Eventually different functions and inaccessible cardinals
type: Item
instance of: scholarly article
title: Eventually different functions and inaccessible cardinals
zbMATH Open document ID: 1215.03059
author: Jörg Brendle
author: Benedikt Löwe
published in: Journal of the Mathematical Society of Japan
publication date: 2011-03-28T00:00:00Z
full work available at URL: article
review text: The conditions of eventually different forcing, denoted by \textbf{E}, are pairs \(\langle s,F \rangle\) where \(s\) is a finite sequence of natural numbers and \(F\) is a finite set of reals. The order is given by \(\langle s,F \rangle \leq \langle t, G \rangle\) iff \(t \subseteq s\), \(G \subseteq F\) and for all \(i \in \text{dom}(s \setminus t)\) and all \(g \in G\), we have \(s(i) \neq g(i)\). This forcing generates a topology \(\mathcal E\), the eventually different topology, and refines the standard topology on Baire spaces. The authors investigate the Baire property of the eventually different topology. They obtain a characterization of the \(\mathcal{E}\)-Baire property of the \({\pmb \Sigma}^1_2\) sets by showing (Theorem 7) that the following are equivalent: (i) every \({\pmb \Sigma}^1_2\) set has the \(\mathcal{E}\)-Baire property (i.e., \({\pmb \Sigma}^1_2({\mathbf E}))\), (ii) for every \(x\), the set of \({\mathbf E}\)-generics over \(L[x]\) is \(\mathcal{E}\)-comeager, and (iii) \(\omega_1\) is inaccessible by reals. Further on, the authors show: Let \(G\) be \(L\)-generic for the \(\omega_1\)-stage finite support iteration of Hechler forcing. Then, in \(L[G]\), \({\pmb \Delta}^1_2({\mathbf D})\) holds while \({\pmb \Delta}^1_2({\mathbf E})\) fails.
reviewed by: Martin Weese
Mathematics Subject Classification ID: 03E15
Mathematics Subject Classification ID: 03E40
Mathematics Subject Classification ID: 03E55
zbMATH DE Number: 5871033
zbMATH Keywords: Baire property
zbMATH Keywords: eventually different topology
zbMATH Keywords: projective hierarchy
zbMATH Keywords: inaccessible by reals
zbMATH Keywords: dominating topology
zbMATH Keywords: Hechler forcing
MaRDI profile type: Publication
OpenAlex ID: W2167819478
cites work: Adjoining dominating functions
cites work: Silver Measurability and its relation to other regularity properties
cites work: Q4844091
cites work: Solovay-type characterizations for forcing-algebras
cites work: Combinatorial properties of classical forcing notions
cites work: Cardinal invariants of the continuum and combinatorics on uncountable cardinals
cites work: Forcing absoluteness and regularity properties
cites work: \(\Delta^ 1_ 2\)-sets of reals
cites work: Topology and descriptive set theory
cites work: Q4361603
cites work: Why Solovay real produces Cohen real
cites work: A model of set-theory in which every set of reals is Lebesgue measurable
DOI: 10.2969/JMSJ/06310137
description: scientific article; zbMATH DE number 5871033