Hostname: page-component-7dd5485656-gs9qr Total loading time: 0 Render date: 2025-10-26T12:27:35.990Z Has data issue: false hasContentIssue false
  • English
  • Français

UNIFORMITY NUMBERS OF THE NULL-ADDITIVE AND MEAGER-ADDITIVE IDEALS

Part of: Set theory

Published online by Cambridge University Press:  07 August 2025

MIGUEL ANTONIA CARDONA Open the ORCID record for MIGUEL ANTONIA CARDONA [Opens in a new window] ,
DIEGO ALEJANDRO MEJÍA Open the ORCID record for DIEGO ALEJANDRO MEJÍA [Opens in a new window]  and
ISMAEL RIVERA-MADRID Open the ORCID record for ISMAEL RIVERA-MADRID [Opens in a new window]
MIGUEL ANTONIA CARDONA
Affiliation:
EINSTEIN INSTITUTE OF MATHEMATICS EDMOND J. SAFRA CAMPUS, GIVAT RAM THE HEBREW UNIVERSITY OF JERUSALEM JERUSALEM, 91904, ISRAEL E-mail: miguel.cardona@mail.huji.ac.il URL: https://sites.google.com/view/miacardonamo
DIEGO ALEJANDRO MEJÍA*
Affiliation:
GRADUATE SCHOOL OF SYSTEM INFORMATICS, KOBE UNIVERSITY 1-1 ROKKODAI-CHO, NADA-KU, KOBE, HYOGO 657-8501 JAPAN URL: https://researchmap.jp/mejia?lang=en
ISMAEL RIVERA-MADRID
Affiliation:
FACULTY OF ENGINEERING INSTITUCIÓN UNIVERSITARIA PASCUAL BRAVO CALLE 73 NO. 73A-226, MEDELLÍN, COLOMBIA E-mail: ismael.rivera@pascualbravo.edu.co
*
E-mail: damejiag@people.kobe-u.ac.jp
Get access Rights & Permissions [Opens in a new window]

Abstract

Denote by $\mathcal {NA}$ and $\mathcal {MA}$ the ideals of null-additive and meager-additive subsets of $2^{\omega }$, respectively. We prove in ZFC that $\mathrm {add}(\mathcal {NA})=\mathrm {non}(\mathcal {NA})$ and introduce a new (Polish) relational system to reformulate Bartoszyński’s and Judah’s characterization of the uniformity of $\mathcal {MA}$, which is helpful to understand the combinatorics of $\mathcal {MA}$ and to prove consistency results. As for the latter, we prove that $\mathrm {cov}(\mathcal {MA})<\mathfrak {c}$ (even $\mathrm {cov}(\mathcal {MA})<\mathrm {non}(\mathcal {N})$) is consistent with ZFC, as well as several constellations of Cichoń’s diagram with $\mathrm {non}(\mathcal {NA})$, $\mathrm {non}(\mathcal {MA}),$ and $\mathrm {add}(\mathcal {SN})$, which include $\mathrm {non}(\mathcal {NA})<\mathfrak {b}< \mathrm {non}(\mathcal {MA})$ and $\mathfrak {b}< \mathrm {add}(\mathcal {SN})<\mathrm {cov}(\mathcal {M})<\mathfrak {d}=\mathfrak {c}$.

Keywords

null-additive setsmeager-additive setsstrong measure zerocardinal characteristics of the continuumuf-extendable matrix iterations

MSC classification

Primary: 03E17: Cardinal characteristics of the continuum
Secondary: 03E05: Other combinatorial set theory 03E35: Consistency and independence results 03E40: Other aspects of forcing and Boolean-valued models

Information

Type
Article
Information
The Journal of Symbolic Logic , First View , pp. 1 - 37
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

