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.     

A superatomic Boolean algebra with few automorphisms

Assuming GCH, we prove that for every successor cardinal μ > ω₁, there is a superatomic Boolean algebra B such that |B| = 2^μ and |Aut B| = μ. Under ◊_ω1, the same holds for μ = ω₁. This answers Monk's Question 80 in [Mo]. Received: 1 January 1998 / Revised version: 18 May 1999 / Published online: 21 December 2000     

Universal WCG banach spaces and universal Eberlein Compacts

We prove that for cardinalsτ satisfying τ^ω=τ and forτ=ω ₁, there do not exist universal Eberlein Compacts of weightτ, or universal WCG spaces of density characterτ. Ifτ is a strong limit cardinal of countable cofinality such universal spaces do exist. Thus under GCH universal spaces exist forτ iff cof(τ)=ω. The research of the second author was supported by a grant from the United States-Israel Binational Science Foundation and the Fund for the Promotion of Research at the Technion.     

A consistency result on cardinal sequences of scattered Boolean spaces

We prove that if GCH holds and τ = 〈κ_α : α < η 〉 is a sequence of infinite cardinals such that κ_α ≥ |η | for each α < η, then there is a cardinal‐preserving partial order that forces the existence of a scattered Boolean space whose cardinal sequence is τ. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A gap 1 cardinal transfer theorem

We extend the gap 1 cardinal transfer theorem (κ ⁺, κ ) → (λ ⁺, λ ) to any language of cardinality ≤λ, where λ is a regular cardinal. This transfer theorem has been proved by Chang under GCH for countable languages and by Silver in some cases for bigger languages (also under GCH). We assume the existence of a coarse (λ, 1)‐morass instead of GCH. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On some questions concerning strong compactness

A question of Woodin asks if κ is strongly compact and GCH holds below κ, then must GCH hold everywhere? One variant of this question asks if κ is strongly compact and GCH fails at every regular cardinal δ?< κ, then must GCH fail at some regular cardinal δ?≥ κ? Another variant asks if it is possible for GCH to fail at every limit cardinal less than or equal to a strongly compact cardinal κ. We get a negative answer to the first of these questions and positive answers to the second of these questions for a supercompact cardinal κ in the context of the absence of the full Axiom of Choice.     

Lebesgue constants of Gaussian cardinal interpolation operators from l (ℤ) into L (ℝ)

Summary In recent years, with the attention to the radial-basis function by mathematicians, more and more research is concentrated on the Gaussian cardinal interpolation. The main purpose of this paper is to discuss the asymptotic behavior of Lebesgue constants of the Gaussian cardinal interpolation operator ℒ_λ from l (ℤ) into L (ℝ), that is, ∥ℒ_λ∥₁. We obtain the strong asymptotic estimate of the Lebesgue constants which improves the results of Riemenschneider and Sivakumar in [11].     

Σ 3 1 absoluteness and the second uniform indiscernible

We show that that if every real has a sharp and there are Δ ₂ ¹ -definable prewellorderings of ℝ of ordinal ranks unbounded inω ₂, then there is an inner model for a strong cardinal. Similarly, assuming the same sharps, the Core ModelK is Σ ₃ ¹ -absolute unless there is an inner model for a strong cardinal.     

