Exact equiconsistency results for \(\Delta_ 3^ 1\)-sets of reals
type: Item
instance of: scholarly article
title: Exact equiconsistency results for \(\Delta_ 3^ 1\)-sets of reals
zbMATH Open document ID: 0786.03035
DOI: 10.1007/BF01269952
author: Haim Judah
published in: Archive for Mathematical Logic
publication date: 1993-11-28T00:00:00Z
review text: Raisonnier proved that if (a) all \(\Sigma^ 1_ 2\) sets of reals are Lebesgue measurable and (b) all \(\Sigma^ 1_ 3\) sets are \(K_ \sigma\) regular, then \(\aleph_ 1\) is inaccessible in the constructible universe. The author relaxes \(\Sigma^ 1_ 3\) in (b) to \(\Pi^ 1_ 2\) and shows that one cannot relax it to \(\Sigma^ 1_ 2\). He gives a model in which all \(\Sigma^ 1_ 2\) sets are \(K_ \sigma\) regular and all \(\Delta^ 1_ 3\) sets are Lebesgue measurable, Ramsey and have the Baire property. He also gives a model in which all \(\Delta^ 1_ 3\) sets have the Baire property but not all \(\Delta^ 1_ 2\) sets are measurable.
Mathematics Subject Classification ID: 03E35
Mathematics Subject Classification ID: 03E15
zbMATH DE Number: 438611
zbMATH Keywords: sets of reals
zbMATH Keywords: Lebesgue measurable
zbMATH Keywords: Ramsey
zbMATH Keywords: Baire property
reviewed by: Janusz Pawlikowski
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 438611