# Introduction to remarkable cardinals

## Victoria Gitman

### The City University of New York

Ralf Schindler introduced remarkable cardinals because he discovered that they are precisely equiconsistent with the statement that the theory of $L(\mathbb R)$ is absolute for proper forcing. The statement that the theory of $L(\mathbb R)$ is absolute for all set forcing is closely related to whether $L(\mathbb R)\models {\rm AD}$. In contrast, remarkable cardinals sit relatively low in the large cardinal hierarchy; for instance, they are downward absolute to $L$. I will discuss the various equivalent characterizations of remarkable cardinals due to Schindler and show where the remarkable cardinals fit into the large cardinal hierarchy using results due to Schindler, Philip Welch and myself.

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.