Supercompact cardinals and trees of normal ultrafilters

1982 ◽  
Vol 47 (1) ◽  
pp. 89-109
Author(s):  
Julius B. Barbanel
Keyword(s):  
Structural Properties ◽  
Tree Structure ◽  
Partial Ordering ◽  
Supercompact Cardinal ◽  
Compact Cardinal ◽  
Supercompact Cardinals ◽  
Basic Facts ◽  
Normal Ultrafilter

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 normal ultrafilters associated with a fixed supercompact cardinal. Many results already appearing in the literature can be conveniently phrased in terms of structural properties of T (see, e.g. [4] or [6]). In this paper, we establish additional structural facts concerning T.In §1 we standardize our notation and review some of the basic facts and methods that will be used throughout. §2 begins with a presentation of an important technique, due to Solovay, which will be an important tool for us. Also in §2, we begin a detailed study of the structure of T in terms of branching and the existence of many successors to branches at limit levels. §3 contains results proving the existence of many nodes of T which do not have successors above certain levels of T. This complements work of Magidor [6] who established the existence of many nodes which have successors at all higher levels of T.

On measurable limits of compact cardinals

1999 ◽  
Vol 64 (4) ◽  
pp. 1675-1688
Author(s):  
Arthur W. Apter
Keyword(s):  
Supercompact Cardinal ◽  
Compact Cardinal ◽  
Supercompact Cardinals ◽  
Strongly Compact Cardinals

AbstractWe extend earlier work (both individual and joint with Shelah) and prove three theorems about the class of measurable limits of compact cardinals, where a compact cardinal is one which is either strongly compact or supercompact. In particular, we construct two models in which every measurable limit of compact cardinals below the least supercompact limit of supercompact cardinals possesses non-trivial degrees of supercompactness. In one of these models, every measurable limit of compact cardinals is a limit of supercompact cardinals and also a limit of strongly compact cardinals having no non-trivial degree of supercompactness. We also show that it is consistent for the least supercompact cardinal κ to be a limit of strongly compact cardinals and be so that every measurable limit of compact cardinals below κ has a non-trivial degree of supercompactness. In this model, the only compact cardinals below κ with a non-trivial degree of supercompactness are the measurable limits of compact cardinals.

Some results concerning strongly compact cardinals

1985 ◽  
Vol 50 (4) ◽  
pp. 874-880
Author(s):  
Yoshihiro Abe
Keyword(s):  
Order Type ◽  
Supercompact Cardinal ◽  
Strongly Compact Cardinal ◽  
Compact Cardinal ◽  
Supercompact Cardinals ◽  
Identity Function ◽  
Power Set ◽  
Elementary Embeddings ◽  
Strongly Compact Cardinals

This paper consists of two parts. In §1 we mention the first strongly compact cardinal. Magidor proved in [6] that it can be the first measurable and it can be also the first supercompact. In [2], Apter proved that Con(ZFC + there is a supercompact limit of supercompact cardinals) implies Con(ZFC + the first strongly compact cardinal κ is ϕ(κ)-supercompact + no α < κ is ϕ(α)-supercompact) for a formula ϕ which satisfies certain conditions.We shall get almost the same conclusion as Apter's theorem assuming only one supercompact cardinal. Our notion of forcing is the same as in [2] and a trick makes it possible.In §2 we study a kind of fine ultrafilter on Pκλ investigated by Menas in [7], where κ is a measurable limit of strongly compact cardinals. He showed that such an ultrafilter is not normal in some case (Theorems 2.21 and 2.22 in [7]). We shall show that it is not normal in any case (even if κ is supercompact). We also prove that it is weakly normal in some case.We work in ZFC and much of our notation is standard. But we mention the following: the letters α,β,γ… denote ordinals, whereas κ,λ,μ,… are reserved for cardinals. R(α) is the collection of sets rank <α. φM denotes the realization of a formula φ to a class M. Except when it is necessary, we drop “M”. For example, M ⊩ “κ is φ(κ)-supercompact” means “κ is φM(κ)-supercompact in M”. If x is a set, |x| is its cardinality, Px is its power set, and . If also x ⊆ OR, denotes its order type in the natural ordering. The identity function with the domain appropriate to the context is denoted by id. For the notation concerning ultrapowers and elementary embeddings, see [11]. When we talk about forcing, “⊩” will mean “weakly forces” and “p < q” means “p is stronger than q”.

