Regularity properties of definable sets of reals
type: Item
instance of: scholarly article
title: Regularity properties of definable sets of reals
zbMATH Open document ID: 0583.03033
DOI: 10.1016/0168-0072(85)90003-X
author: Jacques Stern
published in: Annals of Pure and Applied Logic
publication date: 1985-01-01T00:00:00Z
review text: Various desirable properties of sets of reals which follow from the axiom of determinateness have been shown to be consistent with ZF. This paper is a further contribution to this subject improving results obtained by R. M. Solovay and S. Shelah. The main theorem states that - assuming the existence of a standard model of ZF - there is a standard model of ZFC in which (i) every set of reals definable from ordinals and reals as parameters has the property of Baire and contains a perfect set or is well-ordered with order type \(\leq \omega_ 1\), and (ii) every ordinal- definable set of reals is Lebesgue-measurable and is Ramsey (and some additional properties hold). The author's main tool is a new concept of forcing with topologically ordered sets satisfying a ''lower bound property''. (The papers by \textit{J. Raisonnier} and \textit{S. Shelah} to which the author refers have been published meanwhile in Isr. J. Math. 48 (1984), pp. 48-56 and pp. 1-47 resp.)
reviewed by: K.Gloede
Mathematics Subject Classification ID: 03E15
Mathematics Subject Classification ID: 03E35
Mathematics Subject Classification ID: 03E45
zbMATH DE Number: 3933056
zbMATH Keywords: ordinal definability
zbMATH Keywords: Lebesgue measurability
zbMATH Keywords: Baire property
zbMATH Keywords: perfect set theorem
zbMATH Keywords: standard model of ZFC
zbMATH Keywords: forcing with topologically ordered sets
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3933056