Full-splitting Miller trees and infinitely often equal reals
type: Item
instance of: scholarly article
title: Full-splitting Miller trees and infinitely often equal reals
zbMATH Open document ID: 1426.03029
author: Yurii Khomskii
author: Giorgio Laguzzi
published in: Annals of Pure and Applied Logic
publication date: 2017-05-18T00:00:00Z
review text: For every tree \(T \subseteq\,^{< \omega}\omega\), \([T]\) denote, as usual, the set of all branches of \(T\), i.e. \([T] = \{s \in\,^\omega \omega: \forall n < \omega\,[s \upharpoonright n \in T]\}\). A tree \(T \subseteq\,^{< \omega}\omega\) is said to be a \textit{full-splitting Miller tree} if every \(t \in T\) has an extension \(s \in T\) which is \textit{full-splitting}, i.e., \(s^\frown\langle n \rangle \in T\) for every \(n < \omega\), and it is said to be an \textit{infinitely often equal tree}, or simply \textit{ioe-tree}, if for every \(t \in T\) there exists some \(N > |t|\) such that for every \(k < \omega\) there exists \(s \in T\) which extends \(t\) and satisfies \(s(N) = k\). Letting \(\mathbb{FM}\) and \(\mathbb{IE}\) denote the partial orders of full-splitting Miller trees and ioe-trees (both ordered by inclusion), respectively, one has \(\mathbb{FM} \subsetneq \mathbb{IE}\). Two reals \(x,y \in\,^\omega \omega\) are \textit{infinitely often equal (ioe)} if \(\exists^\infty n\, [x(n) = y(n)]\), and are \textit{eventually different (evd)} if \(\forall^\infty n\, [x(n) \neq y(n)]\). \(\mathfrak{D}_\omega\) denotes the \(\sigma\)-ideal generated by \(\{D_f: f \in\,^{(^{<\omega} \omega)} \omega\}\), where for a given function \(f:\, ^{<\omega} \omega \to \omega\) one has \(D_f = \{x \in\,^\omega \omega: \forall^\infty n\, [x(n) \neq f(x\upharpoonright n)]\}\), and \(\mathfrak{J}_{\mathrm{ioe}}\) denotes the \(\sigma\)-ideal generated by \(\{K_x: x \in\,^\omega \omega\}\), where for a given real \(x: \omega \to \omega\) one has \(K_x = \{y \in\,^\omega \omega: \forall^\infty n\, [x(n) \neq y(n)]\}\). One has \(\mathfrak{J}_{\mathrm{ioe}} \subsetneq \mathfrak{D}_\omega \subsetneq \mathcal{M}\), where \(\mathcal{M}\) denotes the ideal of meager subsets of \(^\omega \omega\). \(\mathfrak{J}_{\mathrm{ioe}}\)-positive sets are said to be \textit{countably infinitely often equal families}. In the paper under review, the authors investigate dichotomy theorems…
reviewed by: Samuel Gomes da Silva
Mathematics Subject Classification ID: 03E15
Mathematics Subject Classification ID: 03E17
Mathematics Subject Classification ID: 03E35
zbMATH DE Number: 6720136
zbMATH Keywords: descriptive set theory
zbMATH Keywords: idealized forcing
zbMATH Keywords: dichotomy theorems
zbMATH Keywords: regularity properties
MaRDI profile type: Publication
full work available at URL: j.apal.2017.02.001
OpenAlex ID: W2588727999
cites work: Q4844091
cites work: Solovay-type characterizations for forcing-algebras
cites work: Silver Measurability and its relation to other regularity properties
cites work: Q5342877
cites work: Cichoń's diagram, regularity properties and \(\varDelta^1_3\) sets of reals
cites work: Forcing absoluteness and regularity properties
cites work: Q4805123
cites work: On a Notion of Smallness for Subsets of the Baire Space
cites work: On the separation of regularity properties of the reals
cites work: Q4693160
cites work: The Ideal Determined by the Unsymmetric Game
cites work: On game ideals
cites work: Perfect set theorems
cites work: Descriptive set theory and definable forcing
cites work: Q5444715
cites work: Dimension theory and forcing
DOI: 10.1016/J.APAL.2017.02.001
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description: scientific article; zbMATH DE number 6720136