Stationary reflections for uncountable cofinality

1986 ◽  
Vol 51 (1) ◽  
pp. 147-151 ◽  
Author(s):  
Péter Komjáth
Keyword(s):  
Partial Order ◽  
Large Cardinal ◽  
Supercompact Cardinal ◽  
Weakly Compact ◽  
Compact Cardinal ◽  
Stationary Reflection ◽  
Supercompact Cardinals ◽  
Stationary Set ◽  
Compactness Theorems ◽  
Weakly Compact Cardinal

It was J. E. Baumgartner who in [1] proved that when a weakly compact cardinal is Lévy-collapsed to ω2 the new ω2 inherits some of the large cardinal properties; e.g. if S is a stationary set of ω-limits in ω2 then for some α < ω2, S ∩ α is stationary in α. Later S. Shelah extended this to the following theorem: if a supercompact cardinal κ is Lévy-collapsed to ω2, then in the resulting model the following holds: if S ⊆ λ is a stationary set of ω-limits and cf(λ) ≥ ω2 then there is an α. < λ such that S ∩ α is stationary in α, i.e. stationary reflection holds for countable cofinality (see [1] and [3]). These theorems are important prototypes of small cardinal compactness theorems; many applications and generalizations can be found in the literature. One might think that these results are true for sets with an uncountable cofinality μ as well, i.e. when an appropriate large cardinal is collapsed to μ++. Though this is true for Baumgartner's theorem, there remains a problem with Shelah's result. The point is that the lemma stating that a stationary set of ω-limits remains stationary after forcing with an ω2-closed partial order may be false in the case of μ-limits in a cardinal of the form λ+ with cf(λ) < μ, as was shown in [8] by Shelah. The problem has recently been solved by Baumgartner, who observed that if a universal box-sequence on the class of those ordinals with cofinality ≤ μ exists, the lemma still holds, and a universal box-sequence of the above type can be added without destroying supercompact cardinals beyond μ.

Laver indestructibility and the class of compact cardinals

1998 ◽  
Vol 63 (1) ◽  
pp. 149-157 ◽  
Author(s):  
Arthur W. Apter
Keyword(s):  
Limit Point ◽  
Supercompact Cardinal ◽  
Generic Extension ◽  
Ground Model ◽  
Strongly Compact Cardinal ◽  
Joint Work ◽  
Compact Cardinal ◽  
Supercompact Cardinals ◽  
The Universe ◽  
Strongly Compact Cardinals

AbstractUsing 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 forcing to give a new proof of the Kimchi-Magidor Theorem in which every compact cardinal in the universe (supercompact or strongly compact) satisfies certain indestructibility properties. Specifically, we show that if K is the class of supercompact cardinals in the ground model, then it is possible to force and construct a generic extension in which the only strongly compact cardinals are the elements of K or their measurable limit points, every κ ∈ K is a supercompact cardinal indestructible under ∈-directed closed forcing, and every κ a measurable limit point of K is a strongly compact cardinal indestructible under κ-directed closed forcing not changing ℘(κ). We then derive as a corollary a model for the existence of a strongly compact cardinal κ which is not κ+ supercompact but which is indestructible under κ-directed closed forcing not changing ℘(κ) and remains non-κ+ supercompact after such a forcing has been done.

WEAK SQUARES AND VERY GOOD SCALES

2018 ◽  
Vol 83 (1) ◽  
pp. 1-12 ◽  
Author(s):  
MAXWELL LEVINE
Keyword(s):  
Supercompact Cardinal ◽  
Supercompact Cardinals ◽  
Good Scale ◽  
Weak Square

AbstractWe assume the existence of a supercompact cardinal and produce a model with weak square but no very good scale at a particular cardinal. This follows work of Cummings, Foreman, and Magidor, but uses a different approach. We produce another model, starting from countably many supercompact cardinals, where □K,<K holds but □K, λ fails for λ < K.

A note on a result of Kunen and Pelletier

1992 ◽  
Vol 57 (2) ◽  
pp. 461-465
Author(s):  
Julius B. Barbanel
Keyword(s):  
Supercompact Cardinal ◽  
Large Collection ◽  
Supercompact Cardinals ◽  
Combinatorial Principle ◽  
Partition Property

AbstractSuppose that U and U′ are normal ultrafilters associated with some supercompact cardinal. How may we compare U and U′? In what ways are they similar, and in what ways are they different? Partial answers are given in [1], [2], [3], [5], [6], and [7]. In this paper, we continue this study.In [6], Menas introduced a combinatorial principle χ(U) of normal ultrafilters U associated with supercompact cardinals, and showed that normal ultrafilters satisfying this property also satisfy a partition property. In [5], Kunen and Pelletier showed that this partition property for U does not imply χ(U). Using results from [3], we present a different method of finding such normal ultrafilters which satisfy the partition property but do not satisfy χ(U). Our method yields a large collection of such normal ultrafilters.

