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.

Rowbottom cardinals and Jonsson cardinals are almost the same

1973 ◽  
Vol 38 (3) ◽  
pp. 423-427 ◽  
Author(s):  
E. M. Kleinberg
Keyword(s):  
Large Cardinal ◽  
Order Logic ◽  
Elementary Substructure ◽  
Similarity Type ◽  
First Order ◽  
Reflection Principles ◽  
Uncountable Cardinals ◽  
The Moment ◽  
Definition Of ◽  
Jonsson Cardinals

Each of the various “large cardinal” axioms currently studied in set theory owes its inspiration to concrete phenomena in various fields. For example, the statement of the well-known compactness theorem for first-order logic can be generalized in various ways to infinitary languages to yield definitions of compact cardinals, and the reflection principles provable in ZF, when modified in the appropriate way, yields indescribable cardinals.In this paper we concern ourselves with two kinds of large cardinals which are probably the two best known of those whose origins lie in model theory. They are the Rowbottom cardinals and the Jonsson cardinals.Let us be more specific. A cardinal κ is said to be a Jonsson cardinal if every structure of cardinality κ has a proper elementary substructure of cardinality κ. (It is routine to see that only uncountable cardinals can be Jonsson. Erdös and Hajnal have shown [2] that for n < ω no ℵn is Jonsson. (In fact, they showed that if κ is not Jonsson then neither is the successor cardinal of κ and that, assuming GCH, no successor cardinal can be Jonsson.) Keisler and Rowbottom first showed that the existence of a Jonsson cardinal contradicts V = L.) The definition of a Rowbottom cardinal is only slightly more intricate. We assume for the moment that our similarity type has a designated one-place relation.

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…

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.

Weak partition relations and measurability

1986 ◽  
Vol 51 (1) ◽  
pp. 33-38
Author(s):  
Mitchell Spector
Keyword(s):  
Measurable Cardinal ◽  
Order Type ◽  
Large Cardinal ◽  
Partition Relation ◽  
Consistency Strength ◽  
Partition Relations ◽  
Inner Model ◽  
Increasing Function ◽  
Natural Way

The concept of "partition relation" has proven to be extremely important in the development of the theory of large cardinals. This is due in good part to the fact that the ordinal numbers which appear as parameters in partition relations provide a natural way to define a detailed hierarchy of the corresponding large cardinal axioms. In particular, the study of cardinals satisfying Ramsey-Erdös-style partition relations has yielded a great number of very interesting large cardinal axioms which lie in strength strictly between inaccessibility and measurability. It is the purpose of this paper to show that this phenomenon does not occur if we use infinite exponent partition relations; no such partition relation has consistency strength strictly between inaccessibility and measurability. We also give a complete determination of which infinite exponent partition relations hold, assuming that there is no inner model of set theory with a measurable cardinal.Our notation is standard. If F is a function and x is a set, then F″x denotes the range of F on x. If X is a set of ordinals and α is an ordinal, then [X]α is the collection of all subsets of X of order type α. We identify a member of [X]α with a strictly increasing function from α to X. If p ∈ [X]α and q ∈ [α]β, then the composition of p with q, which we denote pq, is a member of [X]β.

scholarly journals CLASSIFICATION THEORY FOR ACCESSIBLE CATEGORIES

2016 ◽  
Vol 81 (1) ◽  
pp. 151-165 ◽  
Author(s):  
M. LIEBERMAN ◽  
J. ROSICKÝ
Keyword(s):  
Recent Result ◽  
Large Cardinal ◽  
Classification Theory ◽  
Abstract Elementary Classes ◽  
Robust Version ◽  
Accessible Categories ◽  
Elementary Classes ◽  
Added Benefit ◽  
Precise Role

AbstractWe show that a number of results on abstract elementary classes (AECs) hold in accessible categories with concrete directed colimits. In particular, we prove a generalization of a recent result of Boney on tameness under a large cardinal assumption. We also show that such categories support a robust version of the Ehrenfeucht–Mostowski construction. This analysis has the added benefit of producing a purely language-free characterization of AECs, and highlights the precise role played by the coherence axiom.

Jonsson-like partition relations and j: V → V

2004 ◽  
Vol 69 (4) ◽  
pp. 1267-1281 ◽  
Author(s):  
Arthur W. Apter ◽  
Grigor Sargsyan
Keyword(s):  
Axiom Of Choice ◽  
Elementary Embedding ◽  
Partition Relation ◽  
Partition Relations ◽  
Final Segment ◽  
The Axiom Of Choice ◽  
Jonsson Cardinals

Abstract.Working in the theory ”ZF + There is a nontrivial elementary embedding j : V → V“, we show that a final segment of cardinals satisfies certain square bracket finite and infinite exponent partition relations. As a corollary to this, we show that this final segment is composed of Jonsson cardinals. We then show how to force and bring this situation down to small alephs. A prototypical result is the construction of a model for ZF in which every cardinal μ ≥ ℵ2 satisfies the square bracket infinite exponent partition relation . We conclude with a discussion of some consistency questions concerning different versions of the axiom asserting the existence of a nontrivial elementary embedding j: V → V. By virtue of Kunen's celebrated inconsistency result, we use only a restricted amount of the Axiom of Choice.

Partitioning pairs of countable sets of ordinals

1990 ◽  
Vol 55 (3) ◽  
pp. 1019-1021 ◽  
Author(s):  
Dan Velleman
Keyword(s):  
Partition Theorem ◽  
Partition Relations ◽  
Uncountable Cardinals ◽  
Image Position

In [3], Todorčević showed that ω1 ⇸ [ω1]ω12. In this paper we use similar methods to prove an analogous partition theorem for Pω1(λ), for certain uncountable cardinals λ.Recall that ω1 → [ω1]ω12, means that for every function f: [ω1]2 → ω1 there is a set A ∈ [ω1]ω1 such that f“[A]2 ≠ ω1, and of course ω1 ⇸ [ω1]ω12, is the negation of this statement. For partition relations on Pω1(→) it is customary to partition only those pairs of sets in which the first set is a subset of the second. Thus for A ⊆ Pω1(λ) we defineWe will write Pω1(λ) → [unbdd]λ2 to mean that for every function f: [Pω1(λ)]⊂2 → λ there is an unbounded set A ⊆ Pω1(λ) such that f“[A]⊂2 ≠ λ, and again Pω1(λ) ⇸ [unbdd]λ2 is the negation of this statement.

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.

scholarly journals Transformations of the transfinite plane

2021 ◽  
Vol 9 ◽  
Author(s):  
Assaf Rinot ◽  
Jing Zhang
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Inaccessible Cardinal ◽  
Partition Relation ◽  
Partition Relations ◽  
Uncountable Cardinals ◽  
Stationary Set

Abstract We study the existence of transformations of the transfinite plane that allow one to reduce Ramsey-theoretic statements concerning uncountable Abelian groups into classical partition relations for uncountable cardinals. To exemplify: we prove that for every inaccessible cardinal $\kappa $ , if $\kappa $ admits a stationary set that does not reflect at inaccessibles, then the classical negative partition relation $\kappa \nrightarrow [\kappa ]^2_\kappa $ implies that for every Abelian group $(G,+)$ of size $\kappa $ , there exists a map $f:G\rightarrow G$ such that for every $X\subseteq G$ of size $\kappa $ and every $g\in G$ , there exist $x\neq y$ in X such that $f(x+y)=g$ .

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