We study the general problem of the behaviour of the continuum function in the presence of non-supercompact strongly compact cardinals. We begin by showing that it is possible to force violations of GCH at an arbitrary strongly compact cardinal using only strong compactness as our initial assumption. This result is due to the third author. We then investigate realising Easton functions at and above the least measurable limit of supercompact cardinals starting from an initial assumption of the existence of a measurable limit of supercompact cardinals. By results due to Menas, assuming 2κ = κ+, the least measurable limit of supercompact cardinals κ is provably in ZFC a non-supercompact strongly compact cardinal which is not κ+supercompact. We also consider generalisations of our earlier theorems in the presence of more than one strongly compact cardinal. We conclude with some open questions.
|Publication status||Published - 2019 Jan 16|
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