On the indestructibility aspects of identity

We investigate the indestructibility properties of strongly compact cardinals in universes where strong compactness suffers from identity crisis. We construct an iterative poset that can be used to establish Kimchi-Magidor theorem from [22], i.e., that the first n strongly compact cardinals can be the first n measurable cardinals. As an application, we show that the first n strongly compact cardinals can be the first n measurable cardinals while the strong compactness of each strongly compact cardinal is indestructible under Levy collapses (our theorem is actually more general, see section 3). A further application is that the class of strong cardinals can be nonempty yet coincide with the class of strongly compact cardinals while strong compactness of any strongly compact cardinal κ is indestructible under κ-directed closed posets that force GCH at κ.