Perfect-set properties in \(L(\mathbb{R})[U]\)
It is well known that various forms of the axiom of choice lead to sets of reals with singular properties. One of the most familiar examples is Bernstein's totally imperfect set of reals obtained using a well-ordering of \(\mathbb{R}\), i.e., a set of reals \(X\) which is neither disjoint nor includes a nonempty perfect set of reals. That some form of AC is needed to get such a set was proved much later by Solovay, who produced a model in which every definable set of reals \(X\) is Lebesgue measurable, has the property of Baire, and contains a nonempty perfect set whenever uncountable. Here, ``definable'' means that \(X= \{x\in\mathbb{R}: \varphi(x,r,\alpha)\}\) for some formula \(\varphi\) and parameters \(r\in R\) and \(\alpha\in\text{Ord}\). Thus, a natural model where the pathology of totally imperfect sets of reals might not occur is the model \(L(\mathbb{R})\), the constructible closure of the reals. Solovay constructed his model using a Lévy collapse of an inaccessible cardinal \(\kappa\) to the first uncountable ordinal \(\omega_1\), so it is appropriate to call \(L(\mathbb{R})\) as computed in \(V^{\text{Coll}(\omega,<\kappa)}\) a Solovay model over the ground model \(V\). This turned out to be quite a useful concept especially after W. H. Woodin proved, using a result of Foreman, Magidor, and Shelah, that in the presence of certain large cardinals, the \(L(\mathbb{R})\) of the universe is elementarily equivalent to a Solovay model. Thus any study of Solovay models is likely to lead not only to consistency results but also to results about definable sets of reals in the presence of some large cardinals. It is interesting that in this situation we also have that any selective ultrafilter \(U\) on \(\omega\) is an \(L(\mathbb{R})\)-generic subset of \([\omega]^\omega\) when considered as a forcing notion ordered by \(\subseteq^*\), the inclusion modulo a finite set. This leads us to another interesting model \(L(\mathbb{R})[U]\), considered as the corresponding generic extension. In this paper we show that many of the pleasant properties of the Solovay model \(L(\mathbb{R})\) are preserved by passing to \(L(\mathbb{R})[U]\). By a well-known classical argument of W. Sierpinski, nonprincipal ultrafilters on \(\mathbb{N}\) can naturally be identified with sets of reals which are neither Lebesgue measurable nor have the property of Baire. So one of the interpretations of our preservation results is that the strong perfect set properties which hold in \(L(\mathbb{R})[U]\) are not sufficient for concluding that all sets of reals are Lebesgue measurable and have the property of Baire. However, our main interest in \(L(\mathbb{R})[U]\) comes from our attempts to develop the so-called ``parameterized partition calculus''. Namely, it is not hard to see that any preservation result of a perfect-set property while passing from \(L(\mathbb{R})\) to \(L(\mathbb{R})[U]\) is equivalent to a partition property of the product \([\omega]^\omega\times \mathbb{R}\). Thus, it is not surprising that one of the main tools we use here is Mathias' analysis of the Ramsey property in Solovay models. Instead of working directly with the set of real numbers, we will work with the Baire Space, \(\omega^\omega\), the set of sequences of natural numbers with the product topology obtained giving the discrete topology to the set of natural numbers.
scientific article; zbMATH DE number 1236402
scientific article