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If there is a proper class of Woodin cardinals, then Woodin showed (using stationary towers) that $(\Sigma^2_1)^{\text{uB}}$ statements are generically absolute, where $\text{uB}$ denotes the pointclass of universally Baire sets of reals. This generic absoluteness result has a more local version: if $\lambda$ is a limit of Woodin cardinals, then $(\Sigma^2_1)^{\text{uB}_\lambda}$ statements are generically absolute for posets of size less than $\lambda$, where $\text{uB}_\lambda$ denotes the pointclass of $\lambda$-universally Baire sets of reals (or what some people would call $\mathord{<}\lambda$-universally Baire sets of reals.)

The "local" generic absoluteness for $(\Sigma^2_1)^{\text{uB}_\lambda}$ can be explained in terms of trees (although the proof uses stationary towers instead.) More precisely, let $\varphi(v)$ be a formula in the language of set theory expanded by a unary predicate symbol. For every limit $\lambda$ of Woodin cardinals there is a tree $T_{\varphi,\lambda}$ such that in every generic extension $V[g]$ by a poset of size less than $\lambda$ we have

$$ V[g] \models p[T_{\varphi,\lambda}] = \{x \in \mathbb{R} : \exists A \in \text{uB}_\lambda\,(\text{HC}; \mathord{\in},A) \models \varphi[x]\}.$$

This tree is obtained from the scale property for the pointclass $\Sigma^2_1$ of the derived model of $V$ at $\lambda$. Given these trees $T_{\varphi,\lambda}$, the "local" generic absoluteness follows by a standard argument using the absoluteness of well-foundedness.

My question is, can the "global" generic absoluteness for $(\Sigma^2_1)^{\text{uB}}$ also be explained in terms of trees, assuming that there is a proper class of Woodin cardinals? More precisely, is there a single proper-class-sized tree $T_\varphi$ such that in every generic extension $V[g]$ we have

$$ V[g] \models p[T_{\varphi}] = \{x \in \mathbb{R} : \exists A \in \text{uB}\,(\text{HC}; \mathord{\in},A) \models \varphi[x]\}?$$

I can think of two possible approaches, both with apparently serious problems.

  1. Consider the "derived model at $\text{Ord}$." Problem: this doesn't really exist.

  2. Define $T_\varphi$ as the amalgamation of the trees $T_{\varphi,\lambda}$ for various $\lambda$, e.g. all limits of Woodin cardinals, or all limit of Woodin cardinals above some point. Problem: I don't see any way to show that the projection of such an amalgamated tree in some generic extension $V[g]$ is not too large.

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  • $\begingroup$ Is there a way to define "bigger and bigger" derived models for larger and larger $\lambda$ limits of Woodin cardinals and look at the corresponding trees? Along the way the derived models would have to cohere in some specific way as to ensure that the successive projections of the trees agree. Basically more and more statements would have to be verified. The final tree could be a lim inf of the construction. This just a quick guess which might turned out to be naive. In the 5th line there is a small typo: you meant to write "where $UB_{\lambda}$ denotes". Nice question by the way. $\endgroup$ Nov 19 '13 at 23:51
  • $\begingroup$ @CarloVonSchnitzel Thanks. Fixing any particular generic extension $V[g]$, the trees $T_{\varphi,\lambda}$ for sufficiently large $\lambda$ all have the correct projection. (This is because in $V[g]$ we have $\text{uB}_\lambda = \text{uB}$ for all sufficiently large $\lambda$. Unfortunately the meaning of "sufficiently large" depends on $g$.) So if there were a tree whose projection in any generic extension was the limit of the projections of the trees $T_{\varphi,\lambda}$ as $\lambda \to \text{Ord}$ then the answer to my question would be "yes". $\endgroup$ Nov 20 '13 at 0:49
  • $\begingroup$ ...but I don't know of any general construction of a tree whose projection is the limit (or lim sup or lim inf, if we don't want to assume that the limit exists) of the projections of a given uncountable sequence of trees, however. $\endgroup$ Nov 20 '13 at 0:51
  • $\begingroup$ Also, note that for any given $\lambda$ the tree $T_{\varphi,\lambda}$ is a set, so in sufficiently large generic extensions $V[g]$ it is countable and its projection is analytic and therefore too simple to be the desired $(\Sigma^2_1)^{\text{uB}_\lambda}$ set. But one approach would be trying to show that this analytic set (the projection) is always contained in the desired $(\Sigma^2_1)^{\text{uB}_\lambda}$ set. I have no idea whether this is true. (The analogous containment is true if you consider various sizes of Shoenfield tree for $\Sigma^1_2$, so maybe there is hope.) $\endgroup$ Nov 20 '13 at 0:57
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The answer is yes. Hugh Woodin showed me the following argument, which I post here with his permission.

