Notre Dame Journal of Formal Logic

Decomposable Ultrafilters and Possible Cofinalities

Paolo Lipparini

Abstract

We use Shelah's theory of possible cofinalities in order to solve some problems about ultrafilters. Theorem: Suppose that λ is a singular cardinal, λ ' < λ , and the ultrafilter D is κ -decomposable for all regular cardinals κ with λ ' < κ < λ . Then D is either λ -decomposable or λ + -decomposable. Corollary: If λ is a singular cardinal, then an ultrafilter is ( λ , λ )-regular if and only if it is either cf λ -decomposable or λ + -decomposable. We also give applications to topological spaces and to abstract logics.

Article information

Source
Notre Dame J. Formal Logic, Volume 49, Number 3 (2008), 307-312.

Dates
First available in Project Euclid: 15 July 2008

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1216152553

Digital Object Identifier
doi:10.1215/00294527-2008-014

Mathematical Reviews number (MathSciNet)
MR2428557

Zentralblatt MATH identifier
1152.03041

Subjects
Primary: 03C20: Ultraproducts and related constructions 03E04: Ordered sets and their cofinalities; pcf theory
Secondary: 03C95: Abstract model theory 54D20: Noncompact covering properties (paracompact, Lindelöf, etc.)

Keywords
$λ-decomposable, (μ,λ)-regular (ultra)-filter cofinality of a partial order (productive) [μ,λ]-compactness

Citation

Lipparini, Paolo. Decomposable Ultrafilters and Possible Cofinalities. Notre Dame J. Formal Logic 49 (2008), no. 3, 307--312. doi:10.1215/00294527-2008-014. https://projecteuclid.org/euclid.ndjfl/1216152553


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