A three-lecture mini-course tutorial in set-theoretic geology at the summer school Set Theory and Higher-Order Logic: Foundational Issues and Mathematical Developments, August 1-6, 2011, University of London, Birkbeck.
The technique of forcing in set theory is customarily thought of as a method for constructing outer as opposed to inner models of set theory; one starts in a ground model V and considers the possible forcing extensions V[G] of it. A simple switch in perspective, however, allows us to use forcing to describe inner models, by considering how a given universe V may itself have arisen by forcing. This change in viewpoint leads to the topic of set-theoretic geology, aiming to investigate the structure and properties of the ground models of the universe. In this tutorial, I shall present some of the most interesting initial results in the topic, along with an abundance of open questions, many of which concern fundamental issues.
A ground of the universe V is an inner model W of ZFC over which the universe V=W[G] is a forcing extension. The model V satisfies the Ground Axiom of there are no such W properly contained in V. The model W is a bedrock of V if it is a ground of V and satisfies the Ground Axiom. The mantle of V is the intersection of all grounds of V, and the generic mantle is the intersection of all grounds of all set-forcing extensions. The generic HOD, written gHOD, is the intersection of all HODs of all set-forcing extensions. The generic HOD is always a model of ZFC, and the generic mantle is always a model of ZF. Every model of ZFC is the mantle and generic mantle of another model of ZFC, and this can be proved while also controlling the HOD of the final model, as well as the generic HOD. Iteratively taking the mantle penetrates down through the inner mantles to the outer core, what remains when all outer layers of forcing have been stripped away. Many fundamental questions remain open.