Hostname: page-component-7bb8b95d7b-dtkg6 Total loading time: 0 Render date: 2024-09-20T19:17:51.116Z Has data issue: false hasContentIssue false
  • English
  • Français

More on Cichoń's diagram and infinite games

Published online by Cambridge University Press:  12 March 2014

Masaru Kada
Masaru Kada*
Affiliation:
Department of Mathematics, Osaka Prefecture University, Sakai, Osaka 599-8531, Japan
*
Department of Computer Sciences, Kitami Institute of Technology, Kitami, Hokkaido 090-8507, Japan, E-mail:kada@math.cs.kitami-it.ac.jp
Get access Rights & Permissions [Opens in a new window]

Abstract

Some cardinal invariants from Cichoń's diagram can be characterized using the notion of cut-and-choose games on cardinals. In this paper we give another way to characterize those cardinals in terms of infinite games. We also show that some properties for forcing, such as the Sacks Property, the Laver Property and ωω-boundingness, are characterized by cut-and-choose games on complete Boolean algebras.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 65 , Issue 4 , December 2000 , pp. 1713 - 1724
Copyright
Copyright © Association for Symbolic Logic 2000

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.)

References

REFERENCES

[1]Bartoszyński, T. and Judah, H., Set theory: On the structure of the real line, A. K. Peters, Wellesley>, Massachusetts, 1995.CrossRefGoogle Scholar
[2]Blass, A., Cardinal characteristics and the product of countably many infinite cyclic groups, Journal of Algebra, vol. 169 (1994), pp. 512–540.CrossRefGoogle Scholar
[3]Foreman, M., Games played on Boolean algebras, this Journal, vol. 48 (1983), pp. 714–723.Google Scholar
[4]Ishiu, T., Games of transfinite length on Boolean algebra, preprint, 1997.Google Scholar
[5]Jech, T., More game theoretic properties of Boolean algebras, Annals of Pure and Applied Logic, vol. 26 (1984), pp. 11–29.CrossRefGoogle Scholar
[6]Jech, T. and Shelah, S., On countably closed complete Boolean algebras, this Journal, vol. 61 (1996), pp. 1380–1386.Google Scholar
[7]Kada, M., Consistency results concerning shrinkability for positive sets of reals, Ph.D. thesis, Osaka Prefecture University, 1997.Google Scholar
[8]Kada, M., The Baire category theorem and the evasion number, Proceedings of the American Mathematical Society, vol. 126 (1998), pp. 3381–3383.CrossRefGoogle Scholar
[9]Scheepers, M., Lebesgue measure zero subsets of the real line and an infinite game, this Journal, vol.61 (1996), pp. 246–250.Google Scholar
[10]Scheepers, M., Meager sets and infinite games, Contemporary Mathematics, vol. 192 (1996), pp. 77–89.CrossRefGoogle Scholar
[11]Veličković, B., Playful Boolean algebras, Transactions of the American Mathematical Society, vol. 296 (1986), pp. 727–740.CrossRefGoogle Scholar
[12]Zapletal, J., More on the cut and choose game, Annals of Pure and Applied Logic, vol. 76 (1995), pp. 291–301.CrossRefGoogle Scholar