Co-analytic mad families and definable wellorders
type: Item
instance of: scholarly article
title: Co-analytic mad families and definable wellorders
zbMATH Open document ID: 1303.03076
author: Yurii Khomskii
author: Vera Fischer
author: Sy-David Friedman
published in: Archive for Mathematical Logic
publication date: 2013-11-06T00:00:00Z
review text: It is consistent that \(\mathfrak b=\mathfrak c=\aleph_3\), there exists a~\(\Pi^1_2\)~\(\omega\)-mad family, and there exists a~\(\Delta^1_3\)-definable well-order of the reals [\textit{V. Fischer} et al., Ann. Pure Appl. Logic 162, No. 11, 853--862 (2011; Zbl 1267.03049)]. For any uncountable regular cardinal~\(\kappa\), it is consistent that \(\mathfrak b=\mathfrak c=\kappa\) and there exists a~\(\Pi^1_1\)~mad family [\textit{J. Brendle} and \textit{Y. Khomskii}, J. Symb. Log. 78, No. 4, 1164--1180 (2013; Zbl 1375.03057)]. The paper under review is concerned with the question to what extent these two consistency results can be merged together. The main theorem of the paper shows the consistency of \(\mathfrak b=\mathfrak c=\aleph_3\), the existence of a~\(\Pi^1_1\)~mad family, and the existence of a~\(\Delta^1_3\)-definable well-order of the reals.
reviewed by: Miroslav Repický
Mathematics Subject Classification ID: 03E15
Mathematics Subject Classification ID: 03E17
Mathematics Subject Classification ID: 03E35
zbMATH DE Number: 6223057
zbMATH Keywords: projective well-orders
zbMATH Keywords: projective mad families
zbMATH Keywords: large continuum
MaRDI profile type: Publication
full work available at URL: s00153-013-0345-8
OpenAlex ID: W1990234199
cites work: Adjoining dominating functions
cites work: Maximal almost disjoint families of functions
cites work: Countable Fréchet Boolean groups: An independence result
cites work: Mad Families Constructed from Perfect Almost Disjoint Families
cites work: Cardinal characteristics and projective wellorders
cites work: Cardinal characteristics, projective wellorders and large continuum
cites work: Projective wellorders and mad families with large continuum
cites work: Fine structure and class forcing
cites work: Projective mad families
cites work: Long projective wellorderings
cites work: Set Theory
cites work: Q5626664
cites work: Analytic and coanalytic families of almost disjoint functions
cites work: Set theory. An introduction to independence proofs
cites work: Happy families
cites work: Infinite combinatorics and definability
cites work: Σ12 and Π11 Mad Families
DOI: 10.1007/S00153-013-0345-8
description: scientific article; zbMATH DE number 6223057