Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-10T12:25:55.463Z Has data issue: false hasContentIssue false
  • English
  • Français

The Magidor function and diamond

Published online by Cambridge University Press:  12 March 2014

Pierre Matet
Pierre Matet*
Affiliation:
Université de Caen- CNRS, Laboratoire de Mathematiques, BP 5186, 14032 Caen Cedex, France, E-mail: matet@math.unicaen.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

Let κ be a regular uncountable cardinal and λ be a cardinal greater than κ. We show that if 2<κ ≤ M(κ, λ), then ◇κ,λ holds, where M(κ, λ) equals λℵ0 if cf(λ) ≥ κ, and (λ+)ℵ0 otherwise.

Keywords

Pκ(λ)diamond principle
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 76 , Issue 2 , June 2011 , pp. 405 - 417
Copyright
Copyright © Association for Symbolic Logic 2011

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]Abraham, U. and Magidor, M., Cardinal arithmetic, Handbook of set theory (Foreman, M. and Kanamori, A., editors), Springer, 2010, pp. 1149–1227.CrossRefGoogle Scholar
[2]Cummings, J., A model in which GCH holds at successors but fails at limits. Transactions of the American Mathematical Society, vol. 329 (1992), pp. 1–39.CrossRefGoogle Scholar
[3]Donder, H. D. and Matet, P., Two cardinal versions of diamond, Israel Journal of Mathematics, vol. 83 (1993), pp. 1–43.CrossRefGoogle Scholar
[4]Foreman, M. and Magidor, M., Mutually stationary sequences of sets and the non-saturation of the non-stationary ideal on , Acta Mathematica, vol. 186 (2001), pp. 271–300.CrossRefGoogle Scholar
[5]Foreman, M. and Todorcevic, S., A new Lőwenheim–Skolem Theorem, Transactions of the American Mathematical Society, vol. 357 (2005), pp. 1693–1715.CrossRefGoogle Scholar
[6]Jech, T. J., Some combinatorial problems concerning uncountable cardinals, Annals of Mathematical Logic, vol. 5 (1973), pp. 165–198.CrossRefGoogle Scholar
[7]Jech, T. J., Set theory, the third millenium ed., Springer Monographs in Mathematics, Springer, Berlin, 2002.Google Scholar
[8]Kojman, M., The A, B, Cofpcf. a companion to pcf theory, part I, unpublished, 1995.Google Scholar
[9]König, B., Winning strategies and tactics in club games, preprint.Google Scholar
[10]Kunen, K., Set theory, North-Holland, Amsterdam, 1980.Google Scholar
[11]Liu, A., Bounds for covering numbers, this Journal, vol. 71 (2006), pp. 1303–1310.Google Scholar
[12]Magidor, M., Representing sets of ordinals as countable unions of sets in the core model, Transactions of the American Mathematical Society, vol. 317 (1990), pp. 91–126.CrossRefGoogle Scholar
[13]Matet, P., Concerning stationary subsets of [λ]<κ, Set theory and its applications (Steprāns, J. and Watson, S., editors), Lecture Notes in Mathematics 1401, Springer, Berlin, 1989, pp. 119–127.CrossRefGoogle Scholar
[14]Matet, P., Club-guessing, goodpoints and diamond, Commentationes Universitatis Carolinae, vol. 48 (2007), pp. 211–216.Google Scholar
[15]Matet, P., Guessing with mutually stationary sets, Canadian Mathematical Bulletin, vol. 51 (2008), pp. 579–583.CrossRefGoogle Scholar
[16]Matet, P., Game ideals, Annals of Pure and Applied Logic, vol. 158 (2009), pp. 23–39.CrossRefGoogle Scholar
[17]Shelah, S., More on stationary coding, Around classification theory of models, Lecture Notes in Mathematics 1182, Springer, Berlin, 1986, pp. 224–246.CrossRefGoogle Scholar
[18]Shelah, S., Non-structure theory, to appear.Google Scholar
[19]Shioya, M., Splitting into maximally many stationary sets, Israel Journal of Mathematics, vol. 114 (1999), pp. 347–357.CrossRefGoogle Scholar
[20]Shioya, M., Diamonds on , Computational prospects of infinity, part II (Chong, C., Feng, Q., Slaman, T., Woodin, W. H., and Yang, Y., editors). Lecture Notes Series, Institute of Mathematical Sciences, National University of Singapore, vol. 15, World Scientific, Singapore, 2008, pp. 271–291.CrossRefGoogle Scholar
[21]Solovay, R. M., Real-valued measurable cardinals, Axiomatic set theory (Scott, D. S., editor), Proceedings of Symposia in Pure Mathematics, vol. 13, American Mathematical Society, Providence, 1971, part 1, pp. 397–428.CrossRefGoogle Scholar
[22]Todorcevic, S., personal communication.Google Scholar