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CARDINAL INVARIANTS RELATED TO DENSITY

Part of: Set theory

Published online by Cambridge University Press:  02 September 2025

DAVID VALDERRAMA*
Affiliation:
DEPARTAMENTO DE MATEMÁTICAS https://ror.org/02mhbdp94 UNIVERSIDAD DE LOS ANDES (BOGOTÁ) BOGOTÁ 111711 COLOMBIA

Abstract

We investigate some variants of the splitting, reaping, and independence numbers defined using asymptotic density. Specifically, we give a proof of Con($\mathfrak {i}<\mathfrak {s}_{1/2}$), Con($\mathfrak {r}_{1/2}<\mathfrak {b}$), and Con($\mathfrak {i}_*<2^{\aleph _0}$). This answers two questions raised in [5]. Besides, we prove the consistency of $\mathfrak {s}_{1/2}^{\infty } < $ non$(\mathcal {E})$ and cov$(\mathcal {E}) < \mathfrak {r}_{1/2}^{\infty }$, where $\mathcal {E}$ is the $\sigma $-ideal generated by closed sets of measure zero.

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Type
Article
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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References

Bartoszynski, T. and Judah, H., Set Theory: On the Structure of the Real Line , AK Peters, Wellesley, MA, 1995.Google Scholar
Blass, A., Combinatorial Cardinal Characteristics of the Continuum , Springer, Dordrecht, 2010, pp. 395489.Google Scholar
Brendle, J., Around splitting and reaping . Commentationes Mathematicae Universitatis Carolinae , vol. 39 (1998), no. 2, pp. 269279.Google Scholar
Brendle, J., Forcing and the structure of the real line: The Bogota lectures. 2009.Google Scholar
Brendle, J., Halbeisen, L. J., Klausner, L. D., Lischka, M., and Shelah, S., Halfway new cardinal characteristics . Annals of Pure and Applied Logic , vol. 174 (2023), no. 9, p. 103303.Google Scholar
Farkas, B., Klausner, L. D., and Lischka, M., More on halfway new cardinal characteristics . The Journal of Symbolic Logic , vol. 90 (2025), no. 3, pp. 13241339.Google Scholar
Kamburelis, A. and Węglorz, B., Splittings . Archive for Mathematical Logic , vol. 35 (1996), no. 4, pp. 263277.Google Scholar
Kunen, K., Set Theory: An Introduction to Independence Proofs , North-Holland/Elsevier, Amsterdam, 1980.Google Scholar