Exact equiconsistency results for \(\Delta_ 3^ 1\)-sets of reals
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.
scientific article; zbMATH DE number 438611
scientific article