Adding a closed unbounded set

1976 ◽  
Vol 41 (2) ◽  
pp. 481-482 ◽  
Author(s):  
J. E. Baumgartner ◽  
L. A. Harrington ◽  
E. M. Kleinberg
Keyword(s):  
Regular Cardinal ◽  
Measurable Cardinal ◽  
Axiom Of Choice ◽  
Partition Relations ◽  
Uncountable Cardinal ◽  
Unbounded Subset ◽  
The Axiom Of Choice ◽  
Closed Unbounded Subset

The extreme interest of set theorists in the notion of “closed unbounded set” is epitomized in the following well-known theorem:Theorem A. For any regular cardinal κ > ω, the intersection of any two closed unbounded subsets of κ is closed and unbounded.The proof of this theorem is easy and in fact yields a stronger result, namely that for any uncountable regular cardinal κ the intersection of fewer than κ many closed unbounded sets is closed and unbounded. Thus, if, for κ a regular uncountable cardinal, we let denote {A ⊆ κ ∣ A contains a closed unbounded subset}, then, for any such κ, is a κ-additive nonprincipal filter on κ.Now what about the possibility of being an ultrafilterκ It is routine to see that this is impossible for κ > ℵ1. However, for κ = ℵ1 the situation is different. If were an ultrafilter, ℵ1 would be a measurable cardinal. As is well-known this is impossible if we assume the axiom of choice; however if ZF + “there exists a measurable cardinal” is consistent, then so is ZF + “ℵ1 is a measurable cardinal” [2]. Furthermore, under the assumption of certain set theoretic axioms (such as the axiom of determinateness or various infinite exponent partition relations) can be proven to be an ultrafilter. (See [3] and [5].)

A measurable cardinal with a nonwellfounded ultrapower

1980 ◽  
Vol 45 (3) ◽  
pp. 623-628 ◽  
Author(s):  
Mitchell Spector
Keyword(s):  
Set Theory ◽  
Measurable Cardinal ◽  
Axiom Of Choice ◽  
Partial Ordering ◽  
Partition Relation ◽  
Partition Relations ◽  
Consistency Results ◽  
The Axiom Of Choice ◽  
Measurable Cardinals ◽  
Weakened Form

The usefulness of measurable cardinals in set theory arises in good part from the fact that an ultraproduct of wellfounded structures by a countably complete ultrafilter is wellfounded. In the standard proof of the wellfoundedness of such an ultraproduct, one first shows, without any use of the axiom of choice, that the ultraproduct contains no infinite descending chains. One then completes the proof by noting that, assuming the axiom of choice, any partial ordering with no infinite descending chain is wellfounded. In fact, the axiom of dependent choices (a weakened form of the axiom of choice) suffices. It is therefore of interest to ask whether some use of the axiom of choice is needed in order to prove the wellfoundedness of such ultraproducts or whether, on the other hand, their wellfoundedness can be proved in ZF alone. In Theorem 1, we show that the axiom of choice is needed for the proof (assuming the consistency of a strong partition relation). Theorem 1 also contains some related consistency results concerning infinite exponent partition relations. We then use Theorem 1 to show how to change the cofinality of a cardinal κ satisfying certain partition relations to any regular cardinal less than κ, while introducing no new bounded subsets of κ. This generalizes a theorem of Prikry [5].

Model theory under the axiom of determinateness

1985 ◽  
Vol 50 (3) ◽  
pp. 773-780
Author(s):  
Mitchell Spector
Keyword(s):  
Model Theory ◽  
Axiom Of Choice ◽  
Order Logic ◽  
First Order Logic ◽  
Technical Innovation ◽  
First Order ◽  
Partition Relations ◽  
Hanf Number ◽  
The Axiom Of Choice

AbstractWe initiate the study of model theory in the absence of the Axiom of Choice, using the Axiom of Determinateness as a powerful substitute. We first show that, in this context, is no more powerful than first-order logic. The emphasis then turns to upward Löwenhein-Skolem theorems; ℵ1 is the Hanf number of first-order logic, of , and of a strong fragment of , The main technical innovation is the development of iterated ultrapowers using infinite supports; this requires an application of infinite-exponent partition relations. All our theorems can be proven from hypotheses weaker than AD.

Some consequences of an infinite-exponent partition relation

1977 ◽  
Vol 42 (4) ◽  
pp. 523-526 ◽  
Author(s):  
J. M. Henle
Keyword(s):  
Set Theory ◽  
Measurable Cardinal ◽  
Large Cardinals ◽  
Axiom Of Choice ◽  
Partition Relation ◽  
Basic Theorem ◽  
Axiom Of Determinacy ◽  
Ramsey’S Theorem ◽  
Partition Relations ◽  
Dependent Choice

