# On an unpublished result of Magidor

In 1970th, Magidor proved the following important results:

(1) Assuming the existence of a supercompact cardinal, it is consistent that $\aleph_\omega$ is strong limit and $2^{\aleph_\omega}=\aleph_{\omega+k+1},$ where $0< k \leq \omega$.

(2) Assuming the existence of a supercompact cardinal and a huge cardinal above it, it is consistent that $GCH$ holds below $\aleph_\omega$ and $2^{\aleph_\omega}=\aleph_{\omega+2}$.

On the other hand Shelah has proved the following strengthening of (1).

(3) Assuming the existence of a supercompact cardinal, it is consistent that $\aleph_\omega$ is strong limit and $2^{\aleph_\omega}=\aleph_{\omega+\alpha+1},$ where $0< \alpha < \omega_1$.

There are some references stating that motivated by Shelah's result, Magidor has proved the following:

Theorem. Assuming the existence of very large cardinals, it is consistent that $GCH$ holds below $\aleph_\omega$ and $2^{\aleph_\omega}=\aleph_{\omega+\alpha+1},$ where $0< \alpha < \omega_1$.

The proof seems to me uses a supercompact and $\alpha$-many huge cardinals above it.

I wonder to know if any one knows the basic idea of Magidor's result, or if there is any reference I can find the proof.

Remark. By work of Gitik-Magidor on extender based Prikry forcing we can obtain the above theorem in a simpler way and just using a strong cardinal; but I am still interested to see Magidor's original proof.

• Seeing supercompact cardinals in the assumptions is not at all surprising. :-) – Asaf Karagila Sep 1 '15 at 4:49