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ON CONFIGURATIONS CONCERNING CARDINAL CHARACTERISTICS AT REGULAR CARDINALS

Part of: Set theory

Published online by Cambridge University Press:  10 July 2020

OMER BEN-NERIA
Affiliation:
EINSTEIN INSTITUTE OF MATHEMATICS THE HEBREW UNIVERSITY OF JERUSALEMJERUSALEM91904, ISRAELE-mail: omer.bn@mail.huji.ac.ilE-mail: shimon.garty@mail.huji.ac.il
SHIMON GARTI
Affiliation:
EINSTEIN INSTITUTE OF MATHEMATICS THE HEBREW UNIVERSITY OF JERUSALEMJERUSALEM91904, ISRAELE-mail: omer.bn@mail.huji.ac.ilE-mail: shimon.garty@mail.huji.ac.il

Abstract

We study the consistency and consistency strength of various configurations concerning the cardinal characteristics $\mathfrak {s}_\theta , \mathfrak {p}_\theta , \mathfrak {t}_\theta , \mathfrak {g}_\theta , \mathfrak {r}_\theta $ at uncountable regular cardinals $\theta $ . Motivated by a theorem of Raghavan–Shelah who proved that $\mathfrak {s}_\theta \leq \mathfrak {b}_\theta $ , we explore in the first part of the paper the consistency of inequalities comparing $\mathfrak {s}_\theta $ with $\mathfrak {p}_\theta $ and $\mathfrak {g}_\theta $ . In the second part of the paper we study variations of the extender-based Radin forcing to establish several consistency results concerning $\mathfrak {r}_\theta ,\mathfrak {s}_\theta $ from hyper-measurability assumptions, results which were previously known to be consistent only from supercompactness assumptions. In doing so, we answer questions from [1], [15] and [7], and improve the large cardinal strength assumptions for results from [10] and [3].

Type
Articles
Copyright
© The Association for Symbolic Logic 2020

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