A characterization of lifting generics for Sacks-like forcings
Supercompactness and failures of GCH
Let κ < λ be regular cardinals. We say that an embedding j: V → M with critical point κ is λ-tall if λ < j(κ) and M is closed under κ-sequences in V. Silver showed that GCH can fail at a measurable cardinal κ, starting with κ being κ⁺⁺-supercompact. Later, Woodin improved this result, starting from the optimal hypothesis of a κ⁺⁺-tall measurable cardinal κ. Now more generally, suppose that κ ≤ λ are regular and one wishes the GCH to fail at λ with κ being λ-supercompact. Silver’s methods show...
Easton functions and supercompactness
Suppose that κ is λ-supercompact witnessed by an elementary embedding j: V → M with critical point κ, and further suppose that F is a function from the class of regular cardinals to the class of cardinals satisfying the requirements of Easton’s theorem: (1) ∀α α < cf(F(α)), and (2) α < β ⇒ F(α) ≤ F(β). We address the question: assuming GCH, what additional assumptions are necessary on j and F if one wants to be able to force the continuum function to agree with F globally, while preserving...
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