathcal{P}(\omega_1) \setminus NS$,

(1.1) $A = \{ \beta < \omega_1 : \beta \notin f(\alpha)$ for all $\alpha < \beta \}$

(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 $I_0$.

As stated earlier in the chapter, Foreman, Magidor, and Shelah proved 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), $V[G_\kappa]$ 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.

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 $f : \omega_1 \to \mathcal{P}(\omega_1) \setminus NS$,

(1.1) $A = \{ \beta < \omega_1 : \beta \notin f(\beta)$ for all $\alpha < \beta \}$

(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 $I_0$.

As stated earlier in the chapter, Foreman, Magidor, and Shelah proved that whenever $\delta$ is supercompact and $G \subseteq \mathrm{Col}(\omega_1,<\delta)$ is generic, then in $V[G]$, every maximal antichain $\mathcal P(\omega_1)/NS$ 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), $V[G_\kappa]$ 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.

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 $f : \omega_1 \to \mathcal{P}(\omega_1) \setminus NS$,

(1.1) $A = \{ \beta < \omega_1 : \beta \notin f(\alpha)$ for all $\alpha < \beta \}$

(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 $I_0$.

As stated earlier in the chapter, Foreman, Magidor, and Shelah proved 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), $V[G_\kappa]$ 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.