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Suppose that $\delta$ is a Woodin cardinal and that $\kappa$ is the critical point of the generic embedding $j:V\rightarrow M$ after forcing with the stationary tower ($\kappa$ can be $\omega_1$ or $\omega_2$). I have seen that in some places people mention that $j(\kappa)=\delta$, and in other places that $\delta$ is fixed by $j$. Is this right? Can someone explain the difference, and in which cases we get the first and in which cases the second (for example, does countable or full tower make a difference)? On what conditions do these properties depend?

Unfortunately I have no reference for the tower and my knowlegde comes from some online sources. So perhaps my questions are trivial or vague but I would really appreciate any helpful comments or hints.

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    $\begingroup$ I think $j(\omega_1)=\delta$ holds for the countable stationary tower forcing while $j(\delta)=\delta$ holds for the full stationary tower forcing. $\endgroup$ Oct 19, 2013 at 14:41

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Peter gives the correct answer: in fact for the full tower the map $j$ also fixes cofinally many completely Jonsson cardinals below $\delta$ (if $\delta$ is a limit of completely Jonsson cardinals).

(A cardinal $ \kappa > \omega_1 $ is completely Jonsson if for every stationary set $S$ such that $S ∈ V_{\kappa}$, $\{Y ⊆ V_{\kappa} \mid \kappa = |Y \cap \kappa| \wedge Y \cap \bigcup S ∈ S\}$ is stationary in $P(\kappa)$. )

In fact if $G$ is the generic, then for any cardinal $\kappa<\delta$, if $\{Y\subseteq \kappa \mid|Y|=\kappa\}∈G$, then $j(\kappa) = \kappa$.

See: Paul Larson "The Stationary Tower: Notes on a course of W. Hugh Woodin," AMS University Lecture Note Series, vol 32, 2004.

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  • $\begingroup$ More generally, for $\alpha \leq \beta < \delta$, and $R$ a relation in $\{ \leq, =, \geq\}$, $j(\alpha) R \beta$ if and only if $\{ z \in \mathcal{P}(\beta) : \mathrm{ot}(z \cap \beta) R \alpha\}$ is in the generic filter, where $\mathrm{ot}(x)$ denotes the ordertype of $x$. $\endgroup$ Jul 7, 2014 at 5:32
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In any restricton of the stationary tower when $S=\mathcal{P}_\lambda (V_\delta)$ the critical point is given by the fact: $cp(j)=\mu$ if and only if $\{z\in \mathcal{P}(\mu):z\cap\mu\in \mu\}$ is in the generic.

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