Infinite Exponent Partition Relations Research Articles

Partitions and conservativity

We study the partition properties enjoyed by the “next best thing to a P-point” ultrafilters introduced recently in joint work with Dobrinen and Raghavan. That work established some finite-exponent partition relations, and we now analyze the connections between these relations for different exponents and the notion of conservativity introduced much earlier by Phillips. In addition, we establish some infinite-exponent partition relations for these ultrafilters and also for sums of non-isomorphic selective ultrafilters indexed by selective ultrafilters.

Comparisons of Polychromatic and Monochromatic Ramsey Theory

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.

Jonsson-like partition relations and j: V → V

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.

The calculus of partition sequences, changing cofinalities, and a question of Woodin

We study in this paper polarized infinite exponent partition relations. We apply our results to constructing a model for the theory “ZF + + DC + ω 1 +\omega _1 is the only regular, uncountable cardinal ≤ ω ω 1 + 1 \le \omega _{\omega _1+1} .” This gives a partial answer to a question of Woodin.

Ultrapowers without the axiom of choice

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.

Ultrapowers without the axiom of choice

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.

Weak partition relations and measurability

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.

Model theory under the axiom of determinateness

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.

Infinite subscripts from infinite exponents

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].

A measure representation theorem for strong partition cardinals

There are two main axiomatic extensions of Zermelo-Fraenkel set theory without the axiom of choice, that associated with the axiom of determinateness, and that associated with infinite exponent partition relations. Initially, the axiom of determinateness, henceforth AD, was the sole tool available. Using it, set theorists in the late 1960s produced many remarkable results in pure set theory (e.g. the measurability of ℵ1) as well as in projective set theory (e.g. reduction principles for ). Infinite exponent partition relations were first studied successfully soon after these early consequences of AD. They too produced measurable cardinals and not only were the constructions here easier than those from AD—the results gave a far clearer picture of the measures involved than had been offered by AD. In general, the techniques offered by infinite exponent partition relations became so attractive that a great deal of the subsequent work from AD involved an initial derivation from AD of the appropriate infinite exponent partition relation and then the derivation from the partition relation of the desired result.Since the early 1970s work on choiceless extensions of ZF + DC has split mainly between AD and its applications to projective set theory, and infinite exponent partition relations and their applications to pure set theory. There has certainly been a fair amount of interplay between the two, but for the most part the theories have been pursued independently.Unlike AD, infinite exponent partition relations have shown themselves amenable to nontrivial forcing arguments. For example, Spector has constructed models for interesting partition relations, consequences of AD, in which AD is false. Thus AD is a strictly stronger assumption than are various infinite exponent partition relations. Furthermore, Woodin has recently proved the consistency of infinite exponent partition relations relative to assumptions consistent with the axiom of choice, in particular, relative to the existence of a supercompact cardinal. The notion of doing this for AD is not even considered.

A measurable cardinal with a nonwellfounded ultrapower

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].

On the compactness of ℵ1 and ℵ2

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.

Some consequences of an infinite-exponent partition relation

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 .

Adding a closed unbounded set

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].)

Adding a closed unbounded set

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].)

Infinite exponent partition relations and well-ordered choice

The study of partition relations for cardinal numbers introduced by Erdös and his school in the 1950's has, for the past several years, had a profound impact in logic. Unfortunately, quite early in their development, it was noticed by Rado [1] that the potentially most fruitful class of such relations, infinite exponent partition relations, were always in contradiction with the axiom of choice (AC). As a result, such relations were overlooked. This turned out to be a mistake; for, as has been noticed recently, a close study of infinite exponent partition relations is both interesting and rewarding. For example, there are weakened versions of such relations which are provable in ZF and which have valuable applications in recursion theory and set theory. In addition, the pure theory of these relations, like that of the axiom of determinateness, is fruitful as well as elegant. For more background here one should refer to [3].At any rate, with a more detailed look at infinite exponent partition relations came a more refined version of Rado's original theorem. Specifically, Rado used the full axiom of choice to carefully construct partitions to violate any desired relation—a more sophisticated look at the actual theory of such relations indicated how one could put together some desired partitions using only well-ordered choice [3].The distinction between well-ordered choice and full choice is by no means vacuous in this context. For Mathias has shown [4] that the simplest infinite exponent partition relation, ω → (ω)ω, is consistent with countable choice (well-ordered choice of length ℵ0) and, in fact, is consistent with dependent choice.