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)+.

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.

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.

scholarly journals UNIVERSAL CLASSES NEAR ${\aleph _1}$

2018 ◽  
Vol 83 (04) ◽  
pp. 1633-1643 ◽  
Author(s):  
MARCOS MAZARI-ARMIDA ◽  
SEBASTIEN VASEY
Keyword(s):  
Sufficient Conditions ◽  
List Type ◽  
Type Number ◽  
Abstract Elementary Classes ◽  
Uncountable Cardinal ◽  
Elementary Classes ◽  
Universal Classes ◽  
Uncountable Cardinals ◽  
Image Position

AbstractShelah has provided sufficient conditions for an ${\Bbb L}_{\omega _1 ,\omega } $-sentence ψ to have arbitrarily large models and for a Morley-like theorem to hold of ψ. These conditions involve structural and set-theoretic assumptions on all the ${\aleph _n}$’s. Using tools of Boney, Shelah, and the second author, we give assumptions on ${\aleph _0}$ and ${\aleph _1}$ which suffice when ψ is restricted to be universal:Theorem. Assume ${2^{{\aleph _0}}} < {2^{{\aleph _1}}}$. Let ψ be a universal ${\Bbb L}_{\omega _1 ,\omega } $-sentence.(1)If ψ is categorical in ${\aleph _0}$ and $1 \leqslant {\Bbb L}\left( {\psi ,\aleph _1 } \right) < 2^{\aleph _1 } $, then ψ has arbitrarily large models and categoricity of ψ in some uncountable cardinal implies categoricity of ψ in all uncountable cardinals.(2)If ψ is categorical in ${\aleph _1}$, then ψ is categorical in all uncountable cardinals.The theorem generalizes to the framework of ${\Bbb L}_{\omega _1 ,\omega } $-definable tame abstract elementary classes with primes.

The entire NS ideal on can be precipitous

1997 ◽  
Vol 62 (4) ◽  
pp. 1161-1172 ◽  
Author(s):  
Noa Goldring
Keyword(s):  
Supercompact Cardinal ◽  
Ground Model ◽  
Consistency Strength ◽  
Complete Proof ◽  
Uncountable Cardinal ◽  
New Ideas ◽  
Uncountable Cardinals ◽  
Woodin Cardinal

The main result of this note is showing that if γ and μ are regular uncountable cardinals with γ ≤ μ then the non-stationary ideal (henceforth the NS ideal) on can be precipitous. This strengthens a result of [1] showing, under the same hypotheses, that a restriction of this ideal can be precipitous. See [1, Theorem 29, p. 36]. In fact, we show that even the strongly NS ideal on is precipitous in our model (since the former ideal is a restriction of the latter, the latter's being precipitous is a stronger assertion).More precisely, by starting with a model of “ZFC + ‘κ is a supercompact cardinal’ + ‘μ < κ is a regular uncountable cardinal’ ”, we generate a model of ZFC where all cardinals below and including μ are not collapsed and where the NS and strongly NS ideals on Pγμ are precipitous, for all regular uncountable γ which are less than or equal to μ.As far as consistency strength, we can obtain the same result even if κ is only Woodin in the ground model. However, the proof of this result is more complicated than in the case when κ is a supercompact cardinal. Furthermore, there are essentially no new ideas in adapting the proof relative to a supercompact cardinal to that relative to a Woodin cardinal beyond what appears in, e.g., [2]. We therefore give the complete proof relative to the existence of a supercompact cardinal and then briefly sketch the proof relative to the existence of a Woodin cardinal, using [2] as a reference.

Finest partitions for ultrafilters

1986 ◽  
Vol 51 (2) ◽  
pp. 327-332 ◽  
Author(s):  
Akihiro Kanamori
Keyword(s):  
Extensive Study ◽  
Uncountable Cardinal ◽  
Master Function ◽  
Uncountable Cardinals ◽  
One To One

