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.

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.

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.

A combinatorial property of pκλ

1976 ◽  
Vol 41 (1) ◽  
pp. 225-234
Author(s):  
Telis K. Menas
Keyword(s):  
Large Cardinals ◽  
Combinatorial Property ◽  
Combinatorial Properties ◽  
Partition Property ◽  
Homogeneous Set ◽  
Unbounded Subset

In a paper on combinatorial properties and large cardinals [2], Jech extended several combinatorial properties of a cardinal κ to analogous properties of the set of all subsets of λ of cardinality less than κ, denoted by “pκλ”, where λ is any cardinal ≤κ. We shall consider in this paper one of these properties which is historically rooted in a theorem of Ramsey [10] and in work of Rowbottom [12].As in [2], define [pκλ]2 = {{x, y}: x, y ∈ pκλ and x ≠ y}. An unbounded subset A of pκλ is homogeneous for a function F: [pκλ]2 → 2 if there is a k < 2 so that for all x, y ∈ A with either x ⊊ y or y ⊊ x, F({x, y}) = k. A two-valued measure ü on pκλ is fine if it is κ-complete and if for all α < λ, ü({x ∈ pκλ: α ∈ x}) = 1, and ü is normal if, in addition, for every function f: pκλ → λsuch that ü({x ∈ pκλ: f(x) ∈ x}) = 1, there is an α < λ such that ü({x ∈ pκλ: f(x) = α}) = 1. Finally, a fine measure on pκλ has the partition property if every F: [pκλ]2 → 2 has a homogeneous set of measure one.

scholarly journals On the partition property of measures onPℋλ

1982 ◽  
Vol 5 (4) ◽  
pp. 817-821
Author(s):  
Donald H. Pelletier
Keyword(s):  
Measurable Cardinal ◽  
Combinatorial Property ◽  
Partition Property ◽  
Normal Measure

The partition property for measures onPℋλwas formulated by analogy with a property which Rowbottom [1] proved was possessed by every normal measure on a measurable cardinal. This property has been studied in [2], [3], and [4]. This note summarizes [5] and [6], which contain results relating the partition property with the extendibility of the measure and with an auxiliary combinatorial property introduced by Menas in [4]. Detailed proofs will appear in [5] and [6].

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
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