Selctive ultrafilters and homogeneity
type: Item
instance of: scholarly article
title: Selctive ultrafilters and homogeneity
zbMATH Open document ID: 0649.03036
DOI: 10.1016/0168-0072(88)90027-9
author: Andreas Blass
published in: Annals of Pure and Applied Logic
publication date: 1988-01-01T00:00:00Z
review text: This paper consists, roughly speaking, of three parts: in the first part the author refines older results on the existence of homogeneous sets for partitions of \([\omega]^{\omega}\). An interesting byproduct of these investigations is the result that: if every ccc forcing adds either a Cohen or a random real, then there are no P-points. This sheds some light on the question of Laver and Prikry whether the if-clause is consistent with ZFC. The second part investigates what happens after Lévy-collapsing a Mahlo cardinal to \(\omega_ 1\). In that model non-isomorphic selective ultrafilters are radically different: they are mutually generic on \([\omega]^{\omega}\) over the class of sets hereditarily definable over the ground model and \({\mathbb{R}}.\) In the third part, the author shows that under CH there is a group of autohomeomorphisms of \(\beta\omega\setminus \omega\) that distinguishes non-isomorphic ultrafilters. It is a result of \textit{W. Rudin} [Duke Math. J. 23, 409-419 (1956; Zbl 0073.396)] that under CH any two P-points can be mapped onto each other by an autohomeomorphism of \(\beta\omega\setminus \omega\).
Mathematics Subject Classification ID: 03E05
Mathematics Subject Classification ID: 03E55
Mathematics Subject Classification ID: 03E35
Mathematics Subject Classification ID: 54A25
zbMATH DE Number: 4059385
zbMATH Keywords: partition theorems
zbMATH Keywords: Cohen real
zbMATH Keywords: homogeneous sets
zbMATH Keywords: ccc forcing
zbMATH Keywords: random real
zbMATH Keywords: P-points
zbMATH Keywords: Lévy-collapsing
zbMATH Keywords: Mahlo cardinal
zbMATH Keywords: selective ultrafilters
reviewed by: K. P. Hart
MaRDI profile type: Publication
full work available at URL: 0168-0072(88)90027-9
OpenAlex ID: W1986090861
cites work: Q4087141
cites work: Partition Theorems and Ultrafilters
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cites work: Ultrafilters with small generating sets
cites work: Ultrafilters on a countable set
cites work: Q4095536
cites work: A new proof that analytic sets are Ramsey
cites work: Borel sets and Ramsey's theorem
cites work: Combinatorics on ideals and forcing
cites work: Q3724321
cites work: Ultrafilters and Independent Sets
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cites work: Une méthode topologique pour l'étude de la propriété de Ramsey
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description: scientific article; zbMATH DE number 4059385