scholarly journals ON THE SPLITTING NUMBER AT REGULAR CARDINALS

2015 ◽  
Vol 80 (4) ◽  
pp. 1348-1360 ◽  
Author(s):  
OMER BEN-NERIA ◽  
MOTI GITIK
Keyword(s):  
Core Model ◽  
Generic Extension ◽  
Singular Cardinal ◽  
Ground Model ◽  
The Core ◽  
Regular Cardinals ◽  
Splitting Number ◽  
Uncountable Cardinals

AbstractLetκ, λ be regular uncountable cardinals such that λ >κ+is not a successor of a singular cardinal of low cofinality. We construct a generic extension withs(κ) = λ starting from a ground model in whicho(κ) = λ and prove that assuming ¬0¶,s(κ) = λ implies thato(κ) ≥ λ in the core model.

scholarly journals Diagonal Prikry extensions

2010 ◽  
Vol 75 (4) ◽  
pp. 1383-1402 ◽  
Author(s):  
James Cummings ◽  
Matthew Foreman
Keyword(s):  
Model Theory ◽  
Measurable Cardinal ◽  
Core Model ◽  
Singular Cardinal ◽  
Inner Model Theory ◽  
Regular Cardinals ◽  
Inner Model ◽  
Singular Cardinals ◽  
Free Abelian Groups ◽  
Woodin Cardinal

§1. Introduction. It is a well-known phenomenon in set theory that problems in infinite combinatorics involving singular cardinals and their successors tend to be harder than the parallel problems for regular cardinals. Examples include the behaviour of cardinal exponentiation, the extent of the tree property, the extent of stationary reflection, and the existence of non-free almost-free abelian groups. The explanation for this phenomenon lies in inner model theory, in particular core models and covering lemmas. If W is an inner model of V then1. W strongly covers V if every uncountable set of ordinals is covered by a set of the same V -cardinality lying in W.2. W weakly covers V if W computes the successor of every V-singular cardinal correctly.Strong covering implies weak covering.In inner model theory there are many theorems of the general form “if there is no inner model of large cardinal hypothesis X then there is an L-like inner model Kx which Y covers V”. Here the L-like properties of Kx always include GCH and Global Square. Examples include1. X is “0# exists”, Kx is L, Y is “strongly”.2. X is “there is a measurable cardinal”, Kx is the Dodd-Jensen core model, Y is “strongly”.3. X is “there is a Woodin cardinal”, Kx is the core model for a Woodin cardinal, Y is “weakly”.

scholarly journals Jónsson cardinals, Erdős cardinals, and the core model

1999 ◽  
Vol 64 (3) ◽  
pp. 1065-1086 ◽  
Author(s):  
W. J. Mitchell
Keyword(s):  
Minimal Model ◽  
Independent Interest ◽  
Core Model ◽  
Generic Extension ◽  
The Core ◽  
Steel Core ◽  
Inner Model ◽  
Jonsson Cardinals ◽  
Woodin Cardinal

AbstractWe show that if there is no inner model with a Woodin cardinal and the Steel core model K exists, then every Jónsson cardinal is Ramsey in K, and every δ-Jónsson cardinal is δ5-Erdős in K.In the absence of the Steel core model K we prove the same conclusion for any model L[] such that either V = L[] is the minimal model for a Woodin cardinal, or there is no inner model with a Woodin cardinal and V is a generic extension of L[].The proof includes one lemma of independent interest: If V = L[A], where A ⊂ κ and κ is regular, then Lκ[A] is a Jónsson algebra. The proof of this result. Lemma 2.5, is very short and entirely elementary.

