Introduction

The goal of these lectures is to give an exposition of the concept of an open stationary set, an associated reflection principle (for lack of a better word), and a list of examples of how this sort of consideration arises naturally in the context of modern set theory. We will begin with a list of seemingly unrelated questions.

Question 1.1 Does PFA imply there is a well ordering of P(ω1) which is definable over 〈H(ℵ2), ∈〉 (with parameters)?

Question 1.2 Is it consistent that every Aronszajn line contains a Countryman suborder?

Question 1.3 Is it consistent that for all c : [ω1]2 → 2 there exist A, B ∈ [ω1]ω1 such that c is constant on {{α, β} : α, < β ∧ α ∈ A ∧ β ∈ Bg}?

Let us focus on the second question for a moment. Consider the following analogy. Recall that a forcing Q satisfies the countable chain condition (c.c.c.) if every uncountable collection of conditions in Q contains two compatible conditions. Similarly, Q satisfies Knaster's Condition (Property K) if every uncountable collection of conditions contains an uncountable subcollection of pairwise compatible conditions. It is easily verified that the product of a c.c.c. forcing and one with Property K is c.c.c..