All ultrafilters under consideration here are non-principal ultrafilters on the set omega of natural numbers. We are concerned with the possible cofinalities of ultrapowers of omega with respect to such ultrafilters. We show that no cardinal below the groupwise density number g can occur as such a cofinality and that at most one cardinal below the splitting number s can so occur. The proof for s, when combined with a result of Nyikos, gives the additional information that all P_{kappa}-point ultrafilters, for kappa greater than the bounding number b, are nearly coherent.
We prove that consistently there is a singular cardinal $κ$ of uncountable cofinality such that $2^κ$ is weakly inaccessible, and every regular cardinal strictly between $κ$ and $2^κ$ is the character of some uniform ultrafilter on $κ$.
In this paper we analyse and compare two different notions of regularity for filters on complete Boolean algebras. We also announce two results from a forthcoming paper in preparation, which provide a characterization of Keisler's order in terms of Boolean ultrapowers.
We study the structure of the Rudin-Frolik order on countably complete ultrafilters under the assumption that this order is directed. This assumption, called the Ultrapower Axiom, holds in all known canonical inner models. It turns out that assuming the Ultrapower Axiom, much more about the Rudin-Frolik order can be determined. Our main theorem is that under the Ultrapower Axiom, a countably complete ultrafilter has at most finitely many predecessors in the Rudin-Frolik order. In other words, any wellfounded ultrapower (of the universe) is the ultrapower of at most finitely many ultrapowers.
We show that while the length $ω$ iterated ultrapower by a normal ultrafilter is a Boolean ultrapower by the Boolean algebra of Prikry forcing, it is consistent that no iteration of length greater than $ω$ (of the same ultrafilter and its images) is a Boolean ultrapower. For longer iterations, where different ultrafilters are used, this is possible, though, and we give Magidor forcing and a generalization of Prikry forcing as examples. We refer to the discovery that the intersection of the finite iterates of the universe by a normal measure is the same as the generic extension of the direct li…
In this paper, we characterize the possible cofinalities of the least $λ$-strongly compact cardinal. We show that, on the one hand, for any regular cardinal, $δ$, that carries a $λ$-complete uniform ultrafilter, it is consistent, relative to the existence of a supercompact cardinal above $δ$, that the least $λ$-strongly compact cardinal has cofinality $δ$. On the other hand, provably the cofinality of the least $λ$-strongly compact cardinal always carries a $λ$-complete uniform ultrafilter.
Assuming an abstract comparison principle called the Ultrapower Axiom, which is motivated by the comparison process of inner model theory and generalizes the statement that the Mitchell order is linear on normal ultrafilters, we characterize strongly compact cardinals in terms of supercompactness: a strongly compact cardinal is either supercompact or a limit of supercompact cardinals. Assuming the Ultrapower Axiom and the GCH, we also prove a local result that roughly states that every countably complete ultrafilter factors as a finite iteration of supercompact ultrafilters.
Various questions posed by P. Nyikos concerning ultrafilters on $ω$ and chains in the partial order $(ω,<^*)$ are answered. The main tool is the oracle chain condition and variations of it.
Let $B(H)$ be the algebra of bounded linear operators on a separable infinite-dimensional Hilbert space $H$. We study the commutant of $B(H)$ in its ultrapower. We characterize the class of non-principal ultrafilters for which this commutant is non-trivial. Additionally, we extend the class of ultrafilters for which the commutant is trivial.
We show from a weak comparison principle (the Ultrapower Axiom) that the Mitchell order is linear on certain kinds of ultrafilters: normal ultrafilters, Dodd solid ultrafilters, and assuming GCH, generalized normal ultrafilters. In the process we prove a generalization of Solovay's lemma to singular cardinals.
In standard construction of hyperrational numbers using an ultrapower we assume that the ultrafilter is selective. It makes possible to assign real value to any finite hyperrational number. So, we can consider hyperrational numbers with selective ultrafilter as extension of traditional real numbers. Also proved the existence of strictly monotonic or stationary representing sequence for any hyperrational number.
