Matatyahu Rubin

On the consistency of some partition theorems for continuous colorings, and the structure of ℵ1-dense real order types

We present some techniques in c.c.c. forcing, and apply them to prove consistency results concerning the isomorphism and embeddability relations on the family of ℵ1-dense sets of real numbers. In this direction we continue the work of Baumgartner [2] who proved the axiom BA stating that every two ℵ1-dense subsets of R are isomorphic, is consistent. We e.g. prove Con(BA+(2ℵ0>ℵ2)). Let be the set of order types of ℵ1-dense homogeneous subsets of R with the relation of embeddability. We prove that for every finite model : Con(MA+ ≏ ) iff L is a distributive lattice. We prove that it is consistent that the Magidor-Malitz language is not countably compact. We deal with the consistency of certain topological partition theorems. E.g. We prove that MA is consistent with the axiom OCA which says: “If X is a second countable space of power ℵ1, and {U0,\h.;,Un−1} is a cover of D(X)XxX-} ¦xϵX} consisting of symmetric open sets, then X can be partitioned into {Xi \brvbar; i ϵ ω} such that for every i ϵ ω there is l

Some questions about Boolean algebras

Very recently there has been much progress on some fundamental settheoretic problems concerning Boolean algebras. The purpose of this article is to indicate some problems still left open, put in perspective by what has been shown recently. We have made no a t tempt to completely cover the field with these questions, but hope that for the problems ment ioned here the picture we give is fairly complete. To some extent this is a survey of recent set-theoretical results on Boolean algebras. In particular, part of the information we give here answers questions f rom earlier informal versions of this paper and has been included so as to make clear what no longer is an open problem. We are grateful to R. Bonnet , S. Koppelberg, K. Kunen, R. Laver, R. McKenzie, P. Nyikos, S. Shelah and M. Weese for comments on earlier versions of this article.

Theories of linear order

LetT be a complete theory of linear order; the language ofT may contain a finite or a countable set of unary predicates. We prove the following results. (i) The number of nonisomorphic countable models ofT is either finite or 2ω. (ii) If the language ofT is finite then the number of nonisomorphic countable models ofT is either 1 or 2ω. (iii) IfS1(T) is countable then so isSn(T) for everyn. (iv) In caseS1(T) is countable we find a relation between the Cantor Bendixon rank ofS1(T) and the Cantor Bendixon rank ofSn(T). (v) We define a class of modelsL, and show thatS1(T) is finite iff the models ofT belong toL. We conclude that ifS1(T) is finite thenT is finitely axiomatizable. (vi) We prove some theorems concerning the existence and the structure of saturated models.

The Reconstruction of Trees from Their Automorphism Groups

An extended introduction Some preliminaries concerning interpretations, groups and

Combinatorial problems on trees: Partitions, δ-systems and large free subtrees

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Locally Moving Groups and Reconstruction Problems

-categoricity A new reconstruction theorem for Boolean algebras The completion and the Boolean algebra of a U-tree The statement of the canonization and reconstruction theorems The canonization of trees The reconstruction of the Boolean algebra of a U-tree The reconstruction of

On the reconstruction of boolean algebras from their automorphism groups

PT({\mathrm Exp}(M))

On the Elementary Equivalence of Automorphism Groups of Boolean Algebras; Downward Skolem Lowenheim Theorems and Compactness of Related Quantifiers

Final reconstruction results Observations, examples and discussion Augmented trees The reconstruction of

On well-generated Boolean algebras

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On the automorphism groups of countable Boolean algebras

-categorical trees Nonisomorphic 1-homogeneous chains which have isomorphic automorphism groups Bibliography A list of notations and definitions.