Beginning with Ramsey's theorem of 1930, combinatorists have been intrigued with the notion of large cardinals satisfying partition relations. Years of research have established the smaller ones, weakly ineffable, Erdös, Jonsson, Rowbottom and Ramsey cardinals to be among the most interesting and important large cardinals in set theory. Recently, cardinals satisfying more powerful infinite-exponent partition relations have been examined with growing interest. This is due not only to their inherent qualities and the fact that they imply the existence of other large cardinals (Kleinberg [2], [3]), but also to the fact that the Axiom of Determinacy (AD) implies the existence of great numbers of such cardinals (Martin [5]).That these properties are more often than not inconsistent with the full Axiom of Choice (Kleinberg [4]) somewhat increases their charm, for the theorems concerning them tend to be a little odd, and their proofs, circumforaneous. The properties are, as far as anyone knows, however, consistent with Dependent Choice (DC).Our basic theorem will be the following: If k > ω and k satisfies k→(k)k then the least cardinal δ such that has a δ-additive, uniform ultrafilter. In addition, if ACω is assumed, we will show that δ is greater than ω, and hence a measurable cardinal. This result will be strengthened somewhat when we prove that for any k, δ, if then .

scholarly journals Comparisons of Polychromatic and Monochromatic Ramsey Theory

2013 ◽  
Vol 78 (3) ◽  
pp. 951-968 ◽  
Author(s):  
Justin Palumbo
Keyword(s):  
Classical Result ◽  
Ramsey Theory ◽  
Axiom Of Choice ◽  
Ramsey Theorem ◽  
Partition Relations ◽  
Combinatorial Principles ◽  
The Axiom Of Choice ◽  
Relationship Of ◽  
The Relationship ◽  
Special Classes

AbstractWe compare the strength of polychromatic and monochromatic Ramsey theory in several set-theoretic domains. We show that the rainbow Ramsey theorem does not follow from ZF, nor does the rainbow Ramsey theorem imply Ramsey's theorem over ZF. Extending the classical result of Erdős and Rado we show that the axiom of choice precludes the natural infinite exponent partition relations for polychromatic Ramsey theory. We introduce rainbow Ramsey ultrafilters, a polychromatic analogue of the usual Ramsey ultrafilters. We investigate the relationship of rainbow Ramsey ultrafilters with various special classes of ultrafilters, showing for example that every rainbow Ramsey ultrafilter is nowhere dense but rainbow Ramsey ultrafilters need not be rapid. This entails comparison of the polychromatic and monochromatic Ramsey theorems as combinatorial principles on ω.

On large cardinals and partition relations

1971 ◽  
Vol 36 (2) ◽  
pp. 305-308 ◽  
Author(s):  
E. M. Kleinberg ◽  
R. A. Shore
Keyword(s):  
Set Theory ◽  
Measurable Cardinal ◽  
Order Type ◽  
Large Cardinals ◽  
Infinite Cardinal ◽  
Partition Relation ◽  
Weakly Compact ◽  
Original Result ◽  
Partition Relations ◽  
Uncountable Cardinal

A significant portion of the study of large cardinals in set theory centers around the concept of “partition relation”. To best capture the basic idea here, we introduce the following notation: for x and y sets, κ an infinite cardinal, and γ an ordinal less than κ, we let [x]γ denote the collection of subsets of x of order-type γ and abbreviate with the partition relation for each function F frominto y there exists a subset C of κ of cardinality κ such that (such that for each α < γ) the range of F on [С]γ ([С]α) has cardinality 1. Now although each infinite cardinal κ satisfies the relation for each n and m in ω (F. P. Ramsey [8]), a connection with large cardinals arises when one asks, “For which uncountable κ do we have κ → (κ)2?” Indeed, any uncountable cardinal κ which satisfies κ → (κ)2 is strongly inaccessible and weakly compact (see [9]). As another example one can look at the improvements of Scott's original result to the effect that if there exists a measurable cardinal then there exists a nonconstructible set. Indeed, if κ is a measurable cardinal then κ → (κ)< ω, and as Solovay [11] has shown, if there exists a cardinal κ such that κ → (κ)< ω3 (κ → (ℵ1)< ω, even) then there exists a nonconstructible set of integers.

Some properties of large filters

1988 ◽  
Vol 53 (4) ◽  
pp. 1027-1035
Author(s):  
Chris Freiling ◽  
T. H. Payne
Keyword(s):  
Measurable Cardinal ◽  
Axiom Of Choice ◽  
Baire Property ◽  
Index Set ◽  
The Axiom Of Choice

Dowker [1] raised the question of the existence of filters such that for every coloring (partition) of the underlying index set I with two colors there is a relation R on I which (i) is fat (in the sense that sets of the form {y Є I∣xRy} are in the filter) and (ii) has no bichromatic symmetric pairs (i.e., distinct indices x and y such that x R y and y R x). Additionally, he required that the filter have no anti-symmetric fat relation, for such a relation would vacuously satisfy (i) and (ii). The question of the existence of Dowker filters has been studied more recently by Rudin [3], [4], who conjectures [3] that such filters do not exist.For ZFC the problem remains open. However, Example 2 of this paper shows that one can construct a Dowker filter provided one drops the axiom of choice in favor of the Baire Property (BP) axiom which is known to be incompatible with ZFC but relatively consistent with ZF. In fact, the filter constructed is super-Dowker in the sense that (ii) can be replaced by the requirement that all components of all symmetric pairs have the same color. But, in ZFC the existence of a super-Dowker filter implies the existence of a measurable cardinal.Let F be a filter on an index set I. A set will be called big, small, or medium depending on whether F contains that set, its compliment, or neither, respectively. We define five cardinals associated with F:α denotes the smallest cardinal such that there is a family of α big sets whose intersection is not big.ν denotes the smallest cardinal such that there is a family of ν big sets whose intersection is small.

