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.

Homogeneous changes in cofinalities with applications to HOD

2017 ◽  
Vol 17 (02) ◽  
pp. 1750007 ◽  
Author(s):  
Omer Ben-Neria ◽  
Spencer Unger
Keyword(s):  
Large Cardinals ◽  
Related Question ◽  
New Technique ◽  
Singular Cardinals ◽  
Uncountable Cardinal ◽  
A New Technique

We present a new technique for changing the cofinality of large cardinals using homogeneous forcing. As an application we show that many singular cardinals in [Formula: see text] can be measurable in HOD. We also answer a related question of Cummings, Friedman and Golshani by producing a model in which every regular uncountable cardinal [Formula: see text] in [Formula: see text] is [Formula: see text]-supercompact in HOD.

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.

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 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.

On the compactness of ℵ1 and ℵ2

1978 ◽  
Vol 43 (3) ◽  
pp. 394-401 ◽  
Author(s):  
C. A. di Prisco ◽  
J. Henle
Keyword(s):  
Recent Result ◽  
Large Cardinal ◽  
Partition Relations ◽  
Uncountable Cardinals ◽  
Open Question ◽  
Jonsson Cardinals

In recent years, the Axiom of Determinateness (AD) has yielded numerous results concerning the size and properties of the first ω-many uncountable cardinals. Briefly, these results began with Solovay's discovery that ℵ1 and ℵ2 are measurable [8], [3], continued with theorems of Solovay, Martin, and Kunen concerning infinite-exponent partition relations [6], [3], Martin's proof that ℵn has confinality ℵ2 for 1 < n < ω, and very recently, Kleinberg's proof that the ℵn are Jonsson cardinals [4].This paper was inspired by a very recent result of Martin from AD that ℵ1 is ℵ2-super compact. It was known for some time that AD implies ℵ1 is α-strongly compact for all ℵ < θ (where θ is the least cardinal onto which 2ω cannot be mapped, quite a large cardinal under AD), and that ADR implies that ℵ1, is α-super compact for all α < θ. A key open question had been whether or not ℵ1 is super compact under AD alone.This paper comments on the method of Martin in several different ways. In §2, we will prove that ℵ1 is ℵ2-super compact, and then generalize the method to show that ℵ2 is ℵ3-strongly compact. In addition, we will demonstrate a limitation in the method by showing that the possible measures obtained on are not normal, and that the method cannot be extended to show that ℵ2 is ℵ4-strongly compact.

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.

AXIOM I0 AND HIGHER DEGREE THEORY

2015 ◽  
Vol 80 (3) ◽  
pp. 970-1021 ◽  
Author(s):  
XIANGHUI SHI
Keyword(s):  
Critical Point ◽  
Degree Theory ◽  
Large Cardinal ◽  
Elementary Embedding ◽  
Generic Absoluteness ◽  
Perfect Set ◽  
Singular Cardinals ◽  
Uncountable Cardinal ◽  
Image Position ◽  
Measurable Cardinals

AbstractIn this paper, we analyze structures of Zermelo degrees via a list of four degree theoretic questions (see §2) in various fine structure extender models, or under large cardinal assumptions. In particular we give a detailed analysis of the structures of Zermelo degrees in the Mitchell model for ω many measurable cardinals. It turns out that there is a profound correlation between the complexity of the degree structures at countable cofinality singular cardinals and the large cardinal strength of the relevant cardinals. The analysis applies to general degree notions, Zermelo degree is merely the author’s choice for illustrating the idea.I0(λ) is the assertion that there is an elementary embedding j : L(Vλ+1) → L(Vλ+1) with critical point < λ. We show that under I0(λ), the structure of Zermelo degrees at λ is very complicated: it has incomparable degrees, is not dense, satisfies Posner–Robinson theorem etc. In addition, we show that I0 together with a mild condition on the critical point of the embedding implies that the degree determinacy for Zermelo degrees at λ is false in L(Vλ+1). The key tool in this paper is a generic absoluteness theorem in the theory of I0, from which we obtain an analogue of Perfect Set Theorem for “projective” subsets of Vλ+1, and the Posner–Robinson follows as a corollary. Perfect Set Theorem and Posner–Robinson provide evidences supporting the analogy between $$AD$$ over L(ℝ) and I0 over L(Vλ+1), while the failure of degree determinacy is one for disanalogy. Furthermore, we conjecture that the failure of degree determinacy for Zermelo degrees at any uncountable cardinal is a theorem of $$ZFC$$.

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…

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