All uncountable cardinals can be singular

1980 ◽  
Vol 35 (1-2) ◽  
pp. 61-88 ◽  
Author(s):  
M. Gitik
Keyword(s):  
Uncountable Cardinals

NEGATIVE PARTITION RELATIONS OR UNCOUNTABLE CARDINALS OF COFINALITY

2001 ◽  
Vol 37 (1-2) ◽  
pp. 233-236
Author(s):  
P. Matet
Keyword(s):  
Partition Relations ◽  
Uncountable Cardinals

We modify an argument of Baumgartner to show that…

scholarly journals Freely indecomposable almost free groups with free abelianization

2020 ◽  
Vol 23 (3) ◽  
pp. 531-543
Author(s):  
Samuel M. Corson
Keyword(s):  
Free Groups ◽  
Uncountable Cardinals

AbstractFor certain uncountable cardinals κ, we produce a group of cardinality κ which is freely indecomposable, strongly κ-free, and whose abelianization is free abelian of rank κ. The construction takes place in Gödel’s constructible universe L. This strengthens an earlier result of Eklof and Mekler.

scholarly journals Categoricity and U-rank in excellent classes

2003 ◽  
Vol 68 (4) ◽  
pp. 1317-1336 ◽  
Author(s):  
Olivier Lessmann
Keyword(s):  
Free Groups ◽  
Order Theory ◽  
First Order ◽  
Atomic Models ◽  
First Order Theory ◽  
Uncountable Cardinal ◽  
Order Case ◽  
Uncountable Cardinals

AbstractLet be the class of atomic models of a countable first order theory. We prove that if is excellent and categorical in some uncountable cardinal, then each model is prime and minimal over the basis of a definable pregeometry given by a quasiminimal set. This implies that is categorical in all uncountable cardinals. We also introduce a U-rank to measure the complexity of complete types over models. We prove that the U-rank has the usual additivity properties, that quasiminimal types have U-rank 1, and that the U-rank of any type is finite in the uncountably categorical, excellent case. However, in contrast to the first order case, the supremum of the U-rank over all types may be ω (and is not achieved). We illustrate the theory with the example of free groups, and Zilber's pseudo analytic structures.

scholarly journals Mad families, splitting families and large continuum

2011 ◽  
Vol 76 (1) ◽  
pp. 198-208 ◽  
Author(s):  
Jörg Brendle ◽  
Vera Fischer
Keyword(s):  
Measurable Cardinal ◽  
Finite Support ◽  
Mad Families ◽  
Large Continuum ◽  
Uncountable Cardinals ◽  
Matrix Iteration ◽  
Finite Support Iteration

AbstractLet κ < λ be regular uncountable cardinals. Using a finite support iteration (in fact a matrix iteration) of ccc posets we obtain the consistency of . If μ is a measurable cardinal and μ < κ < λ, then using similar techniques we obtain the consistency of .

Splitting number at uncountable cardinals

1997 ◽  
Vol 62 (1) ◽  
pp. 35-42 ◽  
Author(s):  
Jindřich Zapletal
Keyword(s):  
Large Cardinal ◽  
Inner Models ◽  
Splitting Number ◽  
Uncountable Cardinal ◽  
Uncountable Cardinals

AbstractWe study a generalization of the splitting number s to uncountable cardinals. We prove that 𝔰(κ) > κ+ for a regular uncountable cardinal κ implies the existence of inner models with measurables of high Mitchell order. We prove that the assumption 𝔰(ℵω) > ℵω+1 has a considerable large cardinal strength as well.

Better quasi-orders for uncountable cardinals

1982 ◽  
Vol 42 (3) ◽  
pp. 177-226 ◽  
Author(s):  
Saharon Shelah
Keyword(s):  
Uncountable Cardinals

A hierarchy of filters on regular uncountable cardinals

1987 ◽  
Vol 52 (2) ◽  
pp. 388-395
Author(s):  
Thomas Jech
Keyword(s):  
Uncountable Cardinal ◽  
Uncountable Cardinals

AbstractWe introduce a well-founded relation < between filters on the space of descending sequences of ordinals. For each regular uncountable cardinal κ, the length of the relation is an ordinal o(κ) ≤ (2κ)+.

KNIGHT'S MODEL, ITS AUTOMORPHISM GROUP, AND CHARACTERIZING THE UNCOUNTABLE CARDINALS

2002 ◽  
Vol 02 (01) ◽  
pp. 113-144 ◽  
Author(s):  
GREG HJORTH
Keyword(s):  
Automorphism Group ◽  
Partial Information ◽  
Countable Model ◽  
Dichotomy Theorems ◽  
Analytic Sets ◽  
Uncountable Cardinals ◽  
Countable Structure

We show that every ℵα(α<ω1) can be characterized by the Scott sentence of some countable model; moreover there is a countable structure whose Scott sentence characterizes ℵ1but whose automorphism group fails the topological Vaught conjecture on analytic sets.We obtain some partial information on Ulm type dichotomy theorems for the automorphism group of Knight's model.

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