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APPLICATIONS OF PCF THEORY TO THE STUDY OF IDEALS ON

Part of: Set theory

Published online by Cambridge University Press:  11 January 2022

PIERRE MATET
PIERRE MATET*
Affiliation:
UNIVERSITÉ DE CAEN – CNRS LABORATOIRE DE MATHÉMATIQUES BP 5186, 14032 CAEN CEDEX, FRANCE E-mail: pierre.matet@unicaen.fr
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Abstract

Let $\kappa $ be a regular uncountable cardinal, and a cardinal greater than or equal to $\kappa $. Revisiting a celebrated result of Shelah, we show that if is close to $\kappa $ and (= the least size of a cofinal subset of ) is greater than , then can be represented (in the sense of pcf theory) as a pseudopower. This can be used to obtain optimal results concerning the splitting problem. For example we show that if and , then no $\kappa $-complete ideal on is weakly -saturated.

Keywords

pcf theorypseudopowerPκ(λ)

MSC classification

Primary: 03E04: Ordered sets and their cofinalities; pcf theory
Secondary: 03E35: Consistency and independence results

Information

Type
Article
Information
The Journal of Symbolic Logic , Volume 87 , Issue 3 , September 2022 , pp. 967 - 994
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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