## Journal of Symbolic Logic

### Killing the $GCH$ everywhere with a single real

#### 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 $H(\kappa^{+3})$-strong cardinal; by work of Gitik and Mitchell [6] it is known that more than an $H(\kappa^{++})$-strong cardinal is required.

#### Article information

Source
J. Symbolic Logic, Volume 78, Issue 3 (2013), 803-823.

Dates
First available in Project Euclid: 6 January 2014

https://projecteuclid.org/euclid.jsl/1389032276

Digital Object Identifier
doi:10.2178/jsl.7803060

Mathematical Reviews number (MathSciNet)
MR3135499

Zentralblatt MATH identifier
1328.03049

#### Citation

Friedman, Sy-David; Golshani, Mohammad. Killing the $GCH$ everywhere with a single real. J. Symbolic Logic 78 (2013), no. 3, 803--823. doi:10.2178/jsl.7803060. https://projecteuclid.org/euclid.jsl/1389032276