Exponents of someN-compact spaces

For any topological spaceT, S. Mrówka has defined Exp (T) to be the smallest cardinal κ (if any such cardinals exist) such thatT can be embedded as a closed subset of the productN ^κ of κ copies ofN (the discrete space of cardinality ℵ₀). We prove that forQ, the space of the rationals with the inherited topology, Exp (Q) is equal to a certain covering number, and we show that by modifying some earlier work of ours it can be seen that it is consistent with the usual axioms of set theory including the choice that this number equal any uncountable regular cardinal less than or equal to 2^ℵ ₀. Mrówka has also defined and studied the class ℳ={κ: Exp (N _κ)=κ} whereN _κ is the discrete space of cardinality κ. It is known that the first cardinal not in ℳ must not only be inaccessible but cannot even belong to any of the first ω Mahlo classes. However, it is not known whether every cardinal below 2^ℵ ₀ is contained in ℳ. We prove that if there exists a maximal family of almost-disjoint subsets ofN of cardinality κ, then κ∈ℳ, and we then use earlier work to prove that if it is consistent that there exist cardinals which are not in the first ω Mahlo classes, then it is consistent that there exist such cardinals below 2^ℵ ₀ and that ℳ nevertheless contain all cardinals no greater than 2^ℵ ₀. Finally, we consider the relationship between ℳ and certain “large cardinals”, and we prove, for example, that if μ is any normal measure on a measurable cardinal, then μ(ℳ)=0.     

On a Chang conjecture

We use the core model for the one strong cardinal to show that the Chang Conjecture (ℵ _n+2, ℵ _n+1) ⇒ (ℵ _n+1, ℵ _n ) together with 2^ℵ _{n − 1} = ℵ _n implies, for 1<n<ω, the existence of an inner model with a strong cardinal. An essential step of our proof is an application of the Gitik Game which also admits a presentation.     

Some results on consecutive large cardinals II: Applications of radin forcing

Letκ be a 3 huge cardinal in a countable modelV of ZFC, and letA andB be subsets of the successor ordinals <κ so thatA ⋃B={α<κ:α is a successor ordinal}. Using techniques of Gitik, we construct a choiceless modelN _A of ZF of heightκ so thatN _A ╞“ZF+⌍AC _ω+Forα ∈A, ℵ_a is a Ramsey cardinal+Forβ ∈B, ℵ_β is a singular Rowbottom cardinal which carries a Rowbottom filter+Forγ a limit ordinal, ℵ_y is a Jonsson cardinal which carries a Jonsson filter”. The author wishes to express his thanks to the Rutgers Research Council for a Summer Research Fellowship which partially supported this work. The author also wishes to thank Moti Gitik and Bob Mignone for their useful comments concerning the subject matter of this paper.     

A broader context for monotonically monolithic spaces

This is a sequel of the work done on (strongly) monotonically monolithic spaces and their generalizations. We introduce the notion of monotonically κ-monolithic space for any infinite cardinal κ and present the relevant results. We show, among other things, that any σ-product of monotonically κ-monolithic spaces is monotonically κ-monolithic for any infinite cardinal κ; besides, it is consistent that any strongly monotonically ω-monolithic space with caliber ω ₁ is second countable. We also study (strong) monotone κ-monolithicity in linearly ordered spaces and subspaces of ordinals.     

On the class of measurable cardinals without the axiom of choice

Using techniques of Gitik in conjunction with a large cardinal hypothesis whose consistency strength is strictly in between that of a supercompact and an almost huge cardinal, we obtain the relative consistency of the theory “ZF+⇁AC _w+κ>ω is measurable iffκ is the successor of a singular cardinal”.     

The extent of strength in the club filters

In this paper we consider whether the minimal normal filter onP _κλ, the club filter, can have strong properties like saturation, pre-saturation, or cardinal preserving. We prove in a number of cases that the answer is no. In the case of saturation, Foreman and Magidor prove the answer is always no (except in the caseκ =λ = ℵ₁—and in this case saturation is known to be consistent). The first author was partially supported by NSF grant DMS-9626212.     

More on powers of singular cardinals

We give bounds for where cfδ=ℵ₁, (∀a<δ) , in cases which previously remained opened, including the first such cardinal: theω ₁-th cardinal inC _ω=∩n<ω C _n whereC ₀ is the cardinal andC _n+1 the set of fixed points ofC _n. No knowledge of earlier results is required. A subsequent work generalizing this was applied to many more cardinals ([Sh 7]). The author would like to thank the Canadian NSERC for supporting this research by Grant A3040 and the Israel Academy of Science for supporting it.     

