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Aronszajn Trees and the SCH Itay Neeman

Summary: Aronszajn Trees and the SCH
Itay Neeman
and Spencer Unger
February 28, 2009
1 Introduction
These notes are based on results presented by Itay Neeman at the Appalachian
Set Theory workshop on February 28, 2009. Spencer Unger was the official
note-taker and based these notes closely on Neeman's lectures. The purpose of
the workshop was to present a recent theorem due to Neeman [16].
Theorem 1. From large cardinals, it is consistent that there is a singular strong
limit cardinal of cofinality such that the Singular Cardinal Hypothesis fails
at and the tree property holds at +
The purpose of these notes is to give the reader the flavor of the argument
without going into the complexities of the final 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 2. Let be a regular cardinal and be a cardinal.


Source: Andrews, Peter B. - Department of Mathematical Sciences, Carnegie Mellon University


Collections: Mathematics