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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  and
SHIMON GARTI
OMER BEN-NERIA
Affiliation:
EINSTEIN INSTITUTE OF MATHEMATICS THE HEBREW UNIVERSITY OF JERUSALEM JERUSALEM 91904, ISRAEL E-mail: omer.bn@mail.huji.ac.il E-mail: shimon.garty@mail.huji.ac.il
SHIMON GARTI
Affiliation:
EINSTEIN INSTITUTE OF MATHEMATICS THE HEBREW UNIVERSITY OF JERUSALEM JERUSALEM 91904, ISRAEL E-mail: omer.bn@mail.huji.ac.il E-mail: shimon.garty@mail.huji.ac.il
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Abstract

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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].

Keywords

splitting numberreaping numberpseudointersectiongroupwise densitysupercompact cardinalsconsistency strengthextender-based Radin forcing

MSC classification

Primary: 03E17: Cardinal characteristics of the continuum
Secondary: 03E50: Continuum hypothesis and Martin's axiom 03E55: Large cardinals

Information

Type
Articles
Information
The Journal of Symbolic Logic , Volume 85 , Issue 2 , June 2020 , pp. 691 - 708
Copyright
© The Association for Symbolic Logic 2020

References

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