-definability at uncountable regular cardinals

2012 ◽  
Vol 77 (3) ◽  
pp. 1011-1046 ◽  
Author(s):  
Philipp Lücke
Keyword(s):  
Partial Order ◽  
Regular Cardinal ◽  
Short Proof ◽  
Chain Condition ◽  
Generic Extension ◽  
Infinite Cardinal ◽  
Ground Model ◽  
Generic Absoluteness ◽  
Regular Cardinals ◽  
Quasi Order

AbstractLet κ be an infinite cardinal. A subset of (κκ)n is a -subset if it is the projection p[T] of all cofinal branches through a subtree T of (>κκ)n+1 of height κ. We define and -subsets of (κκ)n as usual.Given an uncountable regular cardinal κ with κ = κ<κ and an arbitrary subset A of κκ, we show that there is a <κ-closed forcing ℙ that satisfies the κ+-chain condition and forces A to be a -subset of κκ in every ℙ-generic extension of V. We give some applications of this result and the methods used in its proof.(i) Given any set x, we produce a partial order with the above properties that forces x to be an element of L.(ii) We show that there is a partial order with the above properties forcing the existence of a well-ordering of κκ whose graph is a -subset of κκ × κκ.(iii) We provide a short proof of a result due to Mekler and Väänänen by using the above forcing to add a tree T of cardinality and height κ such that T has no cofinal branches and every tree from the ground model of cardinality and height κ without a cofinal branch quasi-order embeds into T.(iv) We will show that generic absoluteness for -formulae (i.e., formulae with parameters which define -subsets of κκ) under <κ-closed forcings that satisfy the κ+-chain condition is inconsistent.In another direction, we use methods from the proofs of the above results to show that - and -subsets have some useful structural properties in certain ZFC-models.

BLOWING UP THE POWER OF A SINGULAR CARDINAL OF UNCOUNTABLE COFINALITY

2019 ◽  
Vol 84 (4) ◽  
pp. 1722-1743 ◽  
Author(s):  
MOTI GITIK
Keyword(s):  
Blow Up ◽  
New Method ◽  
Core Model ◽  
Singular Cardinal ◽  
The Core ◽  
Blowing Up

AbstractA new method for blowing up the power of a singular cardinal is presented. It allows to blow up the power of a singular in the core model cardinal of uncountable cofinality. The method makes use of overlapping extenders.

scholarly journals ON SINGULAR STATIONARITY II (TIGHT STATIONARITY AND EXTENDERS-BASED METHODS)

2019 ◽  
Vol 84 (1) ◽  
pp. 320-342
Author(s):  
OMER BEN-NERIA
Keyword(s):  
Generic Extension ◽  
Ground Model ◽  
Regular Cardinals ◽  
Consistency Results ◽  
Image Position ◽  
Model Sequence

AbstractWe study the notion of tightly stationary sets which was introduced by Foreman and Magidor in [8]. We obtain two consistency results showing that certain sequences of regular cardinals ${\langle {\kappa _n}\rangle _{n < \omega }}$ can have the property that in some generic extension, every ground-model sequence of fixed-cofinality stationary sets ${S_n} \subseteq {\kappa _n}$ is tightly stationary. The results are obtained using variations of the short-extenders forcing method.

The consistency strength of the free-subset property for ωω

1984 ◽  
Vol 49 (4) ◽  
pp. 1198-1204 ◽  
Author(s):  
Peter Koepke
Keyword(s):  
Measurable Cardinal ◽  
Core Model ◽  
Reverse Direction ◽  
Ground Model ◽  
Consistency Strength ◽  
The Core ◽  
Inner Model ◽  
Free Subset ◽  
Coherent Sequence ◽  
Measurable Cardinals

A subset X of a structure S is called free in S if ∀x ∈ Xx ∉ S[X − {x}]; here, S[Y] is the substructure of S generated from Y by the functions of S. For κ, λ, μ cardinals, let Frμ(κ, λ) be the assertion:for every structure S with κ ⊂ S which has at most μ functions and relations there is a subset X ⊂ κ free in S of cardinality ≥ λ.We show that Frω(ωω, ω), the free-subset property for ωω, is equiconsistent with the existence of a measurable cardinal (2.2,4.4). This answers a question of Devlin [De].In the first section of this paper we prove some combinatorial facts about Frμ(κ, λ); in particular the first cardinal κ such that Frω(κ, ω) is weakly inaccessible or of cofinality ω (1.2). The second section shows that, under Frω(ωω, ω), ωω is measurable in an inner model. For the convenience of readers not acquainted with the core model κ, we first deduce the existence of 0# (2.1) using the inner model L. Then we adapt the proof to the core model and obtain that ωω is measurable in an inner model. For the reverse direction, we essentially apply a construction of Shelah [Sh] who forced Frω(ωω, ω) over a ground model which contains an ω-sequence of measurable cardinals. We show in §4 that indeed a coherent sequence of Ramsey cardinals suffices. In §3 we obtain such a sequence as an endsegment of a Prikry sequence.

