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.

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.

Supercompact cardinals, trees of normal ultrafilters, and the partition property

1986 ◽  
Vol 51 (3) ◽  
pp. 701-708
Author(s):  
Julius B. Barbanel
Keyword(s):  
Large Cardinal ◽  
Supercompact Cardinal ◽  
Supercompact Cardinals ◽  
Partition Property

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. We investigate the distribution throughout T of normal ultrafilters with and normal ultrafilters without the partition property.

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.

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.

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.

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”.

scholarly journals Supercompact cardinals and a partition property

1977 ◽  
Vol 25 (1) ◽  
pp. 46-55 ◽  
Author(s):  
Carlos Augusto Di Prisco
Keyword(s):  
Supercompact Cardinals ◽  
Partition Property

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.

On the relationship between the partition property and the weak partition property for normal ultrafilters on Pκλ

1993 ◽  
Vol 58 (1) ◽  
pp. 119-127
Author(s):  
Julius B. Barbanel
Keyword(s):  
Measurable Cardinal ◽  
Large Cardinal ◽  
The Other ◽  
Supercompact Cardinal ◽  
Partition Property ◽  
Other Hand ◽  
The One ◽  
Measurable Cardinals ◽  
The Relationship ◽  
Normal Ultrafilter

AbstractSuppose κ is a supercompact cardinal and λ > κ. We study the relationship between the partition properly and the weak partition properly for normal ultrafilters on Pκλ. On the one hand, we show that the following statement is consistent, given an appropriate large cardinal assumption: The partition property and the weak partition properly are equivalent, there are many normal ultrafilters that satisfy these properties, and there are many normal ultrafilters that do not satisfy these properties. On the other hand, we consider the assumption that, for some λ > κ, there exists a normal ultrafilter U on Pκλ such that U satisfies the weak partition property but does not satisfy the partition property. We show that this assumption is implied by the assertion that there exists a cardinal γ > κ such that γ is γ+-supercompact, and, assuming the GCH, it implies the assertion that there exists a cardinal γ > κ such that γ is a measurable cardinal with a normal ultrafilter concentrating on measurable cardinals.

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