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.
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.
| Original language | English |
|---|---|
| Pages (from-to) | 284–308 |
| Number of pages | 25 |
| Journal | Annals of Pure and Applied Logic |
| Volume | 167 |
| Issue number | 3 |
| Early online date | 21 Dec 2015 |
| DOIs | |
| Publication status | Published - Mar 2016 |
Keywords
- Iterated forcing
- Club-guessing principles
- Side conditions
- Polychromatic Ramsey theory