scholarly journals LOCAL CLUB CONDENSATION AND L-LIKENESS

2015 ◽  
Vol 80 (4) ◽  
pp. 1361-1378
Author(s):  
PETER HOLY ◽  
PHILIP WELCH ◽  
LIUZHEN WU
Keyword(s):  
Regular Cardinal ◽  
Large Cardinal ◽  
Ground Model ◽  
The Third ◽  
Chang’S Conjecture ◽  
Chang's Conjecture ◽  
Generalized Condensation

AbstractWe present a forcing to obtain a localized version of Local Club Condensation, a generalized Condensation principle introduced by Sy Friedman and the first author in [3] and [5]. This forcing will have properties nicer than the forcings to obtain this localized version that could be derived from the forcings presented in either [3] or [5]. We also strongly simplify the related proofs provided in [3] and [5]. Moreover our forcing will be capable of introducing this localized principle at κ while simultaneously performing collapses to make κ become the successor of any given smaller regular cardinal. This will be particularly useful when κ has large cardinal properties in the ground model. We will apply this to measure how much L-likeness is implied by Local Club Condensation and related principles. We show that Local Club Condensation at κ+ is consistent with ¬☐κ whenever κ is regular and uncountable, generalizing and improving a result of the third author in [14], and that if κ ≥ ω2 is regular, CC(κ+) - Chang’s Conjecture at κ+ - is consistent with Local Club Condensation at κ+, both under suitable large cardinal consistency assumptions.

scholarly journals Some two-cardinal results for O-minimal theories

1998 ◽  
Vol 63 (2) ◽  
pp. 543-548 ◽  
Author(s):  
Timothy Bays
Keyword(s):  
Minimal Theory ◽  
Chang’S Conjecture ◽  
Chang's Conjecture

AbstractWe examine two-cardinal problems for the class of O-minimal theories. We prove that an O-minimal theory which admits some (κ, λ) must admit every (κ′, λ′). We also prove that every “reasonable” variant of Chang's Conjecture is true for O-minimal structures. Finally, we generalize these results from the two-cardinal case to the δ-cardinal case for arbitrary ordinals δ.

Precipitous ideals

1980 ◽  
Vol 45 (1) ◽  
pp. 1-8 ◽  
Author(s):  
T. Jech ◽  
M. Magidor ◽  
W. Mitchell ◽  
K. Prikry
Keyword(s):  
Large Cardinals ◽  
Large Cardinal ◽  
Ground Model ◽  
Generic Set ◽  
Precipitous Ideal ◽  
Precipitous Ideals ◽  
The Ideal

The properties of small cardinals such as ℵ1 tend to be much more complex than those of large cardinals, so that properties of ℵ1 may often be better understood by viewing them as large cardinal properties. In this paper we show that the existence of a precipitous ideal on ℵ1 is essentially the same as measurability.If I is an ideal on P(κ) then R(I) is the notion of forcing whose conditions are sets x ∈ P(κ)/I, with x ≤ x′ if x ⊆ x′. Thus a set D R(I)-generic over the ground model V is an ultrafilter on P(κ) ⋂ V extending the filter dual to I. The ideal I is said to be precipitous if κ ⊨R(I)(Vκ/D is wellfounded).One example of a precipitous ideal is the ideal dual to a κ-complete ultrafilter U on κ. This example is trivial since the generic ultrafilter D is equal to U and is already in the ground model. A generic set may be viewed as one that can be worked with in the ground model even though it is not actually in the ground model, so we might expect that cardinals such as ℵ1 that cannot be measurable still might have precipitous ideals, and such ideals might correspond closely to measures.

Weakly normal closures of filters on Pκλ

1993 ◽  
Vol 58 (1) ◽  
pp. 55-63 ◽  
Author(s):  
Masahiro Shioya
Keyword(s):  
Regular Cardinal ◽  
Natural Generalization ◽  
Large Cardinal ◽  
The Other ◽  
Disjoint Family ◽  
Straightforward Generalization ◽  
Weak Normality ◽  
Natural Formulation

The study of filters on Pκλ started by Jech [5] as a natural generalization of that of filters on an uncountable regular cardinal κ. Several notions including weak normality have been generalized. However, there are two versions proposed as weak normality for filters on Pκλ. One is due to Abe [1] as a straightforward generalization of weak normality for filters on κ due to Kanamori [6] and the other is due to Mignone [8]. While Mignone's version is weaker than normality, Kanamori-Abe's version is not in general. In fact, Abe [2] has proved, generalizing Kanamori [6], that a filter is weakly normal in the sense of Abe iff it is weakly normal in the sense of Mignone and there exists no disjoint family of cfλ-many positive sets. Therefore Kanamori-Abe's version is essentially a large cardinal property and Mignone's version seems to be the most natural formulation of “weak” normality.In this paper, we study weak normality in the sense of Mignone. In [8], Mignone studies weak normality of canonically defined filters. We complement his chart and try to find the weakly normal closures of these filters (i.e., the minimal weakly normal filters extending them). Therefore our result is a natural refinement of Carr [4].It is now well known that combinatorics on Pκλ is not a naive generalization of that on κ. For example, Menas [7] showed that stationarity on Pκλ can be characterized by 2-dimensional regressive functions, but not by 1-dimensional ones when λ is strictly larger than κ. We show in terms of weak normality that combinatorics on Pκλ vary drastically with respect to cfλ.

On A Variant of Weak Chang's Conjecture

1991 ◽  
Vol 37 (19-22) ◽  
pp. 289-292
Author(s):  
Yasuo Kanai
Keyword(s):  
Chang’S Conjecture ◽  
Chang's Conjecture

scholarly journals Combined Maximality Principles up to large cardinals

2009 ◽  
Vol 74 (3) ◽  
pp. 1015-1046 ◽  
Author(s):  
Gunter Fuchs
Keyword(s):  
Regular Cardinal ◽  
Large Cardinals ◽  
Large Cardinal ◽  
Consistency Strength ◽  
Regular Cardinals ◽  
Maximality Principles

AbstractThe motivation for this paper is the following: In [4] I showed that it is inconsistent with ZFC that the Maximality Principle for directed closed forcings holds at unboundedly many regular cardinals κ (even only allowing κ itself as a parameter in the Maximality Principle for <κ-closed forcings each time). So the question is whether it is consistent to have this principle at unboundedly many regular cardinals or at every regular cardinal below some large cardinal κ (instead of ∞), and if so, how strong it is. It turns out that it is consistent in many cases, but the consistency strength is quite high.

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