If a uniform ultrafilter U over an uncountable cardinal κ is not outright countably complete, probably the next best thing is that it have a finest partition: a master function f:κ → ω with ƒ−({n}) ∉ U for each n ϵ ω such that for any g: κ → κ, either (a) it is one-to-one on a set in U, or (b) it factors through ƒ (mod U), i.e. for some function h, {α < κ ∣ h(f(α)) = g(α)} ϵ U. In this paper, it is shown that recent contructions of irregular ultrafilters over ω1 can be amplified to incorporate a finest partition.Henceforth, let us assume that all ultrafilters are uniform.There has been an extensive study of substantial hypotheses, which are nonetheless weaker than countable completeness, on ultrafilters over uncountable cardinals. To survey some results and to establish a context, let us first recall the Rudin-Keisler (RK) ordering on ultrafilters: If Ui is an ultrafilter over Iii for i = 1, 2, then U1 ≤RKU2 iff there is a projecting function Ψ:I2 → I1 such that U1 = Ψ*(U2) = {X ⊆, I1∣ Ψ−1(X) ϵ U2}· U1, =RKU2 iff U1, ≤RK and U2 and U2≤RKU1; and U1<RKU2 iff U1≤RKU2 yet U1 ≠RKU2. In terms of this ordering, if an ultrafilter U has a finest partition ƒ, then ƒ*(U) over ω is maximum amongst all RK predecessors of U: for any g:κ → κ, if g*(U) <RKU, then g is not one-to-one on a set in U, so since g factors through ƒ with some h,g*(U) = h*(ƒ*(U)). Say now that an ultrafilter U over κ > ω is indecomposable iff whenever ω < λ < κ, there is no V ≤RKU such that V is a (uniform) ultrafilter over λ.

AD and patterns of singular cardinals below Θ

1996 ◽  
Vol 61 (1) ◽  
pp. 225-235 ◽  
Author(s):  
Arthur W. Apter
Keyword(s):  
Recent Result ◽  
Singular Cardinals ◽  
Uncountable Cardinal ◽  
Uncountable Cardinals

AbstractUsing Steel's recent result that assuming AD, in L[ℝ] below Θ, κ is regular iff κ is measurable, we mimic below Θ certain earlier results of Gitik. In particular, we construct via forcing a model in which all uncountable cardinals below Θ are singular and a model in which the only regular uncountable cardinal below Θ is ℵ1.

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.

Co-stationarity of the ground model

2006 ◽  
Vol 71 (3) ◽  
pp. 1029-1043 ◽  
Author(s):  
Natasha Dobrinen ◽  
Sy-David Friedman
Keyword(s):  
Large Cardinals ◽  
Partial Ordering ◽  
Proper Class ◽  
Ground Model ◽  
Covering Theorem ◽  
Uncountable Cardinal ◽  
Cohen Forcing ◽  
Measurable Cardinals

AbstractThis paper investigates when it is possible for a partial ordering ℙ to force Pk(Λ)\V to be stationary in Vℙ. It follows from a result of Gitik that whenever ℙ adds a new real, then Pk(Λ)\V is stationary in Vℙ for each regular uncountable cardinal κ in Vℙ and all cardinals λ ≥ κ in Vℙ [4], However, a covering theorem of Magidor implies that when no new ω-sequences are added, large cardinals become necessary [7]. The following is equiconsistent with a proper class of ω1-Erdős cardinals: If ℙ is ℵ1-Cohen forcing, then Pk(Λ)\V is stationary in Vℙ, for all regular κ ≥ ℵ2and all λ ≩ κ. The following is equiconsistent with an ω1-Erdős cardinal: If ℙ is ℵ1-Cohen forcing, then is stationary in Vℙ. The following is equiconsistent with κ measurable cardinals: If ℙ is κ-Cohen forcing, then is stationary in Vℙ.

The spectrum of resplendency

1990 ◽  
Vol 55 (2) ◽  
pp. 626-636
Author(s):  
John T. Baldwin
Keyword(s):  
Homogeneous Model ◽  
Order Theory ◽  
First Order ◽  
First Order Theory ◽  
Uncountable Cardinal ◽  
Recursive Theory ◽  
Finite Language

AbstractLet T be a complete countable first order theory and λ an uncountable cardinal. Theorem 1. If T is not superstable, T has 2λ resplendent models of power λ. Theorem 2. If T is strictly superstable, then T has at least min(2λ, ℶ2) resplendent models of power λ. Theorem 3. If T is not superstable or is small and strictly superstable, then every resplendent homogeneous model of T is saturated. Theorem 4 (with Knight). For each μ ∈ ω ∪ {ω, 2ω} there is a recursive theory in a finite language which has μ resplendent models of power κ for every infinite κ.

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