REFERENCES

Bartoszyński, T. and Judah, H., Borel images of sets of reals . Real Analysis Exchange, vol. 20 (1994/95), no. 2, pp. 536–558.10.2307/44152538CrossRefGoogle Scholar
Bartoszyński, T. and Judah, H., Set Theory: On the Structure of the Real Line. A K Peters, Wellesley, 1995.10.1201/9781439863466CrossRefGoogle Scholar
Bartoszyński, T., Just, W., and Scheepers, M., Covering games and the Banach–Mazur game: $K$ -tactics . Canadian Journal of Mathematics, vol. 45 (1993), no. 5, pp. 897–929.10.4153/CJM-1993-051-4CrossRefGoogle Scholar
Bartoszyński, T. and Shelah, S., Closed measure zero sets . Annals of Pure and Applied Logic, vol. 58 (1992), no. 2, pp. 93–110.10.1016/0168-0072(92)90001-GCrossRefGoogle Scholar
Blass, A., Combinatorial cardinal characteristics of the continuum , Handbook of Set Theory, (M. Foreman and A. Kanamori, editors), vols. 1, 2, 3. Springer, Dordrecht, 2010, pp. 395–489.10.1007/978-1-4020-5764-9_7CrossRefGoogle Scholar
Blass, A. and Shelah, S., Ultrafilters with small generating sets . Israel Journal of Mathematics, vol. 65 (1989), no. 3, pp. 259–271.10.1007/BF02764864CrossRefGoogle Scholar
Brendle, J., Larger cardinals in Cichoń’s diagram . The Journal of Symbolic Logic, vol. 56 (1991), no. 3, pp. 795–810.Google Scholar
Brendle, J., Shattered iterations, preprint, 2023, arXiv:2302.05069.Google Scholar
Brendle, J., Cardona, M. A., and Mejía, D. A., Filter-linkedness and its effect on preservation of cardinal characteristics . Annals of Pure and Applied Logic, vol. 172 (2021), no. 1, Article No. 102856, 30 pp.10.1016/j.apal.2020.102856CrossRefGoogle Scholar
Brendle, J., Cardona, M. A., and Mejía, D. A., Separating cardinal characteristics of the strong measure zero ideal . Journal of Mathematical Logic, 2025, pp. 1–52.Google Scholar
Brendle, J. and Fischer, V., Mad families, splitting families and large continuum . The Journal of Symbolic Logic, vol. 76 (2011), no. 1, pp. 198–208.10.2178/jsl/1294170995CrossRefGoogle Scholar
Cardona, M. A., On cardinal characteristics associated with the strong measure zero ideal . Fundamenta Mathematicae, vol. 257 (2022), no. 3, pp. 289–304.10.4064/fm83-11-2021CrossRefGoogle Scholar
Cardona, M. A., A friendly iteration forcing that the four cardinal characteristics of can be pairwise different . Colloquium Mathematicum, vol. 173 (2023), no. 1, pp. 123–157.10.4064/cm8917-2-2023CrossRefGoogle Scholar
Cardona, M. A., Cardinal invariants associated with the combinatorics of the uniformity number of the ideal of meager-additive sets . Kyōto Daigaku Sūrikaiseki Kenkyūsho Kōkyūroku, vol. 2315 (2025), pp. 1–24.Google Scholar
Cardona, M. A. and Mejía, D. A., On cardinal characteristics of Yorioka ideals . MLQ, vol. 65 (2019), no. 2, pp. 170–199.10.1002/malq.201800034CrossRefGoogle Scholar
Cardona, M. A. and Mejía, D. A., Forcing constellations of Cichoń’s diagram by using the Tukey order . Kyōto Daigaku Sūrikaiseki Kenkyūsho Kōkyūroku, vol. 2213 (2022), pp. 14–47.Google Scholar
Cardona, M. A. and Mejía, D. A., Localization and anti-localization cardinals . Kyōto Daigaku Sūrikaiseki Kenkyūsho Kōkyūroku, vol. 2261 (2023), pp. 47–77.Google Scholar
Cardona, M. A. and Mejía, D. A., More about the cofinality and the covering of the ideal of strong measure zero sets . Annals of Pure and Applied Logic, vol. 176 (2025), no. 4, p. 103537.10.1016/j.apal.2024.103537CrossRefGoogle Scholar
Cardona, M. A., Mejía, D. A., and Rivera-Madrid, I. E., The covering number of the strong measure zero ideal can be above almost everything else . Archive for Mathematical Logic, vol. 61 (2022), nos. 5–6, pp. 599–610.10.1007/s00153-021-00808-0CrossRefGoogle Scholar
