On Jónsson cardinals with uncountable cofinality

For a regular cardinal κ a Jónsson model of size κ⁺ is presented. We notice that every singular Jónsson cardinal κ with uncountable cofinality is the limit of some continuous sequence of smaller Jónsson cardinals. An analogous statement holds if κ is an inaccessible Jónsson cardinal unless κ is Mahlo. But we prove that the first Mahlo cardinal cannot be Jónsson. Some additional remarks are included.     

On universal locally finite groups

We deal with the question of existence of a universal object in the category of universal locally finite groups; the answer is negative for many uncountable cardinalities; for example, for 2^ℵ ₀, and assuming G.C.H. for every cardinal whose confinality is >ℵ₀. However, if λ>κ when κ is strongly compact and of λ=ℵ₀, then there exists a universal locally finite group of cardinality λ. The idea is to use the failure of the amalgamation property in a strong sense. We shall also prove the failure of the amalgamation property for universal locally finite groups by transferring the kind of failure of the amalgamation property from LF into ULF. We would like to thank Simon Thomas for reading carefully a preliminary version of this paper, proving Lemma 20 and making valuable remarks. Also we thank the United States—Israel Binational Science Foundation for partially supporting this work.     

On the least strongly compact cardinal

We prove that under the assumption of a supercompact cardinal κ which is a limit of supercompact cardinals, for any increasing Σ₂ function φ the set {∂<κ:∂ is at least φ(∂) supercompact, is strongly compact, yet is not fully supercompact} is unbounded in κ. We then use ideas of Magidor to show that under the hypotheses of a supercompact cardinal which is a limit of supercompact cardinals it is consistent for the least strongly compact cardinal κ₀ to be at least φ(κ₀) supercompact yet not to be fully supercompact, where φ is again an increasing Σ₂ function which also meets certain other technical restrictions. The author wishes to thank Menachem Magidor for helpful conversations and suggestions in method which were used in the proof of Theorem 2.     

A combinatorial principle and endomorphism rings I: Onp-groups

Two lines of research are involved here. One is a combinatorial principle, proved in ZFC for many cardinals (e.g., any λ = λ^ℵ ₀) enabling us to prove things which have been proven using the diamond or for strong limit cardinals of uncountable cofinality. The other direction is building abelian groups with few endomorphisms and/or prescribed rings of endomorphisms. We prove that for a ringR, whose additive group is thep-adic completion of a freep-adic module,R is isomorphic to the endomorphism ring of some separable abelianp-groupG divided by the ideal of small endomorphisms, withG of power λ for any λ = λ^ℵ ₀≧|R|. Dedicated to the memory of Abraham Robinson on the tenth anniversary of his death The author would like to thank the United States-Israel Binational Science Foundation for partially supporting this research.     

An infinite combinatorial statement with a poset parameter

We introduce an extension, indexed by a partially ordered set P and cardinal numbers κ,λ, denoted by (κ,<λ)⇝P, of the classical relation (κ,n,λ)→ρ in infinite combinatorics. By definition, (κ,n,λ)→ρ holds if every map F: [κ] ⁿ →[κ]^<λ has a ρ-element free set. For example, Kuratowski’s Free Set Theorem states that (κ,n,λ)→n+1 holds iff κ ≥ λ ⁺ⁿ , where λ ⁺ⁿ denotes the n-th cardinal successor of an infinite cardinal λ. By using the (κ,<λ)⇝P framework, we present a self-contained proof of the first author’s result that (λ ⁺ⁿ ,n,λ)→n+2, for each infinite cardinal λ and each positive integer n, which solves a problem stated in the 1985 monograph of Erdős, Hajnal, Máté, and Rado. Furthermore, by using an order-dimension estimate established in 1971 by Hajnal and Spencer, we prove the relation $(\lambda ^{ + (n - 1)} ,r,\lambda ) \to 2^{\left\lfloor {\tfrac{1} {2}(1 - 2^{ - r} )^{ - n/r} } \right\rfloor } $(\lambda ^{ + (n - 1)} ,r,\lambda ) \to 2^{\left\lfloor {\tfrac{1} {2}(1 - 2^{ - r} )^{ - n/r} } \right\rfloor } , for every infinite cardinal λ and all positive integers n and r with 2≤r<n. For example, (ℵ₂₁₀,4,ℵ₀)→32,768. Other order-dimension estimates yield relations such as (ℵ₁₀₉,4,ℵ₀) → 257 (using an estimate by Füredi and Kahn) and (ℵ₇,4,ℵ₀)→10 (using an exact estimate by Dushnik).     

The consistency strength of “every stationary set reflects”

The consistency strength of a regular cardinal so that every stationary set reflects is the same as that of a regular cardinal with a normal idealI so that everyI-positive set reflects in aI-positive set. We call such a cardinal areflection cardinal and such an ideal areflection ideal. The consistency strength is also the same as the existence of a regular cardinal κ so that every κ-free (abelian) group is κ⁺-free. In L, the first reflection cardinal is greater than the first greatly Mahlo cardinal and less than the first weakly compact cardinal (if any). Research supported by NSERC grant # A8948. Publication # 367. Research partially supported by the BSF.     

Large superuniversal metric spaces

For every uncountable cardinal κ define a metric spaceS to be κ-superuniversal iff for every metric spaceU of cardinality κ, every partial isometry intoS from a subset ofU of cardinality less than κ can be extended to all ofU. We prove that any such space must have cardinality at least , and for each regular uncountable cardinal κ, we construct a κ-superuniversal metric space of cardinality , It is proved that there is a unique κ-superuniversal metric space of cardinality κ iff . Several decomposition theorems are also proved, e.g., every κ-superuniversal space contains a family of disjoint κ-superuniversal subspaces. Finally, we consider some applications to more general topological spaces, to graph theory, and to category theory, and we conclude with a list of open problems.     

