An ℵ1-dense ideal on ℵ2

This paper establishes the consistency of a countably complete, uniform, ℵ₁-dense ideal on ℵ₂. As a corollary, it is consistent that there exists a uniform ultrafilterD on ω₂ such that |ω ₁ ^ω2 D|=ω₁. A general “transfer” result establishes the consistency of countably complete uniform ideal K on ω₂ such thatP(ω₂)/K≅P(ω₁)/ {countable sets}. Partially supported by an NSF grant.     

Non-special Aronszajn trees on ℵω+1

We continue our research on the relative strength ofL-like combinatorial principles for successors of singular cardinals. In [3] we have shown that the existence of a λ⁺-special Aronszajn tree does not follow from that of a λ⁺-Souslin tree. It follows from [5], [4] and [6] that under G.C.H. □_λ does imply the existence of a λ⁺-Souslin tree. In [2] we show that □_λ does not follow from the existence of a λ⁺-special Aronszajn tree. Here we show that the existence of such a tree implies that of an ‘almost Souslin’ λ⁺-tree. It follows that the statement “All λ⁺-Aronszajn trees are special” implies that there are no λ⁺-Aronszajn trees.     

Instances of the Conjecture of Chang

We consider various forms of the Conjecture of Chang. Part A constitutes an introduction. Donder and Koepke have shown that if ρ is a cardinal such that ρ ≧ ω₁, and (ρ⁺⁺,ρ⁺↠(ρ⁺, ρ), then 0⁺ exists. We obtain the same conclusion in Part B starting from some other forms of the transfer hypothesis. As typical corollaries, we get: Theorem A.Assume that there exists cardinals λ, κ, such that λ ≧ K ⁺ ≧ω₂ and (λ⁺, λ)↠(K ⁺,K. Then 0⁺ exists. Theorem B.Assume that there exists a singularcardinal κ such that(K ⁺,K↠(ω₁, ω₀. Then 0⁺ exists. Theorem C.Assume that (λ ⁺⁺, λ). Then 0⁺ exists (also ifK=ω ₀. Remark. Here, as in the paper of Donder and Koepke, “O⁺ exists” is a matter of saying that the hypothesis is strictly stronger than “L(μ) exists”. Of course, the same proof could give a few more sharps overL(μ), but the interest is in expecting more cardinals, coming from a larger core model. Theorem D.Assume that (λ ⁺⁺, λ)↠(K ⁺, K) and thatK≧ω ₁. Then 0⁺ exists. Remark 2. Theorem B is, as is well-known, false if the hypothesis “κ is singular” is removed, even if we assume thatK≧ω ₂, or that κ is inaccessible. We shall recall this in due place. Comments. Theorem B and Remark 2 suggest we seek the consistency of the hypothesis of the form:K ⁺, K↠(ω_n +1, ω_n), for κ singular andn≧0. 0266 0152 V 3 The consistency of several statements of this sort—a prototype of which is (N _ω+1,N _ω)↠(ω₁, ω₀) —have been established, starting with an hypothesis slightly stronger than: “there exists a huge cardinal”, but much weaker than: “there exists a 2-huge cardinal”. These results will be published in a joint paper by M. Magidor, S. Shelah, and the author of the present paper.     

Violating the singular cardinals hypothesis without large cardinals

We extend a transitive model V of ZFC+GCH cardinal preservingly to a model N of ZF + “GCH holds below ℵ _ω ” + “there is a surjection from the power set of ℵ _ω onto λ”, where λ is an arbitrarily high fixed cardinal in V. The construction can be described as follows: add ℵ _n +1 many Cohen subsets of ℵ _n+1 for every n < ω, and adjoin λ many subsets of ℵ _ω which are unions of ω-sequences of those Cohen subsets; then let N be a choiceless submodel generated by equivalence classes of the λ subsets of ℵ _ω modulo an appropriate equivalence relation.     