In the first edition of Classification Theory, the second author characterized the stable theories in terms of saturation of ultrapowers. Prior to this theorem, stability had already been defined in terms of counting types, and the unstable formula theorem was known. A contribution of the ultrapower characterization was that it involved sorting out the global theory, and introducing nonforking, seminal for the development of stability theory. Prior to the present paper, there had been no such characterization of an unstable class. In the present paper, we first establish the existence of so-ca…
The splitting number can be singular. The key method is to construct a forcing poset with finite support matrix iterations of ccc posets introduced by Blass and the second author "Ultrafilters with small generating sets", Israel J. Math., 65, (1989)
We prove a strong dichotomy for the number of ultrapowers of a given countable model associated with nonprincipal ultrafilters on N. They are either all isomorphic, or else there are $2^{2^{\aleph_0}}$ many nonisomorphic ultrapowers. We prove the analogous result for metric structures, including C*-algebras and II$_1$ factors, as well as their relative commutants and include several applications. We also show that the C*-algebra B(H) always has nonisomorphic relative commutants in its ultrapowers associated with nonprincipal ultrafilters on N.
We show that the character spectrum $Sp_χ(λ)$, for a singular cardinal $λ$ of countable cofinality, may include any prescribed set of regular cardinals between $λ$ and $2^λ$.
We study various orders on countably complete ultrafilters on ordinals that coincide and are wellorders under a hypothesis called the Ultrapower Axiom. Our main focus is on the relationship between the Ultrapower Axiom and the linearity of these orders.
Boolean ultrapowers extend the classical ultrapower construction to work with ultrafilters on any complete Boolean algebra, rather than only on a power set algebra. When they are well-founded, the associated Boolean ultrapower embeddings exhibit a large cardinal nature, and the Boolean ultrapower construction thereby unifies two central themes of set theory---forcing and large cardinals---by revealing them to be two facets of a single underlying construction, the Boolean ultrapower.
Our results in this paper increase the model-theoretic precision of a widely used method for building ultrafilters, and so advance the general problem of constructing ultrafilters whose ultrapowers have a precise degree of saturation. We begin by showing that any flexible regular ultrafilter makes the product of an unbounded sequence of finite cardinals large, {thus} saturating any stable theory. We then prove directly that a "bottleneck" in the inductive construction of a regular ultrafilter on $λ$ (i.e. a point after which all antichains of $P(λ)/D$ have cardinality less than $λ$) essentiall…
We prove that there exists a nonprincipal ultrafilter $\mathcal U$ on $\mathbb N$ such that for every countable (or separable) structure $B$ in a countable language the quotient map from the reduced product associated with the Fréchet filter onto the ultrapower has a right inverse. The proof uses the Continuum Hypothesis. We characterize the ultrafilters $\mathcal U$ with this property, and show that consistently with ZFC such ultrafilters need not exist. We also prove a similar ZFC result sufficiently strong to obtain all concrete applications of the existence of a right inverse to the quotie…
The logic L^1_θintroduced in [Sh:797]; it is the maximal logic below L_theta theta in which a well ordering is not definable. We investigate it for theta a compact cardinal. We prove it satisfies several parallel of classical theorems on first order logic, strengthening the thesis that it is a natural logic. In particular, two models are L^1_theta-equivalent iff for some omega-sequence of theta-complete ultrafilters, the iterated ultra-powers by it of those two models are isomorphic.
