# Details for Woodin's forcing argument for a saturated ideal from the Levy collapse

Theorem 2.65 in Woodin's book 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!

• The argument Hugh refers to can be seen in (more) detail at the beginning of Paul Larson's book on stationary tower forcing. May 30, 2015 at 12:00
• Thanks Andres, but could you be a little more specific? May 30, 2015 at 12:14
• And could you let me know if I am on the right track for reconstructing the details? May 30, 2015 at 12:22
• @AndresCaicedo, I do not see how anything in the first chapter addresses my question. May 31, 2015 at 10:59
• Yes, nothing there is explicitly what you need. But the construction is essentially as in 1.1.18-1.1.22, iterated. If I can think of a more explicit place, I'll post a reference. May 31, 2015 at 12:46

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.

• There's some unicode issue in the block quote. (I see three "characters", after $A$ in the first line, and after "some" and before "is strongly" in the third line.) Jun 15, 2015 at 9:00
• It doesn't show up for me, but I tweaked it a bit. Maybe a Mac vs PC thing. Do you still see it? Feel free to fix it. Jun 15, 2015 at 9:03
• I removed that, sorry in advance if something got messed up. Jun 15, 2015 at 9:05