Killing the GCH everywhere with a single real

TL;DR

This paper demonstrates that adding a single real to a model of GCH can violate GCH at all infinite cardinals, assuming the existence of a sufficiently strong large cardinal.

Contribution

It extends previous results by showing GCH can be violated at all infinite cardinals with a single real, under the assumption of an H(κ^{+3})-strong cardinal.

Findings

GCH can be violated at all infinite cardinals by adding one real.

The result depends on the existence of an H(κ^{+3})-strong cardinal.

Strengthens prior work by Shelah-Woodin on GCH failure with a single real.

Abstract

Shelah-Woodin 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 it is known that more than an $H(\kappa^{++})$-strong cardinal is required.