Banach spaces and groups — Order properties and universal models

We deal with two natural examples of almost-elementary classes: the class of all Banach spaces (over ℝ or ℂ) and the class of all groups. We show that both of these classes do not have the strict order property, and find the exact place of each one of them in Shelah’sSOP _n (strong order property of ordern) hierarchy. Remembering the connection between this hierarchy and the existence of universal models, we conclude, for example, that there are “few” universal Banach spaces (under isometry) of regular density characters. This publication is numbered 789 in the list of publications of Saharon Shelah. The research was supported by The Israel Science Foundation.     

The depth of ultraproducts of Boolean algebras

We show that if μ is a compact cardinal then the depth of ultraproducts of less than μ many Boolean algebras is at most μ plus the ultraproduct of the depths of those Boolean algebras. Received May 18, 2004; accepted in final form December 9, 2004.     

More on cardinal arithmetic

Done 1990. Partially supported by the Basic Research Fund, Israel Academy of Science. I thank Alice Leonhardt for the beautiful typing Publ. 410     

Isomorphism types of Aronszajn trees

We study the isomorphism types of Aronszajn trees of height ω₁ and give diverse results on this question (mainly consistency results). The second author would like to thank the United States-Israel Binational Science Foundation for partially supporting this research by a grant.     

Intersection of ultrafilters may have measure zero

We show that it is consistent with ZFC that the intersection of some family of less than ultrafilters have measure zero. This answers a question of D. Fremlin.The author thanks the Lady Davis Fellowship Trust for full supportPartially supported by Basic Research Fund, Israel Academy of Sciences, publication 436     

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.     

Notes on monadic logic. part A. Monadic theory of the real line

The second-order theory of the continuum in the Cohen extension of a set-theoretic universe is interpreted in the monadic theory of the real line and may be interpreted in the monadic topology of Cantor’s discontinuum as well.     

The monadic theory and the “next world”

Suppose thatV is a model of ZFC andU ∈ V is a topological space or a richer structure for which it makes sense to speak about the monadic theory. LetB be the Boolean algebra of regular open subsets ofU. If the monadic theory ofU allows one to speak in some sense about a family ofκ everywhere dense and almost disjoint sets, then the second-orderV ^B-theory of ϰ is interpretable in the monadicV-theory ofU; this is our Interpretation Theorem. Applying the Interpretation Theorem we strengthen some previous results on complexity of the monadic theories of the real line and some other topological spaces and linear orders. Here are our results about the real line. Letr be a Cohen real overV. The second-orderV[r]-theory of ℵ₀ is interpretable in the monadicV-theory of the real line. If CH holds inV then the second-orderV[r]-theory of the real line is interpretable in the monadicV-theory of the real line. Dedicated to the memory of Abraham Robinson on the tenth anniversary of his death The author thanks the United States-Israel Binational Science Foundation for supporting the research.