Set-Theoretic Geology

TL;DR

This paper explores the structure of ground models, mantles, and HODs in set theory, establishing new results about their properties and relationships through forcing extensions and inner models.

Contribution

It introduces the concept of the outer core and demonstrates how to control the HOD and generic HOD in models of ZFC, advancing understanding of set-theoretic geology.

Findings

Every model of ZFC can be realized as the mantle of another model.

Iterative mantle operations reveal a layered structure called the outer core.

The generic mantle and HOD are always models of ZF and ZFC, respectively.

Abstract

A ground of the universe V is a transitive proper class W subset V, such that W is a model of ZFC and V is obtained by set forcing over W, so that V = W[G] for some W-generic filter G subset P in W . The model V satisfies the ground axiom GA if there are no such W properly contained in V . The model W is a bedrock of V if W is a ground of V and satisfies the ground axiom. The mantle of V is the intersection of all grounds of V . The generic mantle of V is the intersection of all grounds of all set-forcing extensions of V . 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. We prove this theorem 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 what we call the outer core, what remains when all outer layers of forcing have been stripped away. Many fundamental questions remain open.