Kurt Gödel Research Center

Selected topics in mathematical logic

Time: Tue, Thu 2:00pm–3:00pm
starts 2016‑03‑01
Place: KGRC

In this course I'll explore three central combinatorial properties of set theory and their interaction. The Tree Property (TP) at a regular cardinal $\kappa$ asserts the nonexistence of a $\kappa$-tree with no $\kappa$-branch. The Reflection Property (RP) at the successor $\kappa^+$ of a regular cardinal $\kappa$ asserts that any stationary subset of $\kappa^+$ consisting of ordinals of cofinality less than $\kappa$ has a stationary proper initial segment. And the Approachability Property (AP) at a regular cardinal $\kappa$ asserts the existence of a sequence $(a_i | i < \kappa)$ of bounded subsets of $\kappa$ such that for club-many $\alpha < \kappa$, there is a cofinal sequence in $\alpha$ of ordertype cof$(\alpha)$ all of whose proper initial segments are of the form $a_i$ for some $i < \alpha$. We'll develop the tools needed for the "Eightfold Way Theorem", the result asserting that all 8 Boolean combinations of TP,RP,AP are possible at double successor cardinals. The situation at successors of limit cardinals is not yet well-understood.