Iterated forcing and normal ideals onω 1

We prove that suitable iteration does not collapse ℵ₁ [and does not add reals], i.e., that in such iteration, certain sealing of maximal antichains of stationary subsets ofω ₁ is allowed. As an application, e.g., we prove from supercompact hypotheses, mainly, the consistency of: ZFC + GCH + “for some stationary setS ⊆ω ₁, {ie345-1}(ω ₁)/(D ω ₁ +S) is the Levy algebra” (i.e., the complete Boolean Algebra corresponding to the Levy collapse Levy (ℵ₀,<ℵ₂) (and we can add “a variant of PFA”) and the consistency of the same, with “Ulam property” replacing “Levy algebra”). The paper assumes no specialized knowledge (if you agree to believe in the semi-properness iteration theorem and RCS iteration). This research was partially supported by the NSF. This paper was largely written during the author’s visit at Cal Tech around the end of April 1985. The author would like to thank M. Foreman, A. Kekris and H. Woodin for their hospitality.     

Strongly almost (<Emphasis Type="Italic">ω, λ, q</Emphasis>)-summable sequences

The purpose of this paper is to introduce the spaces of sequences that are strongly almost (ω, λ, q)-summable with respect to a modulus function. We give some relations related to these sequence spaces. It is also shown that if a sequence is strongly (ω, λ, q)-summable with respect to a modulus function, then it is S(λ _q )-statistically convergent.     

Separating ultrafilters on uncountable cardinals

A uniform ultrafilterU on κ is said to be λ-separating if distinct elements of the ultrapower never projectU to the same uniform ultrafilterV on λ. It is shown that, in the presence of CH, an ω-separating ultrafilterU on κ>ω is non-(ω, ω₁)-regular and, in fact, if κ < ℵ_ω thenU is λ-separating for all λ. Several large cardinal consequences of the existence of such an ultrafilterU are derived.     

Iterated forcing in quadratic form theory

In [Sp1] and [B/Sp] it has been shown that the existence of quadratic spaces of uncountable dimension over finite or countable fields sharing the property that every infinite dimensional subspace has its orthogonal complement of at most countable dimension is independent of the axioms of ZFC set theory. Such a space will be called astrong Gross space in the sequel. Cardinal invariants of the continuum decide whether strong Gross spaces exist or not. Namely, when b=ω₁ a strong Gross space of dimension ℵ₁ exists. When p>ω₁ such spaces do not exist. Here we answer the question what happens with strong Gross spaces in case b>ω₁ or p=ω₁. This work forms part of the author’s Habilitationsschrift at the ETH Zürich. The author is supported by the Basic Research Foundation of the Israel Academy of Sciences.     

Invariants forω-categorical,ω-stable theories

In this paper we give a complete solution to the classification problem forω-categorical,ω-stable theories. More explicitly, supposeT isω-categorical,ω-stable with fewer than the maximum number of models in some uncountable power. We associate with each modelM ofT a “simple” invariantI(M), not unlike a vector of dimensions, such thatI(M)=I(N) if and only ifM≅N. The spectrum function,I(−,T), for a first-order theoryT is such that for all infinite cardinals λ,I(λ,T) is the number of nonisomorphic models ofT of cardinality λ. As an application of our “structure theorem” we determine the possible spectrum functions forω-categorical,ω-stable theories.     

Prime ideals on P ω (λ) with the partition property

 We use ideas of Fred Galvin to show that under Martin's axiom, there is a prime ideal on P _ω (λ) with the partition property for every . Received: 6 May 1999 / Revised version: 15 October 2001 Published online: 2 September 2002     

Spectral structure of Anderson type Hamiltonians

We study self adjoint operators of the form?H _ω = H ₀ + ∑λ_ω(n) <δ _n ,·>δ _n ,?where the δ _n ’s are a family of orthonormal vectors and the λ_ω(n)’s are independently distributed random variables with absolutely continuous probability distributions. We prove a general structural theorem saying that for each pair (n,m), if the cyclic subspaces corresponding to the vectors δ _n and δ _m are not completely orthogonal, then the restrictions of H _ω to these subspaces are unitarily equivalent (with probability one). This has some consequences for the spectral theory of such operators. In particular, we show that “well behaved” absolutely continuous spectrum of Anderson type Hamiltonians must be pure, and use this to prove the purity of absolutely continuous spectrum in some concrete cases. Oblatum 27-V-1999 & 6-I-2000?Published online: 8 May 2000     

More on countably compact,locally countable spaces

Following [5], aT ₃ spaceX is called good (splendid) if it is countably compact, locally countable (andω-fair).G(κ) (resp.S(κ)) denotes the statement that a good (resp. splendid) spaceX with |X|=κ exists. We prove here that (i) Con(ZF)→Con(ZFC+MA+2 ^ω is big+S(κ) holds unlessω=cf(κ)<κ); (ii) a supercompact cardinal implies Con(ZFC+MA+2suω>_ω+₁+┐G(ω_ω+₁); (iii) the “Chang conjecture” (ω_ω+₁),→(ω ₁,ω) implies ┐S(κ) for allκ≧k≧ω_ω; (iv) ifP addsω ₁ dominating reals toV iteratively then, in , we haveG(λ^ω) for allλ. Research supported by Hungarian National Foundation for Scientific Research grant no. 1805.     

