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Separating club-guessing principles in the presence of fat forcing axioms

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Abstract

We separate various weak forms of Club Guessing at \(\omega_1\) in the presence of \(2^{\aleph_0}\) large, Martin's Axiom, and related forcing axioms.

We also answer a question of Abraham and Cummings concerning the consistency of the failure of a certain polychromatic Ramsey statement together with the continuum large.

All these models are generic extensions via finite support iterations with symmetric systems of structures as side conditions, possibly enhanced with \(\omega\)-sequences of predicates, and in which the iterands are taken from a relatively small class of forcing notions.

We also prove that the natural forcing for adding a large symmetric system of structures (the first member in all our iterations) adds \(\aleph_1\)-many reals but preserves CH.

Details

Original languageEnglish
Pages (from-to)284–308
JournalAnnals of Pure and Applied Logic
Volume167
Issue number3
Early online date21 Dec 2015
DOIs
Publication statusPublished - Mar 2016
Peer-reviewedYes

Keywords

    Research areas

  • Iterated forcing, Club-guessing principles, Side conditions, Polychromatic Ramsey theory

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