Perfect-set properties in \(L(\mathbb{R})[U]\)
type: Item
instance of: scholarly article
title: Perfect-set properties in \(L(\mathbb{R})[U]\)
zbMATH Open document ID: 0929.03050
author: Carlos A. Di Prisco
author: Stevo Todorčević
published in: Advances in Mathematics
publication date: 1999-09-14T00:00:00Z
review text: 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 correspondi…
Mathematics Subject Classification ID: 03E05
Mathematics Subject Classification ID: 03E45
Mathematics Subject Classification ID: 03E35
Mathematics Subject Classification ID: 03E55
Mathematics Subject Classification ID: 03E02
zbMATH DE Number: 1236402
zbMATH Keywords: Baire property
zbMATH Keywords: parameterized partition calculus
zbMATH Keywords: constructible closure of the reals
zbMATH Keywords: Solovay model
zbMATH Keywords: selective ultrafilter
zbMATH Keywords: generic extension
zbMATH Keywords: Lebesgue measurable
zbMATH Keywords: perfect-set property
zbMATH Keywords: partition property
zbMATH Keywords: Ramsey property
zbMATH Keywords: Baire Space
MaRDI profile type: Publication
full work available at URL: aima.1998.1752
OpenAlex ID: W2057466172
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DOI: 10.1006/AIMA.1998.1752
description: scientific article; zbMATH DE number 1236402