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 that this can be done starting with κ being λ++-supercompact (note that Silver’s result above is the special case when κ = λ). One can ask if there is an analogue of Woodin’s result for λsupercompactness. We answer this question in the following strong sense: starting with the GCH and κ being λ-supercompact and λ++tall, we preserve λ-supercompactness of κ and kill the GCH at λ by directly manipulating the size of 2λ (i.e. we do not force the failure of GCH at λ as a consequence of having 2κ large enough). The direct manipulation of 2λ, where λ can be a successor cardinal, is the first step toward understanding which Easton functions can be realized as the continuum function on regular cardinals while preserving instances of λ-supercompactness. 2010 Mathematics Subject Classification: Primary 03E35; Secondary 03E55.