Uri Abraham

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

Aronszajn trees on ℵ2 and ℵ3

Abstract Assuming the existence of a supercompact cardinal and a weakly compact cardinal above it, we provide a generic extension where there are no Aronszajn trees of height ω 2 or ω 3 . On the other hand we show that some large cardinal assumptions are necessary for such a consistency result.

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).

Self-stabilizing l-exclusion

Our work presents a self-stabilizing solution to the l-exclusion problem. This problem is a well-known generalization of the mutual-exclusion problem in which up to l, but never more than l, processes are allowed simultaneously in their critical sections. Self-stabilization means that even when transient failures occur and some processes crash, the system finally resumes its regular and correct behavior. The model of communication assumed here is that of shared memory, in which processes use single-writer multiple-reader regular registers. Copyright 2001 Elsevier Science B.V.

A Δ22 well-order of the reals and incompactness of L(QMM)

Abstract A forcing poset of size 2 2ℵ1 which adds no new reals is described and shown to provide a Δ 2 2 definable well-order of the reals (in fact, any given relation of the reals may be so encoded in some generic extension). The encoding of this well-order is obtained by playing with products of Aronszajn trees: some products are special while other are Suslin trees. The paper also deals with the Magidor–Malitz logic: it is consistent that this logic is highly noncompact.

Initial segments of the degrees of size ℵ1

We settle a series of questions first raised by Yates at the Jerusalem (1968) Colloquium on Mathematical Logic by characterizing the initial segments of the degrees of unsolvability of size ℵ1: Every upper semi-lattice of size ℵ1 with zero, in which every element has at most countably many predecessors, is isomorphic to an initial segment of the Turing degrees.

A note on Dilworth's theorem in the infinite case

If ℘ is a poset and every antichain is finite, and if the length of the well-founded poset of antichains is less than ω21, then ℘ is the union of countably many chains. We also compute the length of the poset of antichains in the product of two ordinals, αxβ.

ON POSET BOOLEAN ALGEBRAS

Let (P,≤) be a partially ordered set. The poset Boolean algebra of P, denoted F(P), is defined as follows: The set of generators of F(P) is {xp : p∈P}, and the set of relations is {xp⋅xq=xp : p≤q}. We say that a Boolean algebra B is well-generated, if B has a sublattice G such that G generates B and (G,≤B|G) is well-founded. A well-generated algebra is superatomic.THEOREM 1. Let (P,≤) be a partially ordered set. The following are equivalent. (i) P does not contain an infinite set of pairwise incomparable elements, and P does not contain a subset isomorphic to the chain of rational numbers, (ii) F(P) is superatomic, (iii) F(P) is well-generated.The equivalence (i) ⇔ (ii) is due to M. Pouzet. A partially ordered set W is well-ordered, if W does not contain a strictly decreasing infinite sequence, and W does not contain an infinite set of pairwise incomparable elements.THEOREM 2. Let F(P) be a superatomic poset algebra. Then there are a well-ordered set W and a subalgebra B of F(W), such that F(P) is a homomorphic image of B.This is similar but weaker than the fact that every interval algebra of a scattered chain is embeddable in an ordinal algebra. Remember that an interval algebra is a special case of a poset algebra.

Martin's Axiom and First-Countable S- and L-Spaces

Publisher Summary This chapter discusses first-countable spaces. Every point in this space has a countable basis for open neighborhoods. If an uncountable regular Hausdorff space does not possess an uncountable discrete subspace, despite the fact that all initial segments in some ω-enumeration of that space are open, then that space is called a right-separated S-space of type ω1. If an uncountable and regular Hausdorff space does not have an uncountable discrete subspace, despite the fact that all final segments in some ω1-enumeration of the space are open, then that space is called a left-separated L-space of type ω1. The chapter further presents some methods for constructing models of Martins Axiom with some additional properties.

On Jakovlev spaces

A topological spaceX is called weakly first countable, if for every pointx there is a countable family {Cnx |n ∈ω} such thatx ∈Cn+1x ⊆Cnx and such thatU ⊂X is open iff for eachx ∈U someCnx is contained inU. This weakening of first countability is due to A. V. Arhangelskii from 1966, who asked whether compact weakly first countable spaces are first countable. In 1976, N. N. Jakovlev gave a negative answer under the assumption of continuum hypothesis. His result was strengthened by V. I. Malykhin in 1982, again under CH. In the present paper we construct various Jakovlev type spaces under the weaker assumption b=c, and also by forcing.