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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.

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You can use most of the basic $\LaTeX$ commands here using $ to encase them. –  Asaf Karagila Apr 6 at 15:58
    
The main point in Shelah's proof of the projective Baire property is the construction of homogeneous algebras by amalgamating sweet forcing. Once the construction is done, the argument goes along the lines of Solovay's original proof (so it's not clear where universally Baire sets should emerge in the proof). –  Haim Apr 6 at 16:55
    
Thanks! I guess I have a long way to go before I understand anything of Shelah's paper... –  RAD Apr 6 at 23:09
    
By the way, as for the projective Baire part of the paper, this should be much easier than Shelah's original paper: unomaha.edu/logic/papers/juro93/juro93.ps –  Haim Apr 6 at 23:18
    
Feng, Magidor, and Woodin showed that a set $A$ of reals is $\omega$-universally Baire if and only if every continuous preimage of $A$ by a function $\omega^\omega \to \omega^\omega$ has the property of Baire. So if every set of reals has the property of Baire (as in Shelah's model) then every set of reals is $\omega$-universally Baire. –  Trevor Wilson Apr 7 at 0:53

1 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.

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I edited my answer a couple of times to add some more details. –  Carlo Von Schnitzel Apr 6 at 19:04
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Thanks for your very interesting remarks! In the mean time, I realized that by virtue of Theorem 2.1 in Feng-Magidor-Woodin, if every set of reals has the BP then all sets are omega universally Baire. The mystery to me is what kind of tree representation you get. On the face of it, it seems that the tree is going to be on $\omega\times 2^{\omega}$, which seems like it wouldn't be a very interesting tree. I wonder if in the Solovay model, say, one can do better, i.e., get a tree on $\omega\times\lambda$ for some cardinal - ? –  RAD Apr 6 at 23:04
    
In your case of an $\omega$-universally Baire $A$, your trees, let's call them $T$ and $T^*$ as above, would be trees on $\omega \times \omega$ and say $T$ is the one which projects to $A$. So $A$ is a $\Sigma^1_1$ set. Since the complement of $A$ (the projection of $T^*$) is also $\Sigma^1_1$ this makes $A$ a Borel set (all of this takes place in the generic extension, by definition). In general, Borel sets are universally Baire ($\omega$-universally Baire). For $\lambda$-universally Baire the trees would be on $\omega \times \lambda$ in the generic extension. –  Carlo Von Schnitzel Apr 7 at 0:21
    
I'm glad to see my name come up in your answer! But I should mention that it's only the individual norms of the scales on $\Pi^2_1$ sets that can be OD; not, in general, the scale itself (an $\omega$-sequence of norms.) A couple of other things: for $\Sigma^2_1$ to be the largest pointclass with the scale property the assumption should be $\mathsf{AD}^+ + \theta_0 = \Theta$. And under $\mathsf{AD}_{\mathbb{R}}$, not every pointclass has the scale property (as you mentioned, there are gaps,) but the pointclasses with the scale property are cofinal in the Wadge hierarchy. –  Trevor Wilson Apr 7 at 0:43
    
Also, what you call the scale property is usually called the weak scale property; the scale property for $\Gamma$ is usually defined to mean that every set in $\Gamma$ has a $\Gamma$-scale $\vec{\varphi}$ (roughly, the norms $\varphi_i$ are $\Gamma$-norms, uniformly in $i$.) The distinction is especially important when $\Gamma$ has countable Wadge cofinality. –  Trevor Wilson Apr 7 at 0:47

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