The proper and semi-proper forcing axioms for forcing notions that preserve $\aleph _2$ or $\aleph _3$

The Proper and Semi-proper Forcing Axioms for Forcing Notions That Preserve

We prove that the PFA lottery preparation of a strongly unfoldable cardinal κ under ¬0 forces PFA(א2-preserving), PFA(א3-preserving) and PFAא2 , with 2 ω = κ = א2. The method adapts to semi-proper forcing, giving SPFA(א2-preserving), SPFA(א3-preserving) and SPFAא2 from the same hypothesis. It follows by a result of Miyamoto that the existence of a strongly unfoldable cardinal is equiconsistent ...

Simple Forcing Notions and Forcing Axioms

The Proper Forcing Axiom

The Proper Forcing Axiom is a powerful extension of the Baire Category Theorem which has proved highly effective in settling mathematical statements which are independent of ZFC. In contrast to the Continuum Hypothesis, it eliminates a large number of the pathological constructions which can be carried out using additional axioms of set theory. Mathematics Subject Classification (2000). Primary...

Hypergraphs and proper forcing

Given a Polish space X and a countable collection of analytic hypergraphs on X, I consider the σ-ideal generated by Borel anticliques for the hypergraphs in the family. It turns out that many of the quotient posets are proper. I investigate the forcing properties of these posets, certain natural operations on them, and prove some related dichotomies.

Proper Forcing Remastered

We present the method introduced by Neeman of generalized side conditions with two types of models. We then discuss some applications: a variation of the Friedman-Mitchell poset for adding a club with finite conditions, the consistency of the existence of an ω2 increasing chain in (ω1 1 , <fin), originally proved by Koszmider, and the existence of a thin very tall superatomic Boolean algebra, o...