6 - Aronszajn trees and the SCH

Summary

The eighth Appalachian Set Theory workshop was held at Carnegie Mellon University in Pittsburgh on February 28, 2009. The lecturer was Itay Neeman. As a graduate student Spencer Unger assisted in writing this chapter, which is based on the workshop lectures.

Introduction

The purpose of the workshop was to present a recent theorem due to Neeman [16].

Theorem 1.1From large cardinals, it is consistent that there is a singular strong limit cardinal κ of cofinality ω such that the Singular Cardinal Hypothesis fails at κ and the tree property holds at κ⁺.

The notes are intended to give the reader the flavor of the argument without going into the complexities of the full proof in [16]. Having read these notes, the motivated reader should be prepared to understand the full argument.We begin with a discussion of trees, which are natural objects in infinite combinatorics. One topic of interest is whether a tree has a cofinal branch. For completeness we recall some definitions.

Definition 1.2 Let λ be a regular cardinal and κ be a cardinal.

1. A λ-tree is a tree of height λ with levels of size less than λ.

2. A cofinal branch through a tree of height λ is a linearly ordered subset of order type λ.

3. A λ-Aronszajn tree is a λ-tree with no cofinal branch.

4. A κ⁺-tree is special if there is a function f : T → κ such that for all x; y ∈ T, if x T y then f(x) ≠ f(y).