Continuous tree-like scales

Answering a question raised by Luis Pereira, we show that a continuous tree-like scale can exist above a supercompact cardinal. We also show that the existence of a continuous tree-like scale at ℵ _ω is consistent with Martin’s Maximum.     

Antichains in partially ordered sets of singular cofinality

In their paper from 1981, Milner and Sauer conjectured that for any poset , if , then P must contain an antichain of size κ. We prove that for λ > cf(λ) = κ, if there exists a cardinal μ < λ such that cov(λ, μ, κ, 2) = λ, then any poset of cofinality λ contains λ ^κ antichains of size κ. The hypothesis of our theorem is very weak and is a consequence of many well-known axioms such as GCH, SSH and PFA. The consistency of the negation of this hypothesis is unknown.      

The size of epimorphic extensions

A sharp bound is given for the size of epimorphic extensions in categories of models defined over elementary logic andL _κκ where κ is strongly compact. For fragments ofL _ω1ω an example is given of a category which has a countable model with epimorphic extensions whose cardinalities approach and include the first measurable cardinal. If no measurable cardinal exists then this category has a countable model with epimorphic extensions of unbounded cardinality. This work was supported in part by the National Research Council of Canada under grant numbers A8599, A5603 and A8190. Presented by J. D. Monk.     

The <Emphasis Type="Italic">p</Emphasis>-rank of Ext<Subscript>ℤ</Subscript>(<Emphasis Type="Italic">G</Emphasis>, ℤ) in certain models of <Emphasis Type="Italic">ZFC</Emphasis>

We prove that if the existence of a supercompact cardinal is consistent with ZFC, then it is consistent with ZFC that the p-rank of Ext _ℤ(G, ℤ) is as large as possible for every prime p and for any torsion-free Abelian group G. Moreover, given an uncountable strong limit cardinal μ of countable cofinality and a partition of Π (the set of primes) into two disjoint subsets Π₀ and Π₁, we show that in some model which is very close to ZFC, there is an almost free Abelian group G of size 2^μ = μ⁺ such that the p-rank of Ext _ℤ(G, ℤ) equals 2^μ = μ⁺ for every p ∈ Π₀ and 0 otherwise, that is, for p ∈ Π₁. Number 874 in Shelah’s list of publications. Supported by the German-Israeli Foundation for Scientific Research & Development project No. I-706-54.6/2001. Supported by a grant from the German Research Foundation DFG. __________ Translated from Algebra i Logika, Vol. 46, No. 3, pp. 369–397, May–June, 2007.     

Indestructibility and measurable cardinals with few and many measures

If κ < λ are such that κ is indestructibly supercompact and λ is measurable, then we show that both A = {δ < κ | δ is a measurable cardinal which is not a limit of measurable cardinals and δ carries the maximal number of normal measures} and B = {δ < κ | δ is a measurable cardinal which is not a limit of measurable cardinals and δ carries fewer than the maximal number of normal measures} are unbounded in κ. The two aforementioned phenomena, however, need not occur in a universe with an indestructibly supercompact cardinal and sufficiently few large cardinals. In particular, we show how to construct a model with an indestructibly supercompact cardinal κ in which if δ < κ is a measurable cardinal which is not a limit of measurable cardinals, then δ must carry fewer than the maximal number of normal measures. We also, however, show how to construct a model with an indestructibly supercompact cardinal κ in which if δ < κ is a measurable cardinal which is not a limit of measurable cardinals, then δ must carry the maximal number of normal measures. If we weaken the requirements on indestructibility, then this last result can be improved to obtain a model with an indestructibly supercompact cardinal κ in which every measurable cardinal δ < κ carries the maximal number of normal measures. A. W. Apter’s research was partially supported by PSC-CUNY grants and CUNY Collaborative Incentive grants. In addition, the author wishes to thank the referee, for helpful comments, corrections, and suggestions which have been incorporated into the current version of the paper.