## 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 א_{1}-dense sets of real numbers. In this direction we continue the work of Baumgartner [2] who proved the axiom BA stating that every two א_{1}-dense subsets of R are isomorphic, is consistent. We e.g. prove Con(BA+(2^{א0}>א_{2})). Let <K^{H},<> be the set of order types of א_{1}-dense homogeneous subsets of R with the relation of embeddability. We prove that for every finite model <L, <->: Con(MA+ <K^{H}, <-> {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 א_{1}, and {U_{0},\h.;,U_{n-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 {X_{i} \brvbar; i ε{lunate} ω} such that for every i ε{lunate} ω there is l<n such that D(X_{i})⊇U_{l}". We also prove that MA+OCA [xrArr] 2 א_{0} = א_{2}.

Original language | English |
---|---|

Pages (from-to) | 123-206 |

Number of pages | 84 |

Journal | Annals of Pure and Applied Logic |

Volume | 29 |

Issue number | 2 |

DOIs | |

State | Published - 1 Jan 1985 |

## ASJC Scopus subject areas

- Logic

## Fingerprint

Dive into the research topics of 'On the consistency of some partition theorems for continuous colorings, and the structure of א_{1}-dense real order types'. Together they form a unique fingerprint.