On the structure of \(\Delta^ 1_ 4\)-sets of reals
type: Item
instance of: scholarly article
title: On the structure of \(\Delta^ 1_ 4\)-sets of reals
zbMATH Open document ID: 0842.03036
published in: Archive for Mathematical Logic
publication date: 1996-07-24T00:00:00Z
review text: It is shown that assuming the consistency of an inaccessible cardinal, it is consistent that every \({underset\sim\to\Delta}^1_4\) set of reals is measurable but not every \({\underset\sim\Delta}^1_4\) set of reals has the Baire property. Solovay in 1970 showed that assuming the consistency of an inaccessible cardinal it is consistent to have every projective set of reals be both measurable and have the Baire property. Shelah in 1984 showed that the use of an inaccessible is necessary to obtain Solovay's result, in fact, every \({\underset\sim\Sigma}^1_3\) set of reals measurable implies that \(\omega_1\) is inaccessible in \(L\). The technique used in this paper is to modify a model of \textit{R. David}, ``\({\underset\sim\Delta}^1_3\) reals'' [Ann. Math. Logic 23, 121-125 (1982; Zbl 0519.03039)]. David's model has the property that \[ x \in L[y] \text{ iff \(x\) is } \Delta^1_3(y) \] for every two reals \(x\) and \(y\).
Mathematics Subject Classification ID: 03E15
Mathematics Subject Classification ID: 03E35
Mathematics Subject Classification ID: 03E55
zbMATH DE Number: 851901
zbMATH Keywords: Lebesgue measurable sets of reals
zbMATH Keywords: projective hierarchy
zbMATH Keywords: consistency
zbMATH Keywords: inaccessible cardinal
zbMATH Keywords: Baire property
author: Haim Judah
author: Otmar Spinas
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 851901