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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 $Read more