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 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.

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.

scholarly journals Combinatorial Dichotomies in Set Theory

2011 ◽  
Vol 17 (1) ◽  
pp. 1-72 ◽  
Author(s):  
Stevo Todorcevic
Keyword(s):  
Set Theory ◽  
Large Cardinal ◽  
Uncountable Cardinal

AbstractWe give an overview of a research line concentrated on finding to which extent compactness fails at the level of first uncountable cardinal and to which extent it could be recovered on some other perhaps not so large cardinal. While this is of great interest to set theorists, one of the main motivations behind this line of research is in its applicability to other areas of mathematics. We give some details about this and we expose some possible directions for further research.

The well-foundedness of the Mitchell order

1993 ◽  
Vol 58 (3) ◽  
pp. 931-940 ◽  
Author(s):  
J. R. Steel
Keyword(s):  
Critical Point ◽  
Elementary Proof ◽  
Large Cardinal ◽  
Canonical Embedding ◽  
Inner Models ◽  
Normal Ultrafilter ◽  
Well Foundedness

Let E ⊲ F iff E and F are extenders and E ∈ Ult(V, F). Intuitively, E ⊲ F implies that E is weaker—embodies less reflection—than F. The relation ⊲ was first considered by W. Mitchell in [M74], where it arises naturally in connection with inner models and coherent sequences. Mitchell showed in [M74] that the restriction of ⊲ to normal ultrafilters is well-founded.The relation ⊲ is now known as the Mitchell order, although it is not actually an order. It is irreflexive, and its restriction to normal ultrafilters is transitive, but under mild large cardinal hypotheses, it is not transitive on all extenders. Here is a counterexample. Let κ be (λ + 2)-strong, where λ > κ and λ is measurable. Let E be an extender with critical point κ and let U be a normal ultrafilter with critical point λ such that U ∈ Ult(V, E). Let i: V → Ult(V, U) be the canonical embedding. Then i(E) ⊲ U and U ⊲ E, but by 3.11 of [MS2], it is not the case that i(E) ⊲ E. (The referee pointed out the following elementary proof of this fact. Notice that i ↾ Vλ+2 ∈ Ult(V, E) and X ∈ Ea ⇔ X ∈ i(E)i(a). Moreover, we may assume without loss of generality that = support(E). Thus, if i(E) ∈ Ult(V, E), then E ∈ Ult(V, E), a contradiction.)By going to much stronger extenders, one can show the Mitchell order is not well-founded. The following example is well known. Let j: V → M be elementary, with Vλ ⊆ M for λ = joω(crit(j)). (By Kunen, Vλ+1 ∉ M.) Let E0 be the (crit(j), λ) extender derived from j, and let En+1 = i(En), where i: V → Ult(V, En) is the canonical embedding. One can show inductively that En is an extender over V, and thereby, that En+1 ⊲ En for all n < ω. (There is a little work in showing that Ult(V, En+1) is well-founded.)

scholarly journals Inner models with large cardinal features usually obtained by forcing

2011 ◽  
Vol 51 (3-4) ◽  
pp. 257-283 ◽  
Author(s):  
Arthur W. Apter ◽  
Victoria Gitman ◽  
Joel David Hamkins
Keyword(s):  
Large Cardinal ◽  
Inner Models

Fine structure for tame inner models

1996 ◽  
Vol 61 (2) ◽  
pp. 621-639 ◽  
Author(s):  
E. Schimmerling ◽  
J. R. Steel
Keyword(s):  
Fine Structure ◽  
Large Cardinal ◽  
Structure Theory ◽  
Uniqueness Problem ◽  
Covering Property ◽  
Strong Cardinal ◽  
Woodin Cardinals ◽  
Inner Models ◽  
Normal Measure ◽  
Consistency Results

In this paper, we solve the strong uniqueness problem posed in [St2]. That is, we extend the full fine structure theory of [MiSt] to backgrounded models all of whose levels are tame (defined in [St2] and below). As a consequence, more powerful large cardinal properties reflect to fine structural inner models. For example, we get the following extension to [MiSt, Theorem 11.3] and [St2, Theorem 0.3].Suppose that there is a strong cardinal that is a limit of Woodin cardinals. Then there is a good extender sequence such that(1) every level of is a sound, tame mouse, and(2) ⊨ “There is a strong cardinal that is a limit of Woodin cardinals”.Recall that satisfies GCH if all its levels are sound. Another consequence of our work is the following covering property, an extension to [St1, Theorem 1.4] and [St3, Theorem 1.10].Suppose that fi is a normal measure on Ω and that all premice are tame. Then Kc, the background certified core model, exists and is a premouse of height Ω. Moreover, for μ-almost every α < Ω.Ideas similar to those introduced here allow us to extend the fine structure theory of [Sch] to the level of tame mice. The details of this extension shall appear elsewhere. From the extension of [Sch] and Theorem 0.2, new relative consistency results follow. For example, we have the following application.If there is a cardinal κ such that κ is κ+-strongly compact, then there is a premouse that is not tame.

Deconstructing inner model theory

2002 ◽  
Vol 67 (2) ◽  
pp. 721-736 ◽  
Author(s):  
Ralf-Dieter Schindler ◽  
John Steel ◽  
Martin Zeman
Keyword(s):  
Model Theory ◽  
Initial Segment ◽  
Large Cardinal ◽  
Inner Model Theory ◽  
Inner Models ◽  
Inner Model

In this paper we shall repair some errors and fill some gaps in the inner model theory of [2]. The problems we shall address affect some quite basic definitions and proofs.We shall be concerned with condensation properties of canonical inner models constructed from coherent sequences of extenders as in [2]. Condensation results have the general form: if x is definable in a certain way over a level , then either x ∈ , or else from x we can reconstruct in a simple way.The first condensation property considered in [2] is the initial segment condition, or ISC. In section 1 we show that the version of this condition described in [2] is too strong, in that no coherent in which the extenders are indexed in the manner of [2], and which is such that L[] satisfies the mild large cardinal hypothesis that there is a cardinal which is strong past a measurable, can satisfy the full ISC of [2]. It follows that the coherent sequences constructed in [2] do not satisfy the ISC of [2]. We shall describe the weaker ISC which these sequences do satisfy, and indicate the small changes in the arguments of [2] this new condition requires.

Projective well-orderings and bounded forcing axioms

2005 ◽  
Vol 70 (2) ◽  
pp. 557-572
Author(s):  
Andrés Eduardo Caicedo
Keyword(s):  
Large Cardinal ◽  
Forcing Axioms ◽  
Woodin Cardinals ◽  
Inner Models

AbstractIn the absence of Woodin cardinals, fine structural inner models for mild large cardinal hypotheses admit forcing extensions where bounded forcing axioms hold and yet the reals are projectively well-ordered.

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.

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