# Destroying the Mahloness of a cardinal with $\kappa$.c.c. forcing

Question: Is it possible to have a Mahlo cardinal $\kappa$ such that there is a $\kappa$.c.c. forcing that makes it non-Mahlo?

If this is possible then this forcing must change the cofinality of all but non stationary many inaccessible cardinals below $\kappa$, so at least large cardinals in the region of measurable cardinals of large Mitchell order seem to be unavoidable.

In order to avoid trivialities (such as $Add(\omega,\kappa), Col(\omega, <\kappa)$ ) I require, as Mohhamad suggested, that the forcing preserves the inaccessiblity of $\kappa$.

I'm also interested in the cases where the forcing make $\kappa$ weakly inaccessible, but still destroy its weak Mahloness.

• I think you should require the forcing preserves inaccessibility of $\kappa,$ as otherwise $Add(\omega, \kappa)$ is as required. May 29, 2014 at 3:06

## 1 Answer

I asked this question from Prof. Woodin during the summer school in Singapore, and he gave me the following answer:

The answer is consistency yes. If we assume the existence of a Woodin cardinal and if we assume enough iterability, then we can use the extender algebra to turn a Woodin cardinal $\kappa$ into the least inaccessible cardinal using a $\kappa-c.c.$ forcing notion.

• I don't know the details of the argument, but I hope someone expert in such topics gives more details about the proof. Jul 9, 2014 at 7:42