Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-08T19:27:14.775Z Has data issue: false hasContentIssue false
  • English
  • Français

DENSE IDEALS AND CARDINAL ARITHMETIC

Published online by Cambridge University Press:  14 September 2016

MONROE ESKEW
MONROE ESKEW*
Affiliation:
DEPARTMENT OF MATHEMATICS AND APPLIED MATHEMATICS VIRGINIA COMMONWEALTH UNIVERSITY RICHMOND, VA, USAE-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

From large cardinals we show the consistency of normal, fine, κ-complete λ-dense ideals on ${{\cal P}_\kappa }\left( \lambda \right)$ for successor κ. We explore the interplay between dense ideals, cardinal arithmetic, and squares, answering some open questions of Foreman.

Keywords

Primary 03E35Secondary 03E55generic large cardinalsdense idealscardinal arithmetic
Type
Articles
Information
The Journal of Symbolic Logic , Volume 81 , Issue 3 , September 2016 , pp. 789 - 813
Copyright
Copyright © The Association for Symbolic Logic 2016 

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

Burke, D. and Matsubara, Y., Ideals and combinatorial properties, this Journal, vol. 62 (1997), no. 1, pp. 117–122.Google Scholar
Burke, D. and Matsubara, Y., The extent of strength in the club filters . Israel Journal of Mathematics, vol. 114 (1999), pp. 253–263.Google Scholar
Chang, C. C. and Keisler, H. J., Model Theory, third ed., Studies in Logic and the Foundations of Mathematics, vol. 73, North-Holland, Amsterdam, New York, Oxford, 1990.Google Scholar
Cummings, J., Foreman, M., and Magidor, M., Squares, scales and stationary reflection . Journal of Mathematical Logic, vol. 1 (2001), no. 1, pp. 35–98.Google Scholar
Donder, H-D., Regularity of ultrafilters and the core model . Israel Journal of Mathematics, vol. 63 (1988), no. 3, pp. 289–322.Google Scholar
Erdős, P. and Tarski, A., On families of mutually exclusive sets . Annals of Mathematics, vol. 44 (1943), no. 2, pp. 315–329.Google Scholar
Foreman, M., Ideals and generic elementary embeddings , Handbook of Set Theory, vol. 2 (Foreman, M. and Kanamori, A., editors), Springer, Dordrecht, 2010, pp. 885–1147.Google Scholar
Foreman, M., Calculating quotient algebras of generic embeddings . Israel Journal of Mathematics, vol. 193 (2013), no. 1, pp. 309–341.Google Scholar
Huberich, M., A note on Boolean algebras with few partitions modulo some filter . Mathematical Logic Quarterly, vol. 42 (1996), no. 2, pp. 172–174.CrossRefGoogle Scholar
Jech, T., Set Theory, third millennium ed., Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003.Google Scholar
Kanamori, A., The Higher Infinite, Springer-Verlag, Berlin, 2003.Google Scholar
Keisler, H. J., On cardinalities of ultraproducts . Bulletin of the American Mathematical Society, vol. 70 (1964), pp. 644–647.CrossRefGoogle Scholar
Kunen, K., Set Theory, Studies in Logic, vol. 34, College Publications, London, 2011.Google Scholar
Kunen, K. and Prikry, K., On descendingly incomplete ultrafilters, this Journal, vol. 36 (1971), pp. 650–652.Google Scholar
Shelah, S., Proper Forcing, Lecture Notes in Mathematics, vol. 940, Springer-Verlag, Berlin, New York, 1982.CrossRefGoogle Scholar
Solovay, R., Strongly compact cardinals and the GCH, Proceedings of the Tarski Symposium ( Proceedings of Symposia in Pure Mathematics ), American Mathematical Society, 1971, pp. 365–372.Google Scholar