This is a continuation of the earlier Introduction to remarkable cardinals lecture. The speaker will continue to discuss the various equivalent characterizations of remarkable cardinals and their relationship to other large cardinal notions.

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.