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I am reading "The Stationary Tower" trying to understand the proof of Theorem 1.5.12: "If $\delta$ is a Woodin cardinal, $Z$ is a set, and $T$ is a tree on $\omega\times Z$, then there is an $\alpha<\delta$ such that the forcing Coll($\omega,\alpha$) makes $T$ $<\delta$-weakly homogeneous."

It starts by showing that we can assume without loss of generality that $|T|\geq\delta$. Then it says "Fix a regular cardinal $\eta>\delta$ with $T\in V_{\eta}$ and let $T^{*}$ be the subtree of $T$ consisting of all nodes definable in $V_{\eta}$ from $T, \delta$ and parameters in $V_{\delta}$... $T$ and $T^{*}$ have the same projection in any forcing extension by a partial order in $V_{\delta}$."

I can see that this would be true if the set $Z$ had a definable well-ordering because then given any finite sequence of natural numbers $s$ there would be a definable node in $T_{s}$. But I am not sure why it would be true in general.

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  • $\begingroup$ I suspect (but only that!) the problem would be solved if we fix a well-ordering < on $V_{\eta}$, and define $T^*$ as the set of nodes of $T$ definable n $(V_\kappa. \in, <)$ from $T$ and parameters in $V_\delta$. $\endgroup$
    – Ali Enayat
    Commented May 31, 2011 at 1:58

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We say that $G$ is $< \kappa$-generic over $M$ iff $G$ is $M$-generic for some poset $\mathbb{P}$ such that $M\models |\mathbb{P}| < \kappa$.

I haven't read Woodin's paper but I think he is using an elementary submodel argument (also called a Skolem Hull argument) to show that we have a tree $T^∗$ on $\omega × \kappa$ such that $p[T]=p[T^∗]$ in any $< \kappa$ generic extension of $V$. I think the Hull argument should be by induction on the cardinality of $T$. So we construct an elementary chain $M_\delta$ of substructures of $V_\eta$. These elementary submodels are increasing and all the previous ones are included in the next one. Put everything you need in the first one (i.e ordinals, trees etc..., they're sitting in it as points not as objects themselves). For a limit ordinal we take union as usual. If at a certain ordinal we have the same projection then we're done. If not then we go to the next elementary submodel and find a contradiction, which wil follow by elementarity.

This theorem of Woodin is the ZFC counterpart to the AD theorem (Martin and Woodin) which says that assuming AD, if $\kappa$ is less than the sup of the Suslin cardinals then every tree on $\omega × \kappa$ is weakly homogeneous.

For a different proof, see Steel's paper The Derived Model Theorem (page 13).

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