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SMALL $mathfrak{u}_kappa$ AND LARGE $2^kappa$ FOR SUPERCOMPACT $kappa$ (Forcing extensions and large cardinals)

SMALL $mathfrak{u}_kappa$ AND LARGE $2^kappa$ FOR SUPERCOMPACT $kappa$ (Forcing extensions and large cardinals)
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There are many normal ultrafiltres corresponding to a supercompact cardinal

Israel Journal of Mathematics, 1971
It is proved that ifκ is supercompact, there are at least (2• P κ(β)•)+normal ultrafilters overP k (β) and ifV=H.O.D. exactly (22• P κ(β)•) normal ultrafilters.
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On the role of supercompact and extendible cardinals in logic

Israel Journal of Mathematics, 1971
It is proved that the existence of supercompact cardinal is equivalent to a certain Skolem-Lowenheim Theorem for second order logic, whereas the existence of extendible cardinal is equivalent to a certain compactness theorem for that logic. It is also proved that a certain axiom schema related to model theory implies the existence of many extendible ...
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7 The least supercompact cardinal

2022
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Supercompact cardinals and trees of normal ultrafilters

Journal of Symbolic Logic, 1982
Supercompact cardinals are usually defined in terms of the existence of certain normal ultrafilters. It is well known that there is a natural partial ordering on the collection of all normal ultrafilters associated with a super-compact cardinal, that of normal ultrafilter restriction. Using this notion, we define a tree structure T on the collection of
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Supercompactness and measurable limits of strong cardinals

Journal of Symbolic Logic, 2001
AbstractIn this paper, two theorems concerning measurable limits of strong cardinals and supercompactness are proven. This generalizes earlier work, both individual and joint with Shelah.
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Supercompact cardinals, elementary embeddings and fixed points

Journal of Symbolic Logic, 1982
Supercompactness is usually defined in terms of the existence of certain ultrafilters. By the well-known procedure of taking ultrapowers of V (the universe of sets) and transitive collapses, one obtains transitive inner models of V and corresponding elementary embeddings from V into these inner models.
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Some structural results concerning supercompact cardinals

Journal of Symbolic Logic, 2001
Abstract.We show how the forcing of [5] can be iterated so as to get a model containing supercompact cardinals in which every measurable cardinal δ is δ+ supercompact. We then apply this iteration to prove three additional theorems concerning the structure of the class of supercompact cardinals.
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On a combinatorial property of menas related to the partition property for measures on supercompact cardinals

Journal of Symbolic Logic, 1983
AbstractT.K. Menas [4, pp. 225–234] introduced a combinatorial property Χ(μ) of a measure μ on a supercompact cardinal κ and proved that measures with this property also have the partition property. We prove here that Menas' property is not equivalent to the partition property.
Kenneth Kunen, Donald H. Pelletier
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Supercompact cardinals, trees of normal ultrafilters, and the partition property

Journal of Symbolic Logic, 1986
AbstractSuppose κ is a supercompact cardinal. It is known that for every λ ≥ κ, many normal ultrafilters on Pκ(λ) have the partition property. It is also known that certain large cardinal assumptions imply the existence of normal ultrafilters without the partition property. In [1], we introduced the tree T of normal ultrafilters associated with κ.
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