Forcing with proper classes

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...

I. Proper forcing

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.

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...

Proper Classes

?1. Historical introduction. The modern problem of infinity was first raised by Aristotle who held (at least on the popular interpretation2) that infinite sets exist potentially (i.e. one more number can always be counted, one more division can always be made in a line segment) but not actually (i.e. the numbers or divisions cannot all exist at one time). In fact, Aristotle not only held that c...