MSC (2010) 03E35, 03E45, 03E55We develop a new method for coding sets while preserving GCH in the presence of large cardinals, particularly supercompact cardinals. We will use the number of normal measures carried by a measurable cardinal as an oracle, and therefore, in order to code a subset A of κ, we require that our model contain κ many measurable cardinals above κ. Additionally we will describe some of the applications of this result.

We force a property of cardinals first proved relatively consistent by Sargsyan, that of being supercompact but not \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathrm{HOD}$\end{document}‐supercompact, starting from a model of set theory which does not satisfy \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${GCH}$\end{document} and that contains supercompact cardinals.

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We provide a model where u(κ) = κ + < 2 κ for a supercompact cardinal κ.[10] provides a sketch of how to obtain such a model by modifying the construction in [6]. We provide here a complete proof using a different modification of [6] and further study the values of other natural generalizations of classical cardinal characteristics in our model. For this purpose we generalize some standard facts that hold in the countable case as well as some classical forcing notions and their properties.

In an attempt to extend the property of being supercompact but not hod-supercompact to a proper class of indestructibly supercompact cardinals, a theorem is discovered about a proper class of indestructibly supercompact cardinals which reveals a surprising incompatibility. However, it is still possible to force to get a model in which the property of being supercompact but not hod-supercompact holds for the least supercompact cardinal κ 0 , κ 0 is indestructibly supercompact, the strongly compact and supercompact cardinals coincide except at measurable limit points, and level by level equivalence between strong compactness and supercompactness holds above κ 0 but fails below κ 0. Additionally, we get the property of being supercompact but not hod-supercompact at the least supercompact cardinal, in a model where level by level equivalence between strong compactness and supercompactness holds.