Notes on singular cardinal combinatorics
type: Item
instance of: scholarly article
title: Notes on singular cardinal combinatorics
zbMATH Open document ID: 1121.03053
author: James Cummings
published in: Notre Dame Journal of Formal Logic
publication date: 2006-03-10T00:00:00Z
review text: This paper, an extended version of the author's 2004 Banff lectures, contains a lovely compendium of combinatorial results on successors of singular cardinals, from various authors. The following are the key words on the theorems which are proved, in most cases completely, in an informal style. GCH implies diamond. Club guessing holds in ZFC. Square implies the existence of Souslin tree. Weak square is equivalent to the existence of a special tree. There is a stationary set with square on it. Properties of Shelah's \(I[\lambda]\). Internally approaching chains. Scales, good points, essential upper bounds. Incompactness of systems of countable sets with transversals. Good scales. Various forms of the square principle are negated by PFA, MM. Shelah's trichotomy theorem, very good scales if SCH fails.
Mathematics Subject Classification ID: 03E05
Mathematics Subject Classification ID: 03E04
Mathematics Subject Classification ID: 03E35
Mathematics Subject Classification ID: 03-02
Mathematics Subject Classification ID: 03E50
zbMATH DE Number: 5011391
zbMATH Keywords: stationary sets
zbMATH Keywords: set-theoretical trees
zbMATH Keywords: square principle
zbMATH Keywords: pcf theory
zbMATH Keywords: proper forcing axiom
zbMATH Keywords: successors of singular cardinals
reviewed by: Péter Komjáth
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1305/ndjfl/1125409326
OpenAlex ID: W2065054336
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DOI: 10.1305/NDJFL/1125409326
description: scientific article; zbMATH DE number 5011391