Forcing Axioms, Supercompact Cardinals, Singular Cardinal Combinatorics

2008 ◽  
Vol 14 (1) ◽  
pp. 99-113
Author(s):  
Matteo Viale
Keyword(s):  
Large Cardinals ◽  
Singular Cardinal ◽  
Strongly Compact Cardinal ◽  
Compact Cardinal ◽  
Forcing Axioms ◽  
Supercompact Cardinals ◽  
Saturation Properties ◽  
Proper Forcing Axiom ◽  
Singular Cardinals ◽  
Proper Forcing

The purpose of this communication is to present some recent advances on the consequences that forcing axioms and large cardinals have on the combinatorics of singular cardinals. I will introduce a few examples of problems in singular cardinal combinatorics which can be fruitfully attacked using ideas and techniques coming from the theory of forcing axioms and then translate the results so obtained in suitable large cardinals properties.The first example I will treat is the proof that the proper forcing axiom PFA implies the singular cardinal hypothesis SCH, this will easily lead to a new proof of Solovay's theorem that SCH holds above a strongly compact cardinal. I will also outline how some of the ideas involved in these proofs can be used as means to evaluate the “saturation” properties of models of strong forcing axioms like MM or PFA.The second example aims to show that the transfer principle (ℵω+1, ℵω) ↠ (ℵ2, ℵ1) fails assuming Martin's Maximum MM. Also in this case the result can be translated in a large cardinal property, however this requires a familiarity with a rather large fragment of Shelah's pcf-theory.Only sketchy arguments will be given, the reader is referred to the forthcoming [25] and [38] for a thorough analysis of these problems and for detailed proofs.

Strongly compact cardinals, elementary embeddings and fixed points

1984 ◽  
Vol 49 (3) ◽  
pp. 808-812
Author(s):  
Yoshihiro Abe
Keyword(s):  
Fixed Points ◽  
Elementary Embedding ◽  
Inner Model ◽  
Power Set ◽  
Elementary Embeddings ◽  
Basic Facts ◽  
The Universe ◽  
Normal Ultrafilter ◽  
Strongly Compact Cardinals

J. Barbanel [1] characterized the class of cardinals fixed by an elementary embedding induced by a normal ultrafilter on Pκλ assuming that κ is supercompact. In this paper we shall prove the same results from the weaker hypothesis that κ is strongly compact and the ultrafilter is fine.We work in ZFC throughout. Our set-theoretic notation is quite standard. In particular, if X is a set, ∣X∣ denotes the cardinality of X and P(X) denotes the power set of X. Greek letters will denote ordinals. In particular γ, κ, η and γ will denote cardinals. If κ and λ are cardinals, then λ<κ is defined to be supγ<κγγ. Cardinal exponentiation is always associated from the top. Thus, for example, 2λ<κ means 2(λ<κ). V denotes the universe of all sets. If M is an inner model of ZFC, ∣X∣M and P(X)M denote the cardinality of X in M and the power set of X in M respectively.We review the basic facts on fine ultrafilters and the corresponding elementary embeddings. (For detail, see [2].)Definition. Assume κ and λ are cardinals with κ ≤ λ. Then, Pκλ = {X ⊂ λ∣∣X∣ < κ}.It is important to note that ∣Pκλ∣ = λ< κ.

The least measurable can be strongly compact and indestructible

1998 ◽  
Vol 63 (4) ◽  
pp. 1404-1412 ◽  
Author(s):  
Arthur W. Apter ◽  
Moti Gitik
Keyword(s):  
Measurable Cardinal ◽  
Supercompact Cardinal ◽  
Strongly Compact Cardinal ◽  
Compact Cardinal ◽  
Strong Compactness

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

On a combinatorial property of menas related to the partition property for measures on supercompact cardinals

1983 ◽  
Vol 48 (2) ◽  
pp. 475-481 ◽  
Author(s):  
Kenneth Kunen ◽  
Donald H. Pelletier
Keyword(s):  
Supercompact Cardinal ◽  
Combinatorial Property ◽  
Supercompact Cardinals ◽  
Partition Property

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. We also show that if a is the least cardinal greater than κ such that Pκα bears a measure without the partition property, then α is inaccessible and -indescribable.

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