Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-25T18:59:30.908Z Has data issue: false hasContentIssue false
  • English
  • Français

HOD, V AND THE GCH

Published online by Cambridge University Press:  21 March 2017

MOHAMMAD GOLSHANI
MOHAMMAD GOLSHANI*
Affiliation:
SCHOOL OF MATHEMATICS INSTITUTE FOR RESEARCH IN FUNDAMENTAL SCIENCES (IPM) P.O. BOX: 19395-5746 TEHRAN, IRANE-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Starting from large cardinals we construct a model of ZFC in which the GCH fails everywhere, but such that GCH holds in its HOD. The result answers a question of Sy Friedman. Also, relative to the existence of large cardinals, we produce a model of ZFC + GCH such that GCH fails everywhere in its HOD.

Keywords

HODGCHstrong cardinalextender based Radin forcing
Type
Articles
Information
The Journal of Symbolic Logic , Volume 82 , Issue 1 , March 2017 , pp. 224 - 246
Copyright
Copyright © The Association for Symbolic Logic 2017 

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

Brooke-Taylor, A. D., Large Cardinals and definable well-orders on the universe, this Journal, vol. 74 (2009), no. 2, pp. 641–654.Google Scholar
Cummings, J., David Friedman, S., and Golshani, M., Collapsing the cardinals of HOD . Journal of Mathematical Logic. vol. 15 (2015), no. 2, 32 pages.Google Scholar
Friedman, S.-D., Personal communication.Google Scholar
Friedman, S.-D. and Golshani, Mohammad, Killing GCH everywhere by a cofinality-preserving forcing notion over a model of GCH . Fundamenta Mathematicae, vol. 223 (2013), no. 2, pp. 171–193.Google Scholar
Fuchs, G., David Hamkins, J., and Reitz, J., Set-theoretic geology . Annals of Pure and Applied Logic, vol. 166 (2015), no. 4, pp. 464–501.Google Scholar
Gitik, M., Prikry-type forcings , Handbook of Set Theory, Springer, Dordrecht, 2010, pp. 1351–1447.Google Scholar
Merimovich, C., A power function with a fixed finite gap everywhere, this Journal, vol. 72 (2007), no. 2, pp. 361–417.Google Scholar
Roguski, S., The theory of the class HOD , Set Theory and Hierarchy Theory, V (Proc. Third Conf., Bierutowice, 1976), Lecture Notes in Mathematics, vol. 619, Springer, Berlin, 1977, pp. 251–255.CrossRefGoogle Scholar