ω 1 can be measurable

It is shown that ifZF + the axiom of choice + “there is a measurable cardinal” is consistent thenZF + “ω ₁ is measurable” is consistent. The corresponding model is a symmetric submodel of the Cohen-type extension which collapses the first measurable cardinal onto ω₀.     

Imbedding theorems of Besov spaces in Banach lattices

The paper examines imbeddings of Besov spaces B _{E, θ} ^ω in ideal spaces (Banach lattices) given that ω ∈ S_k). In particular, the symmetric hull of the space B _{E, θ} ^ω is described (E is a symmetric space), an inequality of different metrics is obtained, and imbeddings in Orlicz and Lorentz spaces and in some weighted spaces are studied. Most of the results are easily extended to the anisotropic case. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 159, pp. 69–82, 1987.     

On normal precipitous ideals

An old question of T. Jech and K Prikry asks whether the existence of a precipitous ideal implies the existence of a normal precipitous one. The aim of the paper is to prove some results in the positive direction. In particular, it is shown that under some mild assumptions, the existence of a precipitous ideal over ℵ₁ implies the existence of a normal precipitous ideal over ℵ₁ once a Cohen subset is added to ℵ₂.     

Periodic solutions of “predator–prey” systems with continuous delay and periodic coefficients

We prove the existence of positive ω-periodic solutions for some “predator–prey” systems with continuous delay of the argument for the case where the parameters of these systems are specified by ω-periodic continuous positive functions.     

An extension theorem for separable Banach spaces

We construct a totally disconnected ω^*, norming subsetF of the unit ballB ^* of an arbitrary separable Banach space,X, and an operator fromC(F) toC(B^*) that “amost” commutes with the natural embeddings ofX. This is used to give a new proof of Milutin's theorem and to prove some new results on complemented subspaces ofC[0, 1] with separable dual. In particular we show that a complemented subspace ofC(ω^ω), is either isomorphic toC(ω^ω) or toc _u.     

Points de namioka espaces normants applications à la théorie isométrique de la dualité

We study in this work various aspects of the isometric theory of duality. We show that in wide classes of Banach spaces, dual spaces are characterized by the existence of a retraction fromE″ ontoE. The predual of such spaces is then unique. We study the imbedding of regularly normed spaces into dual spaces. We better the known results on loss of regularity of the norm of dual spaces. We characterize the dual norms on an Asplund space in terms of “bad differentiability”.      

The continuity of superposition operators on some sequence spaces defined by moduli

Let λ and μ be solid sequence spaces. For a sequence of modulus functions Φ = (ϕ k) let λ(Φ) = {x = (x _k ): (ϕk(|x _k |)) ∈ λ}. Given another sequence of modulus functions Ψ = (ψ_k), we characterize the continuity of the superposition operators P _f from λ(Φ) into μ (Ψ) for some Banach sequence spaces λ and μ under the assumptions that the moduli ϕk (k ∈ ℕ) are unbounded and the topologies on the sequence spaces λ(Φ) and μ(Ψ) are given by certain F-norms. As applications we consider superposition operators on some multiplier sequence spaces of Maddox type. This research was supported by Estonian Science Foundation Grant 5376.     

Subspaces ofL p (G) spanned by characters: 0<p<1

LetG be an infinite compact abelian group,μ a Borel measure onG with spectrumE, and 0<p<1. We show that ifμ is not absolutely continuous with respect to Haar measure, thenL _E ^P (G), the closure inL ^p (G) of theE-trigonometric polynomials, does not have enough continuous linear functionals to separate points. Ifμ is actually singular, thenL _E ^p (G) does not have any nontrivial continuous linear functionals at all. Our methods recover the classical F. and M. Riesz theorem, and a related several variable result of Bochner; they reveal the existence of small sets of characters that spanL ^P (T), where T is the unit circle; and they show that theH ^p spaces of the “big disc algebra” have one-dimensional dual.