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

Uri Abraham, Matatyahu Rubin, Saharon Shelah

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74 Scopus citations

Abstract

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+(2א02)). Let <KH,<> 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 <L, <->: Con(MA+ <KH, <-> {difference between} <L, <->) 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){A figure is presented}XxX-}<x,x>|xε{lunate}X} consisting of symmetric open sets, then X can be partitioned into {Xi \brvbar; i ε{lunate} ω} such that for every i ε{lunate} ω there is l<n such that D(Xi)⊇Ul". We also prove that MA+OCA [xrArr] 2 א0 = א2.

Original languageEnglish
Pages (from-to)123-206
Number of pages84
JournalAnnals of Pure and Applied Logic
Volume29
Issue number2
DOIs
StatePublished - 1 Jan 1985

ASJC Scopus subject areas

  • Logic

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