Infinite subscripts from infinite exponents

1984 ◽  
Vol 49 (2) ◽  
pp. 558-562 ◽  
Author(s):  
James E. Baumgartner ◽  
James M. Henle
Keyword(s):  
Axiom Of Choice ◽  
Partition Relations ◽  
The Sixties ◽  
Infinite Case ◽  
The Axiom Of Choice ◽  
The Relationship

Definition. For ordinals α ≤ κ, = [X]α = {p ⊆ X ∣ ot (p) = α}. For α ≤, κ, κ a cardinal, holds iff for every partition F: [Κ]κ → A, there is an X ∈ [Κ]κ with F constant on [X]α. X is called homogeneous for F. When A = 2 the subscript is omitted.It has been known since the sixties that for finite exponents all such properties are equivalent, i.e., κ→ (κ)2 iff for all n < ω, β < κ. The infinite case is much more difficult, though it is not actually known to be different. The usual methods for dealing with partitions fail, and as the weakest of these properties violates the Axiom of Choice, many techniques are unavailable.It is extremely unlikely that an infinite exponent can be increased, but infinite subscripts are generally possible.Theorem. If ∥ α ∥ · 2 ≤ α, then κ → (κ)α implies for all λ < κ.In §1, we show the subscript can be any λ < κ. In §2 we extend this to 2λ. In §3 we discuss the possibilities with limited amounts of well-ordered choice, and apply the theorem to the problem of obliging ordinals. Except in §3, we assume no choice. For the story of finite exponents, see [1]. For the relationship between infinite-exponent partition relations and choice, see [4].

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.

Partitions of products

1993 ◽  
Vol 58 (3) ◽  
pp. 860-871 ◽  
Author(s):  
Carlos A. Di Prisco ◽  
James M. Henle
Keyword(s):  
Finite Number ◽  
Axiom Of Choice ◽  
Inaccessible Cardinal ◽  
Partition Relation ◽  
Partition Relations ◽  
Image Position ◽  
The Axiom Of Choice ◽  
Usual Notation

We will consider some partition properties of the following type: given a function F: ωω →2, is there a sequence H0, H1, … of subsets of ω such that F is constant on ΠiεωHi? The answer is obviously positive if we allow all the Hi's to have exactly one element, but the problem is nontrivial if we require the Hi's to have at least two elements. The axiom of choice contradicts the statement “for all F: ωω→ 2 there is a sequence H0, H1, H2,… of subsets of ω such that {i|(Hi) ≥ 2} is infinite and F is constant on ΠHi”, but the infinite exponent partition relation ω(ω)ω implies it; so, this statement is relatively consistent with an inaccessible cardinal. (See [1] where these partition properties were considered.)We will also consider partitions into any finite number of pieces, and we will prove some facts about partitions into ω-many pieces.Given a partition F: ωω → k, we say that H0, H1…, a sequence of subsets of ω, is homogeneous for F if F is constant on ΠHi. We say the sequence H0, H1,… is nonoverlapping if, for all i ∈ ω, ∪Hi > ∩Hi+1.The sequence 〈Hi: i ∈ ω〉 is of type 〈α0, α1,…〉 if, for every i ∈ ω, ∣Hi∣ = αi.We will adopt the usual notation for polarized partition relations due to Erdös, Hajnal, and Rado.means that for every partition F: κ1 × κ2 × … × κn→δ there is a sequence H0, H1,…, Hn such that Hi ⊂ κi and ∣Hi∣ = αi for every i, 1 ≤ i ≤ n, and F is constant on H1 × H2 × … × Hn.

Ultrapowers without the axiom of choice

1988 ◽  
Vol 53 (4) ◽  
pp. 1208-1219 ◽  
Author(s):  
Mitchell Spector
Keyword(s):  
General Theory ◽  
Set Theory ◽  
Fundamental Theorem ◽  
Axiom Of Choice ◽  
Partition Relations ◽  
Omitting Types ◽  
The Axiom Of Choice ◽  
Almost All ◽  
Models Of Set Theory ◽  
Theory And Model

AbstractA new method is presented for constructing models of set theory, using a technique of forming pseudo-ultrapowers. In the presence of the axiom of choice, the traditional ultrapower construction has proven to be extremely powerful in set theory and model theory; if the axiom of choice is not assumed, the fundamental theorem of ultrapowers may fail, causing the ultrapower to lose almost all of its utility. The pseudo-ultrapower is designed so that the fundamental theorem holds even if choice fails; this is arranged by means of an application of the omitting types theorem. The general theory of pseudo-ultrapowers is developed. Following that, we study supercompactness in the absence of choice, and we analyze pseudo-ultrapowers of models of the axiom of determinateness and various infinite exponent partition relations. Relationships between pseudo-ultrapowers and forcing are also discussed.

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