October 26
Zach Norwood,
Cornell University
The Tree Reflection Principle
In the presence of large cardinals, the $L(\mathbb{R})$ of any proper forcing extension is elementarily equivalent to the ground-model $L(\mathbb{R})$ (Neeman-Zapletal). Schindler showed that this generic absoluteness assertion is equiconsistent with a remarkable cardinal and can therefore be forced over $L$. Recently, Itay Neeman & I improved 'proper' to 'sigma-closed $\ast$ ccc' in Schindler's theorem, giving a totally different lower-bound argument. The proof naturally suggests a compactness property of trees on $\omega_1$, which is the subject of this talk.
I will outline the proof of the generic-absoluteness theorem (joint with Neeman), explain how the Tree Reflection Principle offers an alternative to traditional coding by reshaping, and discuss some applications to forcing axioms. In particular, we will discuss an expansion of the Bagaria-Gitman-Schindler analysis of the Weak Proper Forcing Axiom.
Remarkable cardinals will be defined in the talk. No extensive knowledge of proper forcing will be assumed.