Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-25T07:20:44.406Z Has data issue: false hasContentIssue false
  • English
  • Français

Approachability and Games on Posets

Published online by Cambridge University Press:  12 March 2014

Yasuo Yoshinobu
Yasuo Yoshinobu*
Affiliation:
Graduate School of Human Informatics, Nagoya University, Nagoya 464-8601, Japan, E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that for any infinite cardinal κ, every strongly (κ + 1 )-strategically closed poset is strongly κ+-strategically closed if and only if APκ (the approachability property) holds, answering the question asked in [5]. We also give a complete classification of strengths of strategic closure properties and that of strong strategic closure properties respectively.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 68 , Issue 2 , June 2003 , pp. 589 - 606
Copyright
Copyright © The International Society for Twin Studies 1963

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]Cummings, J., Foreman, M., and Magidor, M., Squares, scales and stationary reflection, Journal of Mathematical Logic, vol. 1 (2001), no. 1, pp. 35–98.CrossRefGoogle Scholar
[2]Foreman, M., Games played on boolean algebras, this Journal, vol. 48 (1983), pp. 714–723.Google Scholar
[3]Foreman, M. and Magidor, M., A very weak square principle, this Journal, vol. 62 (1997), no. 1, pp. 175–196.Google Scholar
[4]Ishiu, T., On Axiom A and games on boolean algebras, Master's thesis, Waseda University, 1997, in Japanese.Google Scholar
[5]Ishiu, T. and Yoshinobu, Y., Directive trees and games on posets, Proceedigs of the American Mathematical Society, to appear.Google Scholar
[6]Jech, T., A game theoretic property of Boolean algebras, Logic Colloquium '77 (Macintyre, A., Pacholski, L., and Paris, J., editors), North-Holland, Amsterdam, 1978, pp. 135–144.CrossRefGoogle Scholar
[7]Jensen, R., The fine structure of the constructible hierarchy, Annals of Mathematical Logic, vol. 4 (1972), pp. 229–308.CrossRefGoogle Scholar
[8]Mitchell, W., Aronszajn trees and independence of transfer property, Annals of Mathematical Logic, vol. 5 (1972), pp. 21–46.CrossRefGoogle Scholar
[9]Shelah, S., On successors of singular cardinals, Logic Colloquium '78 (Boffa, M., van Dalen, D., and McAloon, K., editors), North-Holland, Amsterdam, 1979, pp. 357–380.Google Scholar
[10]Veličkovič, B., Jensens □ principles and the Novák number of partially ordered sets, this Journal, vol. 51 (1986), pp. 47–58.Google Scholar
[11]Velleman, D., On a generalization of Jensen's □κ, and strategic closure of partial orders, this Journal, vol. 48 (1983), pp. 1046–1052.Google Scholar