In earlier work of the second and third author the equivalence of a finite square principle square^fin_{lambda,D} with various model theoretic properties of structures of size lambda and regular ultrafilters was established. In this paper we investigate the principle square^fin_{lambda,D}, and thereby the above model theoretic properties, at a regular cardinal. By Chang's Two-Cardinal Theorem, square^fin_{lambda,D} holds at regular cardinals for all regular filters D if we assume GCH. In this paper we prove in ZFC that for certain regular filters that we call "doubly^+ regular", square^fin_{la…
The following pcf results are proved: 1. Assume that kappa > aleph_0 is a weakly compact cardinal. Let mu > 2^kappa be a singular cardinal of cofinality kappa. Then for every regular lambda < pp^+_{Gamma(kappa)} (mu) there is an increasing sequence (lambda_i | i < kappa) of regular cardinals converging to mu such that lambda = tcf(prod_{i < kappa} lambda_i, <_{J^{bd}_kappa}). 2. Let mu be a strong limit cardinal and theta a cardinal above mu. Suppose that at least one of them has an uncountable cofinality. Then there is sigma_* < mu such that for every chi < theta the following holds: theta > …
For an ultrafilter $D$ on a cardinal $κ,$ we wonder for which pair $(θ_1, θ_2)$ of regular cardinals, we have: for any $(θ_1+θ_2)^+-$saturated dense linear order $J, J^κ/ D$ has a cut of cofinality $(θ_1, θ_2).$ We deal mainly with the case $θ_1, θ_2 > 2^κ.$
A construction of the real number system based on almost homomorphisms of the integers Z was proposed by Schanuel, Arthan, and others. We combine such a construction with the ultrapower or limit ultrapower construction, to construct the hyperreals out of integers. In fact, any hyperreal field, whose universe is a set, can be obtained by such a one-step construction directly out of integers. Even the maximal (i.e., On-saturated) hyperreal number system described by Kanovei and Reeken (2004) and independently by Ehrlich (2012) can be obtained in this fashion, albeit not in NBG. In NBG, it can be…
We prove that for every singular cardinal mu of cofinality omega, the complete Boolean algebra compP_mu(mu) contains as a complete subalgebra an isomorphic copy of the collapse algebra Comp Col(omega_1,mu^{aleph_0}). Consequently, adding a generic filter to the quotient algebra P_mu(mu)=P(mu)/[mu]^{<mu} collapses mu^{aleph_0} to aleph_1. Another corollary is that the Baire number of the space U(mu) of all uniform ultrafilters over mu is equal to omega_2. The corollaries affirm two conjectures by Balcar and Simon. The proof uses pcf theory.
This paper introduces the seed order, a partial order of the class of uniform countably complete ultrafilters that generalizes the Mitchell order on normal measures. Like that order, the seed order is consistently a linear ordering even under strong large cardinal assumptions. In fact, the linearity of the seed order is a feature of all known canonical inner models for large cardinal axioms. We develop the theory of the seed order under the assumption that it is linear. Augmented by large cardinal hypotheses currently out of reach of inner model theory (namely supercompactness), this linearity…
This paper establishes a number of constraints on the structure of large cardinals under strong compactness assumptions. These constraints coincide with those imposed by the Ultrapower Axiom, a principle that is expected to hold in Woodin's hypothesized Ultimate \(L\), providing some evidence for the Ultimate \(L\) Conjecture. We show that every regular cardinal above the first strongly compact that carries an indecomposable ultrafilter is measurable, answering a question of Silver for large enough cardinals. We show that any successor almost strongly compact cardinal of uncountable cofinality…
We continue our study of the class $\mathscr{C}(D)$, where $D$ is a uniform ultrafilter on a cardinal $κ$ and $\mathscr{C}(D)$ is the class of all pairs $(θ_1, θ_2),$ where $(θ_1, θ_2)$ is the cofinality of a cut in $J^κ/D$ and $J$ is some $(θ_1+θ_2)^+$-saturated dense linear order. We give a combinatorial characterization of the class $\mathscr{C}(D)$. We also show that if $(θ_1, θ_2) \in \mathscr{C}(D)$ and $D$ is $\aleph_1$-complete or $θ_1 + θ_2 > 2^κ,$ then $θ_1=θ_2.$
