Summary: Aronszajn Trees and the SCH
and Spencer Unger
February 28, 2009
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 .
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 . 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.