$\omega$ universally Baire sets, tree representations

Question

I've recently encountered the notion of a universally Baire set, and I've tried to look at the paper by Feng, Magidor and Woodin where this notion is studied. There are several points that confuse me.

(1) First of all, what is the extend to which we know that various subsets of the reals have tree representations, i.e., are $\kappa$-Souslin for some cardinal $\kappa$? Is it correct that under AD, all projective set have a tree representation? What about in Solovay's model? Do all subsets of the reals there have tree representations? (It is not at all clear to me that being definable from a sequence of ordinals has anything to do with having a tree representation, but perhaps I am missing something obvious.)

(2) These points aside, what I am really interested in is trying to understand the relationship between the notion of a universally Baire set and the paper by Shelah called "Can you take Solovay's inaccessible away?" (Isr. J. Math, 1984). It at some point seemed to me that what maybe really is going on there (in the section about a model of ZF where all sets have the Baire property) is that Shelah is proving that there is a model of ZF in which all sets are $\omega$ universally Baire, or something close to this fact.

(Here, when I say $\omega$ universally Baire, I mean a subset $A$ of Baire space where there are trees $T$ and $T^*$ such that $p[T]=A$, $p[T^*]=A^c$, and in any forcing extension by a countable poset (Cohen forcing) we have $p[T]\cup p[T^{*}]=\omega^\omega$.)

So, basically my question is: Is there a model of ZF where all subsets of the reals are $\omega$ universally Baire? And if yes, does this require a large cardinal hypothesis?

I hope someone can help, as I am completely new to much of this. Even just some references where some basic things are more clearly explained would help a lot. Thanks, R.A.D.

Answer 1

Concerning your first question, under $AD$, not every pointclass has the scale property. Recall that having a tree representation or as it is called in descriptive set theory, being $\kappa$-Suslin, for some $\kappa$, is essentially the same as having a scale whose norms have length $\kappa$.

A semi-scale on a set $A$ is defined to be a sequence of norms $\{\phi_n\}$ such that if $\{x_i\}$ is a sequence in $A$, if $x_i \to x$ and for every $n \in \omega$, $\phi_n(x_i)$ is eventually constant (equal to some $\lambda_n$ for sufficiently large $i$) then $x \in A$. If in addition we have for each $n \in \omega$, $\phi_n(x) \leq \lambda_n$ then we say $\{\phi_n\}$ is a scale. (this last condition is the lower semi-continuity property). The idea is to abstract the descriptive set theoretical structure we get on closed sets to all sets of reals.

Under $AD^+ + \theta_0=\Theta$, the pointclass $\Sigma^2_1$ is the largest pointclass with the scale property. Recall that a set $B \subseteq \mathbb{R}$ is $\Sigma^2_1$ if it is written as $\exists A\subseteq \mathbb{R} \psi(x,A)$, where $\psi$ is a projective condition. A pointclass $\Gamma$ has the scale property is every set in $\Gamma$ has a scale whose norms are $\Gamma$-norms. This means that under $AD$ there is a $\Pi^2_1$ set of reals such that scales on it are more complicated than $\Pi^2_1$. Or if you want (since scales imply uniformization), there is a $\Pi^2_1$ set of reals which can't be uniformized by a $\Pi^2_1$ function under $AD$. A classical result is the 2nd periodicity theorem of Moschovakis which states that under $AD$ (or locally Projective determinacy) every $\Pi^1_{2n+1}$ and every $\Sigma^1_{2n+2}$ have the scale property. The partern of the scale property is described in the famous article of Steel "Scales in $L(\mathbb{R})$". Basically scales disappear in the $\Sigma_1$-gaps. These are gaps where no new $\Sigma_1$ fact is true between some levels of the $J(\mathbb{R})$ hierarchy. The first non-trivial gap occur past $\kappa^{\mathbb{R}}$, the least $\mathbb{R}$-admissible ordinal. An example of a gap is $[\delta^2_1, \Theta$].

However once we assume $AD(\mathbb{R})$ there are cofinally in $\Theta$ many pointclasses $\Gamma$ with the scale property. Actually assume $AD+ \theta_0 < \Theta$ we can define scales whose norms of optimal complexity on $\Pi^2_1$ (ordinal definale scales, this is a result of Trevor Wilson). $AD(\mathbb{R})$ is equivalent to $AD$+ every set of reals is has a Suslin representation (or a tree representation). $AD(\mathbb{R})$ is the axiom of determinacy where the games are using real numbers instead if natural numbers. $AD+ \theta_0 < \Theta$ is axiom at the base of the "Solovay hierarchy". The central conjecture about the Solovay hierarchy is a conjecture of Grigor Sargsyan that the large cardinal hierarchy is covered and captured by the Solovay hierarchy.

Concerning the notion of universally Baire sets, this is a useful notion in descriptive set theory and inner model theory because under the large cardinal hypothesis of a Woodin cardinal $\kappa$ the notion of universally Baire is equivalent to the concepts of homogeneously Suslin and weakly homogeneously Suslin. Homogeneous trees are important for propagating Suslin representation. Often in the Core Model Induction we looking for model operators and iteration strategies $\Sigma$ which are universally Baire (once they're seen as being coded by sets of reals).

This answer is not exhaustive but I hope it gave the idea of how useful Suslin representations and what they're used for. Thanks to Trevor Wilson for the corrections.