Remarkable Laver functions
The City University of New York
Since Laver defined and used a Laver function to show that supercompact cardinals can be made indestructible by all $\lt\kappa$-directed closed forcing, Laver-like functions have been defined for various large cardinals and used for lifting embeddings in indestructibility arguments. Laver-like functions are also inherently interesting as guessing principles with affinity to $\diamondsuit$. Supposing that a large cardinal $\kappa$ can be characterized by the existence of some kind of embeddings, a Laver-like function $\ell:\kappa\to V_\kappa$ has, roughly speaking, the property that for any set $a$ in the universe, there is an embedding $j$ of the type characterizing the cardinal such that $j(\ell)(\kappa)=a$. Although Laver-like functions can be forced to exist for almost any large cardinal, only a few large cardinals including supercompact, strong, and extendible, have them outright. I will define the notion of a remarkable Laver function for a remarkable cardinal and show that every remarkable cardinal has a remarkable Laver function. Remarkable cardinals were introduced by Ralf Schindler who showed that a remarkable cardinal is precisely equiconsistent with the property that the theory of $L(\mathbb R)$ is absolute for proper forcing. Time permitting, I will show how the existence of remarkable Laver functions is used in demonstrating indestructibility for remarkable cardinals. This is joint work with Yong Cheng. An extended abstract can be found here.
Victoria Gitman received her Ph.D. in 2007 from the CUNY Graduate Center, as a student of Joel Hamkins, and is presently a visiting scholar at the CUNY Graduate Center. Her research is in Mathematical Logic, in particular in the areas of Set Theory and Models of Peano Arithmetic.