Blowing up the power of a singular cardinal of any cofinality with collapses

Cardinal arithmetic is one of the central topics that is still active in set theory until now. Specifically, the Singular Cardinal Hypothesis (SCH) is one of the most classical combinatorial principles which concerns cardinal arithmetic. SCH states that for any singular cardinal κ, κ^cf(κ) ≤ max{2^cf(κ), κ⁺}. In particular, if κ is singular strong limit, then 2^κ = κ⁺. It is known that to obtain a failure of SCH, a certain large cardinal assumption is required. In this thesis, we construct an instance of an extender based forcing which violates SCH. The construction is based on a coherent sequence of extenders. We prove that given if κ is a singular cardinal of cofinality η in the ground model, which is also a limit of suitable large cardinals, and η⁺ = ℵ_γη, then there is an extender based forcing such that it preserves cardinals and cofinalities up to and including η such that in the extension, κ becomes ℵ_γη+η, and SCH fails at κ. In particular, if η is not an ℵ-fixed point, then SCH fails at ℵ_η in the extension. Our large cardinal assumption is the existence of a Woodin cardinal. Gitik’s work [8] suggests that the large cardinal assumption can be weakened to the existence of an increasing sequence of cardinals (κ_α : α < ηi such that for each α < η, there is an Eα-extenders such that if jE_α : V → M_α is the derived elementary embedding, then ^καM_α → M_α, M_α computes cardinals correctly up to an including (sup_β<ηκ_β) ++, and for β < α < η, E_β ∈ M_α. In our model, we also obtain some good scales, which exhibit the failures of SCH.