Abstract
We prove several theorems about the cardinal\(\mathfrak{g}\) associated with groupwise density. With respect to a natural ordering of families of nond-ecreasing maps fromω toω, all families of size\(< \mathfrak{g}\) are below all unbounded families. With respect to a natural ordering of filters onω, all filters generated by\(< \mathfrak{g}\) sets are below all non-feeble filters. If\(\mathfrak{u}< \mathfrak{g}\) then\(\mathfrak{b}< \mathfrak{u}\) and\(\mathfrak{g} = \mathfrak{d} = \mathfrak{c}\). (The definitions of these cardinals are recalled in the introduction.) Finally, some consequences deduced from\(\mathfrak{u}< \mathfrak{g}\) by Laflamme are shown to be equivalent to\(\mathfrak{u}< \mathfrak{g}\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Blass, A.: Near coherence of filters. I. Cofinal equivalence of models of arithmetic. Notre Dame J. Formal Logic27, 579–591 (1986)
Blass, A.: Near coherence of filters. II. Applications to operator ideals, the Stone-Čech remainder of a half-line, order ideals of sequences, and slenderness of groups. Trans. Am. Math. Soc.300, 557–581 (1987)
Blass, A., Shelah, S.: There may be simple Pκ and\(P_{\kappa _2 } \) points and the Rudin-Keisler order may be downward directed. Ann. Pure Appl. Logic83, 213–243 (1987)
Blass, A., Shelah, S.: Near coherence of filters. III. A simplified consistency proof. Notre Dame J. Formal Logic30, 530–538 (1989)
Blass, A., Laflamme, C.: Consistency results about filters and the number of inequivalent growth types. J. Symb. Logic54, 50–56 (1989)
Blass, A.: Applications of superperfect forcing and its relatives. In: Steprano, J., Watson, S. (eds.) Set theory and its applications. (Lect. Notes Math. vol. 1401, pp. 18–40) Berlin Heidelberg New York: Springer 1989
van Douwen, E.: The integers and topology. In: Kunen, K., Vaughan, J. (eds.) Handbook of set-theoretic topology. Amsterdam: North-Holland 1984, pp. 111–167
Göbel, R., Wald, B.: Wachstumstypen und schlanke Gruppen. Symp. Math.23, 201–239 (1979)
Göbel, R., Wald, B.: Martin's axiom implies the existence of certain slender groups. Math. Z.172, 107–121 (1980)
Jech, T.: Set theory. New York: Academic Press 1978
Just, W., Laflamme, C.: Classification of measure zero sets with respect to their open covers. (To appear in Trans. Am. Math. Soc.)
Laflamme, C.: Equivalence of families of functions on the natural numbers. (To appear)
Plewik, S.: Intersections and unions of ultrafilters without the Baire property. Bull. Pol. Acad. Sci. Math.35, 805–808 (1987)
Puritz, C.: Ultrafilters and standard functions in non-standard arithmetic. Proc. Lond. Math. Soc. III. Ser.22, 705–733 (1971)
Simon, P.: Private communication. August 1987
Solomon, R.: Families of sets and functions. Czech. Math. J.27, 556–559 (1977)
Talagrand, M.: Filtres non mesurables et compacts de fonctions mesurables. Stud. Math.67, 13–43 (1980)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Blass, A. Groupwise density and related cardinals. Arch Math Logic 30, 1–11 (1990). https://doi.org/10.1007/BF01793782
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01793782