Topic Archive: inner models
Set-theoretic geology, a line of research jointly created by Hamkins, Reitz and myself, introduced some inner models which result from inverting forcing in some sense. For example, the mantle of a model of set theory V is the intersection of all inner models of which V is an extension by set-forcing. It was an initial, naive hope that one might arrive at a model that is in some sense canonical, but one of the main results on set-theoretic geology is that this is not so: every model of set theory V has a class forcing extension V[G] so that the mantle, as computed in V[G], is V. So quite literally, the mantle of a model of set theory can be anything.
In an attempt to arrive at a concept that fits in with the general spirit of set-theoretic geology, but that stands a chance of being canonical, I defined a set to be solid if it cannot be added to an inner model by set-forcing, and I termed the union of all solid sets the “solid core”.
I will present some results on the solid core which were obtained in recent joint work with Ralf Schindler, and which show that the solid core indeed is a canonical inner model, assuming large cardinals (more precisely, if there is an inner model with a Woodin cardinal), but that it is not as canonical as one might have hoped without that assumption.