Kurt Gödel Research Center

2018 seminar talk: The tree property and the continuum function

Talk held by Šárka Stejskalová (KGRC) at the KGRC seminar on 2018-03-08.

Abstract

We will discuss the tree property, a compactness principle which can hold at successor cardinals such as $\aleph_2$ or $\aleph_3$. For a regular cardinal $\kappa$, we say that $\kappa$ has the tree property if there are no $\kappa$-Aronszajn trees. It is known that the tree property has the following non-trivial effect on the continuum function:

(*) If the tree property holds at $\kappa^{++}$, then $2^\kappa > \kappa^+$.

After defining the key notions, we will review some basic constructions related to the tree property and state some original results regarding the tree property which suggest that (*) is the only restriction which the tree property puts on the continuum function in addition to the usual restrictions provable in ZFC.