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.     

Further cardinal arithmetic

We continue the investigations in the author’s book on cardinal arithmetic, assuming some knowledge of it. We deal with the cofinality of (S _{≤ℵ 0}(κ), ⊆) for κ real valued measurable (Section 3), densities of box products (Section 5,3), prove the equality cov(λ,λ,θ⁺,2) in more cases even when cf(λ)=ℵ₀ (Section 1), deal with bounds of pp(λ) for λ limit of inaccessible (Section 4) and give proofs to various claims I was sure I had already written but did not find (Section 6). Done mainly 1–4/1991. I thank Alice Leonhardt for typing and retyping so beautifully and accurately. Partially supported by the Basic Research Fund, Israel Academy of Sciences. Publication number 430.     

Chang’s conjecture for ℵω

We establish, starting from some assumptions of the order of magnitude of a huge cardinal, the consistency of (ℵ_ω+1,ℵ_ω)↠(ω₁,ω₀), as well as of some other transfer properties of the type (κ⁺,κ)↠(α⁺,α), where κ is singular.     

On a problem inspired by determinacy

In this paper, we construct a modelN in which ℵ₁, the only regular uncountable cardinal, is measurable via the club filter. Thus,N is a model for the theory “ZF+κ is regular iffκ is measurable”. This research in this paper was partially supported by NSF Grant DMS-8413736.     

Axioms of determinacy and biorthogonal systems

If allΠ _n ¹ games are determined, every non-norm-separable subspaceX ofl ^∞(N) which is W* —Σ _n ^+1/1 contains a biorthogonal system of cardinality 2^ℵ ₀. In Levy’s model of Set Theory, the same is true of every non-norm-separable subspace ofl ^∞(N) which is definable from reals and ordinals. Under any of the above assumptions,X has a quotient space which does not linearly embed into 1^∞(N).     

Resolvability and monotone normality

A space X is said to be κ-resolvable (resp., almost κ-resolvable) if it contains κ dense sets that are pairwise disjoint (resp., almost disjoint over the ideal of nowhere dense subsets). X is maximally resolvable if and only if it is Δ(X)-resolvable, where Δ(X) = min{|G| : G ≠ open}. We show that every crowded monotonically normal (in short: MN) space is ω-resolvable and almost μ-resolvable, where μ = min{2 ^ω , ω ₂}. On the other hand, if κ is a measurable cardinal then there is a MN space X with Δ(X) = κ such that no subspace of X is ω ₁-resolvable. Any MN space of cardinality < ℵ _ω is maximally resolvable. But from a supercompact cardinal we obtain the consistency of the existence of a MN space X with |X| = Δ(X) = ℵ _ω such that no subspace of X is ω ₂-resolvable. The preparation of this paper was supported by OTKA grant no. 61600     

Metrizable compacta in the space of continuous functions with the topology of pointwise convergence

We prove that every point-finite family of nonempty functionally open sets in a topological space X has the cardinality at most an infinite cardinal κ if and only if w(X) ≦ κ for every Valdvia compact space Y C _p (X). Correspondingly a Valdivia compact space Y has the weight at most an infinite cardinal κ if and only if every point-finite family of nonempty open sets in C _p (Y) has the cardinality at most κ, that is p(C _p (Y)) ≦ κ. Besides, it was proved that w(Y) = p(C _p (Y)) for every linearly ordered compact Y. In particular, a Valdivia compact space or linearly ordered compact space Y is metrizable if and only if p(C _p (Y)) = ℵ₀. This gives answer to a question of O. Okunev and V. Tkachuk.      

On a superatomic Boolean algebra which is not generated by a well-founded sublattice

Let b denote the unboundedness number of ω^ω. That is, b is the smallest cardinality of a subset such that for everyg∈ω^ω there isf ∈ F such that {n: g(n) ≤ f(n)}is infinite. A Boolean algebraB is wellgenerated, if it has a well-founded sublatticeL such thatL generatesB. We show that it is consistent with ZFC that , and there is a Boolean algebraB such thatB is not well-generated, andB is superatomic with cardinal sequence 〈ℵ₀, ℵ₁, ℵ₁, 1〉. This result is motivated by the fact that if the cardinal sequence of a Boolean algebraB is 〈ℵ₀, ℵ₀, λ, 1〉, andB is not well-generated, then λ≥b.     

Counting metric spaces

If κ is a cardinal number, then any class of mutually non-homeomorphic metric spaces of size κ must be a set whose cardinality cannot exceed 2 ^κ . Our main result is a vivid construction of 2 ^κ mutually non-homeomorphic complete and both path connected and locally path connected metric spaces of size κ for each cardinal number κ from continuum up. Additionally we also deal with counting problems concerning countable metric spaces and Euclidean spaces.     

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.     

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.     

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.     

Multi-dimensionality

We prove that ifT is stable, not multi-dimensional theory, then there is an infinite indiscernible set orthogonal to the empty set. This completes the proof that if ℵ_α=ℵ _•T• ^α >ℵ≥_Kr(T), thenT has ≥2^•α−β• non-isomorphic ℵ_β models of cardinality ℵ_α. Originally written November 5, 1988. Publication 429. Partially supported by the Israel-United States Binational Science Foundation; I thank Alice Leonhardt for the beautiful typing.