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 ...

We study the general problem of behaviour continuum function in presence non-supercompact strongly compact cardinals. begin by showing that it is possible to force violations GCH at an arbitrary cardinal using only strong compactness as our initial assumption. This result due third author. then investigate realising Easton functions and above least measurable limit supercompact cardinals starti...

We give an example of a nonsupercompaet continuous image of a supercompact space. 0. Introduction. This paper deals with supercompact spaces. A space is called supercompact (cf. de Groot [7] ) provided it has a closed subbase such that any of its linked subsystems (a system of sets is called linked if any two of its members meet) has nonempty intersection. Much work has been done to show that c...

If κ < λ are such that κ is both supercompact and indestructible under κ-directed closed forcing which is also (κ+,∞)-distributive and λ is 2λ supercompact, then by [3, Theorem 5], {δ < κ | δ is δ+ strongly compact yet δ isn’t δ+ supercompact} must be unbounded in κ. We show that the large cardinal hypothesis on λ is necessary by constructing a model containing a supercompact cardinal κ in whic...

From a suitable large cardinal hypothesis, we provide a model with a supercompact cardinal in which universal indestructibility holds: every supercompact and partially supercompact cardinal κ is fully indestructible by <κ-directed closed forcing. Such a state of affairs is impossible with two supercompact cardinals or even with a cardinal which is supercompact beyond a measurable cardinal. Lave...

We show relative to strong hypotheses that patterns of compact cardinals in the universe, where a compact cardinal is one which is either strongly compact or supercompact, can be virtually arbitrary. Specifically, we prove if V |= “ZFC + Ω is the least inaccessible limit of measurable limits of supercompact cardinals + f : Ω → 2 is a function”, then there is a partial ordering P ∈ V so that for...

We construct a variety of inner models exhibiting features usually obtained by forcing over universes with large cardinals. For example, if there is a supercompact cardinal, then there is an inner model with a Laver indestructible supercompact cardinal. If there is a supercompact cardinal, then there is an inner model with a supercompact cardinal κ for which 2κ = κ+, another for which 2κ = κ++ ...

In this note, we provide a new proof of Magidor's theorem that the least strongly compact cardinal can be the least supercompact cardinal. A classical theorem of Magidor [5] states that it is consistent, relative to the existence of a supercompact cardinal, for the least supercompact cardinal to be the least strongly compact cardinal. There are currently two different proofs of this fact—the or...

Assuming ω supercompact cardinals we force to obtain a model where the tree property holds both at אω+1, and at אn for all 2 ≤ n < ω. A model with the former was obtained by Magidor–Shelah from a huge cardinal and ω supercompact cardinals above it, and recently by Sinapova from ω supercompact cardinals. A model with the latter was obtained by Cummings–Foreman from ω supercompact cardinals. Our ...