# The failure of GCH al $\aleph_\omega$ by nice forcing

There are several ways to force $GCH$ below $\aleph_\omega$ and $2^{\aleph_\omega}> \aleph_{\omega+1},$ say:

1) The Gitik-Magidor's extender based forcing, see Prikry type Forcings.

2) Woodin's method, see Power set at $\aleph_\omega$: On a theorem of Woodin.

In both of the above methods, the forcing is $\kappa^{++}$-c.c., where $\kappa$ is the cardinal which becomes $\aleph_\omega$ in the final extension.

I am wondering if one can do this using a $\kappa^+$-c.c. forcing notion, more precisely,

Question. Starting from a suitable large cardinal $\kappa,$ is it possible to construct a pair $(V_1, V_2), V_1 \subseteq V_2,$ of generic extensions of the universe such that $\kappa$ is inaccessible in $V_1$, $V_2$ is a generic extension of $V_1$ by a $\kappa^+$-c.c. forcing notion and in $V_2, \kappa=\aleph_\omega,~ GCH$ holds below $\aleph_\omega$ and $2^{\aleph_\omega}> \aleph_{\omega+1}?$

• Could you please add more explanations about the motivation? Why is it interesting to you to force this statement using $\kappa^+$-cc forcings rather than $\kappa^{++}$-cc ones? Are you going to use this phenomenon to prove the consistency of the first failure of GCH at $\aleph_{\omega}$ with something else? If so, what is it? – Morteza Azad Dec 6 '17 at 18:42