Canonical inner models have interesting set-theoretic geology. Below a Woodin cardinal, they have no proper forcing grounds at all. But proper class mice of form L[E] always have proper grounds, as long as they have Woodin cardinals. In recent years, in work of Sargsyan, Schindler and the author, the mantles of certain mice L[E] containing Woodin cardinals and strong cardinals have been analysed, with the result that these mantles are strategy mice L[E,\Sigma] containing Woodin cardinals, are fully iterable, and closed under their own iteration strategies. Even though these strategy mice contain Woodin cardinals, they have no proper grounds themselves. The analysis relates heavily to the analysis of HOD in models of determinacy, but new features arise, and the arguments are purely inner model theoretic.
In this tutorial, I will begin with an introduction to some fundamentals of inner model theory, and then describe some of the key ideas which go into the study of these models.
en
Project title:
Determiniertheit und Woodin Limes von Woodin Kardinalzahlen: Y1498 (FWF - Österr. Wissenschaftsfonds)
-
Project (external):
Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)