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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 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
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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].

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

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References

REFERENCES

Ben-Neria, O. and Gitik, M., On the splitting number at regular cardinals , this Journal, vol. 80 (2015), no. 4, pp. 1348–1360.Google Scholar
Blass, A., Combinatorial cardinal characteristics of the continuum , Handbook of Set Theory (Foreman, M. and Kanamori, A., editors), vol. 1, Springer, Dordrecht, 2010, pp. 395–489.CrossRefGoogle Scholar
Brooke-Taylor, A. D., Fischer, V., Friedman, S. D., and Montoya, D. C., Cardinal characteristics at $\;\kappa\;$ in a small $\;u\left(\kappa \right)$ model . Annals of Pure and Applied Logic, vol. 168 (2017), no. 1, pp. 37–49.CrossRefGoogle Scholar
Cummings, J., and Shelah, S., Cardinal invariants above the continuum . Annals of Pure and Applied Logic, vol. 75 (1995), no. 3, pp. 251–268.CrossRefGoogle Scholar
Garti, S., Gitik, M., and Shelah, S., Cardinal characteristics at ${\aleph}_{\omega }$ , Acta Math. Hungarica , vol. 160 (2020), no. 2, pp. 320–336.CrossRefGoogle Scholar
Garti, S. and Shelah, S., Partition calculus and cardinal invariants . Journal of the Mathematical Society of Japan, vol. 66 (2014), no. 2, pp. 425–434.CrossRefGoogle Scholar
Garti, S. and Shelah, S., Open and solved problems concerning polarized partition relations . Fundamenta Mathematicae, vol. 234 (2016), no. 1, pp. 1–14.CrossRefGoogle Scholar
Gitik, M., The negation of the singular cardinal hypothesis from o(K)=K++ , this Journal, vol. 43 (1989), no. 3, pp. 209–234.Google Scholar
Gitik, M., Prikry-type forcings , Handbook of Set Theory, (Foreman, M. and Kanamori, A., editors), vol. 1, Springer, Dordrecht, 2010, pp. 1351–1447.CrossRefGoogle Scholar
Gitik, M. and Shelah, S., On densities of box products . Topology and its Applications, vol. 88 (1998), no. 3, pp. 219–237.CrossRefGoogle Scholar
Kamo, S., Splitting numbers on uncountable regular cardinals, unpublished.Google Scholar
Merimovich, C., Extender-based Magidor-Radin forcing . Israel Journal of Mathematics, vol. 182 (2011), pp. 439–480.CrossRefGoogle Scholar
Radin, L. B., Adding closed cofinal sequences to large cardinals . Annals of Mathematical Logic, vol. 22 (1982), no. 3, pp. 243–261.CrossRefGoogle Scholar
Raghavan, D. and Shelah, S., Two inequalities between cardinal invariants . Fundamenta Mathematicae, vol. 237 (2017), no. 2, pp. 187–200.CrossRefGoogle Scholar
Raghavan, D. and Shelah, S., A small ultrafilter number at smaller cardinals. Archive for Mathematical Logic , vol. 59 (2020), no. 3–4, pp. 325–334.CrossRefGoogle Scholar
Shelah, S. and Spasojević, Z., Cardinal invariants bk and tk . Publications de l’Institut Mathématique, vol. 72 (2002), no. 86, pp. 1–9.CrossRefGoogle Scholar
Suzuki, T., On splitting numbers . Urikaisekikenky usho Koky uroku (1993), no. 818, pp. 118–120, Mathematical logic and applications ’92 (Japanese) (Kyoto, 1992).Google Scholar