We show that supercompactness and strong compactness can be equivalent even as properties of pairs regular cardinals. Specifically, we if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V true"> <mml:semantics> <mml:mrow> <mml:mi>V</mml:mi> <mml:mo>⊨</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">V \models</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ZFC + GCH is a given model (which in...

Abstract Using an idea developed in joint work with Shelah, we show how to redefine Laver's notion of forcing making a supercompact cardinal κ indestructible under κ-directed closed give new proof the Kimchi-Magidor Theorem which every compact universe (supercompact or strongly compact) satisfies certain indestructibility properties. Specifically, that if K is class cardinals ground model, then it possible force and construct generic extension only are elements their measurable limit points,...

Abstract Combining techniques of the first author and Shelah with ideas Magidor, we show how to get a model in which, for fixed but arbitrary finite n , strongly compact cardinals k 1 ..… are so that i ; = 1..… is both th measurable cardinal supercompact. This generalizes an unpublished theorem Magidor answers question Apter Shelah.

Abstract We show the consistency, relative to a supercompact cardinal, of least measurable cardinal being both strongly compact and fully Laver indestructible. also somewhat yet not completely having its strong compactness degree supercompactness

Generalizing some earlier techniques due to the second author, we show that Menas’ theorem which states least cardinal <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="kappa"> <mml:semantics> <mml:mi>κ</mml:mi> <mml:annotation encoding="application/x-tex">\kappa</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a measurable limit of supercompact or strongly compact cardinals but not alttext="2 Superscript kappa">...

Abstract Can a supercompact cardinal κ be Laver indestructible when there is level-by-level agreement between strong compactness and supercompactness? In this article, we show that if sufficiently large above , then no, it cannot. Conversely, one weakens the requirement either by demanding less indestructibility, such as requiring only indestructibility stratified posets. or agreement, on measure sets, yes. can.

This paper discusses models of set theory without the Axiom Choice. We investigate all possible patterns cofinality function and distribution measurability on first three uncountable cardinals. The result relies heavily a strengthening an unpublished Kechris: we prove (under <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif upper A sans-serif D"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi...

We show, assuming the consistency of one measurable cardinal, that it is consistent for there to be exactly <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="kappa Superscript plus"> <mml:semantics> <mml:msup> <mml:mi>κ</mml:mi> <mml:mo>+</mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">\kappa ^+</mml:annotation> </mml:semantics> </mml:math> </inline-formula> many normal measures on least cardinal alttext="kappa">...

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

We prove two theorems concerning strong compactness, measurability, and the class of supercompact cardinals. begin by showing, relative to appropriate hypotheses, that it is consistent non-trivially for every cardinal be limit o

Abstract If κ &lt; λ are such that is indestructibly supercompact and 2 supercompact, it known from [4] δ a measurable cardinal which not limit of cardinals violates level by equivalence between strong compactness supercompactness} must be unbounded in . On the other hand, using variant argument used to establish this fact, possible prove if measurable, then satisfies The two aforementioned phenomena, however, need occur universe with an sufficiently few large cardinals. In particular, we...

Say that the Specker Property holds for a well ordered cardinal N, and write this as SP(N), if power set of N can be written countable union sets cardinality N. Specker's Problem asks whether it is possible to have model in which SP(tt) every N.In paper, we construct two models large class cardinals.In first model, successor K.In second limit certain N's.

We prove two theorems, one concerning level by inequivalence between strong compactness and supercompactness, equivalence supercompactness. first show that in a universe containing

Abstract We construct two models containing exactly one supercompact cardinal in which all non‐supercompact measurable cardinals are strictly taller than they either strongly compact or supercompact. In the first of these models, level by equivalence between strong compactness and supercompactness holds. other, inequivalence Each universe has only contains relatively few large (© 2010 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)