Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-27T12:04:32.106Z Has data issue: false hasContentIssue false
  • English
  • Français

Killing The GCH Everywhere with a Single Real

Published online by Cambridge University Press:  12 March 2014

Sy-David Friedman  and
Mohammad Golshani
Sy-David Friedman
Affiliation:
Kurtgódel Research center, University of Vienna, Vienna, Austria, E-mail: [email protected]
Mohammad Golshani
Affiliation:
Department of Mathematics, Shahid Bahonar University of Kerman, Kerman, Iran School of Mathematics, Institute of Research in Fundamental Sciences (IPM), P.O.BOX:19395-5746, Tehran, Iran, E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Shelah-Woodin [10] investigate the possibility of violating instances of GCH through the addition of a single real. In particular they show that it is possible to obtain a failure of CH by adding a single real to a model of GCH, preserving cofinalities. In this article we strengthen their result by showing that it is possible to violate GCH at all infinite cardinals by adding a single real to a model of GCH. Our assumption is the existence of an H(κ+3)-strong cardinal; by work of Gitik and Mitchell [6] it is known that more than an H(κ++)-strong cardinal is required.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 78 , Issue 3 , September 2013 , pp. 803 - 823
Copyright
Copyright © Association for Symbolic Logic 2013

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] Eslami, E. and Golshani, M., Shelah's strong covering property and CH in V [r], Mathematical Logic Quarterly, vol. 58 (2012), pp. 153–158.Google Scholar
[2] Friedman, S., Fine structure and class forcing, de Gruyter Series in Logic and its Applications, vol. 3, de Gruyter, 2000.CrossRefGoogle Scholar
[3] Friedman, S., Genericity and large cardinals, Journal of Mathematical Logic, vol. 5 (2005), no. 2, pp. 149–166.Google Scholar
[4] Fuchs, G., A characterization of generalized Prikry sequences, Archive for Mathematical Logic, vol. 44 (2005), pp. 935–971.Google Scholar
[5] Gitik, M., Prikry type forcings, Handbook of set theory (Foreman, M. and Kanamori, A., editors), vol. 2, Springer, 2010, pp. 1351–1447.CrossRefGoogle Scholar
[6] Gitik, M. and Mitchell, W., Indiscernible sequences for extenders and the singular cardinal hypothesis, Annals of Pure and Applied Logic, vol. 82 (1996), no. 3, pp. 273–316.CrossRefGoogle Scholar
[7] Gitik, M. and Shelah, S., On certain indestructibility of strong cardinals and a question of Hajnal, Archive for Mathematical Logic, vol. 28 (1989), pp. 35–42.Google Scholar
[8] Magidor, M., How large is the first strongly compact cardinal, Annals of Mathematical Logic, vol. 10 (1976), pp. 33–57.Google Scholar
[9] Merimovich, C., A power function with a fixedfinite gap everywhere, this Journal, vol. 72 (2007), no. 2, pp. 361–417.Google Scholar
[10] Shelah, S. and Woodin, H., Forcing the failure of CH by adding a real, this Journal, vol. 49 (1984), no. 4, pp. 1185–1189.Google Scholar