scholarly journals The cofinality spectrum of the infinite symmetric group

1997 ◽  
Vol 62 (3) ◽  
pp. 902-916 ◽  
Author(s):  
Saharon Shelah ◽  
Simon Thomas
Keyword(s):  
Symmetric Group ◽  
Inaccessible Cardinal ◽  
Singular Cardinal ◽  
Natural Numbers ◽  
Maximum Element ◽  
Infinite Symmetric Group ◽  
Regular Cardinals ◽  
Uncountable Cardinals ◽  
A Chain

AbstractLet S be the group of all permutations of the set of natural numbers. The cofinality spectrum CF(S) of S is the set of all regular cardinals λ such that S can be expressed as the union of a chain of λ proper subgroups. This paper investigates which sets C of regular uncountable cardinals can be the cofinality spectrum of S. The following theorem.is the main result of this paper.Theorem. Suppose that V ⊨ GCH. Let C be a set of regular uncountable cardinals which satisfies the following conditions.(a) C contains a maximum element.(b) If μ is an inaccessible cardinal such thatμ = sup(C ∩ μ), thenμ ∈ C.(c) if μ is a singular cardinal such thatμ = sup(C ∩ μ), thenμ+ ∈ C.Then there exists a c.c.c. notion of forcing ℙ such that Vℙ ⊨ CF(S) = C.We shall also investigate the connections between the cofinality spectrum and pcf theory; and show that CF(S) cannot be an arbitrarily prescribed set of regular uncountable cardinals.

Scales in K(ℝ) at the end of a weak gap

2008 ◽  
Vol 73 (2) ◽  
pp. 369-390 ◽  
Author(s):  
J. R. Steel
Keyword(s):  
Induction Hypothesis ◽  
The Other ◽  
Core Model ◽  
The Core ◽  
Woodin Cardinals ◽  
Other Hand

In this note we shall proveTheorem 0.1. Letbe a countably ω-iterable-mouse which satisfies AD, and [α, β] a weak gap of. Supposeis captured by mice with iteration strategies in ∣α. Let n be least such that ; then we have that believes that has the Scale Property.This complements the work of [5] on the construction of scales of minimal complexity on sets of reals in K(ℝ). Theorem 0.1 was proved there under the stronger hypothesis that all sets definable over are determined, although without the capturing hypothesis. (See [5, Theorem 4.14].) Unfortunately, this is more determinacy than would be available as an induction hypothesis in a core model induction. The capturing hypothesis, on the other hand, is available in such a situation. Since core model inductions are one of the principal applications of the construction of optimal scales, it is important to prove 0.1 as stated.Our proof will incorporate a number of ideas due to Woodin which figure prominently in the weak gap case of the core model induction. It relies also on the connection between scales and iteration strategies with the Dodd-Jensen property first discovered in [3]. Let be the pointclass at the beginning of the weak gap referred to in 0.1. In section 1, we use Woodin's ideas to construct a Γ-full a mouse having ω Woodin cardinals cofinal in its ordinals, together with an iteration strategy Σ which condenses well in the sense of [4, Def. 1.13]. In section 2, we construct the desired scale from and Σ.

scholarly journals The tree property and the failure of the Singular Cardinal Hypothesis at ℵω2

2012 ◽  
Vol 77 (3) ◽  
pp. 934-946 ◽  
Author(s):  
Dima Sinapova
Keyword(s):  
Generic Extension ◽  
Singular Cardinal ◽  
Tree Property ◽  
Supercompact Cardinals ◽  
Singular Cardinal Hypothesis

AbstractWe show that given ω many supercompact cardinals, there is a generic extension in which the tree property holds at ℵω2+ 1 and the SCH fails at ℵω2.

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