 
Summary: SET MAPPING REFLECTION
Appalachian Set Theory Workshop, May 31, 2008
Lectures by Justin Tatch Moore
Notes taken by David Milovich
1. 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(1) which is definable
over H(2), (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: [1]2
2 there exist A, B [1]1
such that c is constant on {{, } : < A B}?
Let us focus on the second question for a moment. Consider the following anal
ogy. 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. Sim
ilarly, a forcing Q satisfies Knaster's Condition (Property K) if every uncountable