Let $\varphi(v)$ be a formula in the language of set theory expanded by a unary predicate symbol. Given a pair of ordinals $(\alpha, \beta)$, working in $V^{\text{Col}(\omega,\alpha)}$ we let $B$ be a universally Baire set of reals having Wadge rank $\beta$ in the model $L(B,\mathbb{R})$, which satisfies $\mathsf{AD}^+$. Note that this model depends only on $\beta$ and not on $B$, and also that every set of reals in $L(B,\mathbb{R})$ is universally Baire because $B^\sharp$ exists and is universally Baire. Let $T_{\alpha,\beta}$ be the tree of a $(\Sigma^2_1)^{L(B,\mathbb{R})}$-scale on the set $$\{x \in \mathbb{R} : \exists C \in L(B,\mathbb{R})\, (\text{HC}; \in, C) \models \varphi[x]\}.$$ By the homogeneity of $\text{Col}(\omega,\alpha)$ this tree is is independent of the choice of generic filter and we have $T_{\alpha,\beta} \in V$. Let $T$ by the amalgamation of all the trees $T_{\alpha,\beta}$, so that $T$ is a tree on $\omega \times \text{Ord}$ and $p[T] = \bigcup_{\alpha,\beta \in \text{Ord}} p[T_{\alpha,\beta}]$ in every generic extension of $V$.

We claim that $$ V^{\text{Col}(\omega,\alpha)} \models p[T] = \{x \in \mathbb{R} : \exists C \in \text{uB}\, (\text{HC}; \in, C) \models \varphi[x]\},$$ for every ordinal $\alpha$. The right-to-left inclusion follows immediately from the definition of the trees $T_{\alpha,\beta}$, so it remains to prove the left-to-right inclusion. Let $G \subset \text{Col}(\omega,\alpha)$ be a $V$-generic filter and let $x \in p[T]^{V[G]}$, say $x \in p[T_{\alpha',\beta'}]$ for ordinals $\alpha'$ and $\beta'$. We want to show
\begin{equation*}\tag{$*$} \exists C \in \text{uB}^{V[G]}\, (\text{HC}^{V[G]}; \in, C) \models \varphi[x]. \end{equation*}

If $\alpha' = \alpha$, this is easy. There are two remaining cases to consider:

  1. $\alpha' > \alpha$.

  2. $\alpha' < \alpha$.

In case (1), we have ($*$) by $(\Sigma^1_2)^{\text{uB}}$ generic absoluteness for $\text{Col}(\omega,\alpha')$. In case (2), we use the fact that if $B \in V[G \restriction \alpha']$ is a universally Baire set as in the definition of the tree $T_{\alpha', \beta'}$, then $B^\sharp$ exists and is universally Baire, so there is an elementary embedding $$ j : L(B, \mathbb{R}^{V[G \restriction \alpha]}) \to L(B^{V[G]}, \mathbb{R}^{V[G]}),$$ and we have $j(T_{\alpha', \beta'}) = T_{\alpha, \beta}$ where $\beta$ is the Wadge rank of $B^{V[G]}$. Considering the pointwise image of a branch witnessing $x \in p[T_{\alpha',\beta'}]$, we have $x \in p[T_{\alpha,\beta}]$ . Therefore ($*$) is witnessed by a set of reals $C \in L(B^{V[G]}, \mathbb{R}^{V[G]})$.

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