Cichoń, J. and Kraszewski, J., On some new ideals on the cantor and Baire spaces . Proceedings of the American Mathematical Society, vol. 126 (1998), no. 5, pp. 1549–1555.CrossRefGoogle Scholar
Galvin, F., Mycielski, J., and Solovay, R., Strong measure zero sets . Notices of the American Mathematical Society, vol. 26 (1979), no. 3, pp. A–280. Abstract 79T-E25.Google Scholar
Gavalová, V. and Mejía, D. A., Lebesgue measure zero modulo ideals on the natural numbers . The Journal of Symbolic Logic, (2023), pp. 1–30. Published online, https://doi.org/10.1017/jsl.2023.97.Google Scholar
Goldstern, M., Mejía, D. A., and Shelah, Saharon, The left side of Cichoń’s diagram . Proceedings of the American Mathematical Society, vol. 144 (2016), no. 9, pp. 4025–4042.10.1090/proc/13161CrossRefGoogle Scholar
Judah, H. and Shelah, S., The Kunen–Miller chart (Lebesgue measure, the Baire property, laver reals and preservation theorems for forcing) . The Journal of Symbolic Logic, vol. 55 (1990), no. 3, pp. 909–927.10.2307/2274464CrossRefGoogle Scholar
Kamburelis, A., Iterations of Boolean algebras with measure . Archive for Mathematical Logic, vol. 29 (1989), no. 1, pp. 21–28.10.1007/BF01630808CrossRefGoogle Scholar
Kraszewski, J., Transitive properties of ideal. Available at http://www.math.uni.wroc.pl/~kraszew/sources/papers/trans7.pdf, 2002.Google Scholar
Kraszewski, J., Transitive properties of the ideal ${S}_2$ . Real Analysis Exchange, vol. 29 (2003/04), no. 2, pp. 629–638.10.14321/realanalexch.29.2.0629CrossRefGoogle Scholar
Mejía, D. A., Matrix iterations and Cichon’s diagram . Archive for Mathematical Logic, vol. 52 (2013), nos. 3–4, pp. 261–278.10.1007/s00153-012-0315-6CrossRefGoogle Scholar
Mejía, D. A., Matrix iterations with vertical support restrictions , Proceedings of the 14th and 15th Asian Logic Conferences. (B. Kim, J. Brendle, G. Lee, F. Liu, R. Ramanujam, S. M. Srivastava, A. Tsuboi and L. Yu, editors), World Science Publishing, Hackensack, 2019, pp. 213–248.10.1142/9789813237551_0012CrossRefGoogle Scholar
Miller, A. W., Some properties of measure and category . Transactions of the American Mathematical Society, vol. 266 (1981), no. 1, pp. 93–114.10.1090/S0002-9947-1981-0613787-2CrossRefGoogle Scholar
Pawlikowski, J., Powers of transitive bases of measure and category . Proceedings of the American Mathematical Society, vol. 93 (1985), no. 4, pp. 719–729.10.1090/S0002-9939-1985-0776210-2CrossRefGoogle Scholar
Pawlikowski, J., Finite support iteration and strong measure zero sets . The Journal of Symbolic Logic, vol. 55 (1990), no. 2, pp. 674–677.10.2307/2274657CrossRefGoogle Scholar
Rothberger, F., Sur les familles indénombrables de suites de nombres naturels et les problèmes concernant la propriété $C$ . Proceedings of the Cambridge Philosophical Society, vol. 37 (1941), pp. 109–126.10.1017/S0305004100021617CrossRefGoogle Scholar
Shelah, S., Every null-additive set is meager-additive . Israel Journal of Mathematics, vol. 89 (1995), nos. 1–3, pp. 357–376.10.1007/BF02808209CrossRefGoogle Scholar
Talagrand, M., Compacts de fonctions mesurables et filtres non mesurables . Studia Mathematica, vol. 67 (1980), no. 1, pp. 13–43.10.4064/sm-67-1-13-43CrossRefGoogle Scholar
Yorioka, T., The cofinality of the strong measure zero ideal . The Journal of Symbolic Logic, vol. 67 (2002), no. 4, pp. 1373–1384.10.2178/jsl/1190150290CrossRefGoogle Scholar
Zindulka, O., Meager-additive sets in topological groups . The Journal of Symbolic Logic, vol. 87 (2022), no. 3, pp. 1046–1064.10.1017/jsl.2021.79CrossRefGoogle Scholar