Set-theoretic geology

Abstract

A ground of the universe V is a transitive proper class W ⊆ V , such that W |= ZFC and V is obtained by set forcing over W , so that V = W [G] for some W -generic filter G ⊆ P ∈ 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.

Subcomplete forcing principles and definable well‐orders

“…To explain this assumption, I shall have to say a few words about set‐theoretic geology. Research in this area, initiated in , is concerned with the structure of grounds of a universe V. An inner model M of is a ground if , for some g which is generic over M for some forcing . The chief object of study is the mantle , i.e., the intersection of all grounds.…”

“…This is maybe an unexpected appearance of an assumption on the set‐theoretic geology (cf. ) of the ambient universe, but it guarantees that the mantle , the intersection of all grounds, is itself a ground model of the universe, hence is very “close to V”. It is known that the mantle is invariant under set forcing, and hence, the assumption of set many grounds provides us with a forcing invariant inner model that is close to V. By work of Usuba (cf.…”

The grounded Martin's axiom

“…Even if the answer to Question turns out to be positive, we conjecture that the answer to this last question is negative. We expect that it is possible to use the methods of or, more generally, in combination with the forcing construction of theorem to produce a model of which has no c.c.c. ground models and in which (and consequently also ) fails.…”

Laver and set theory

“…Reitz [106] investigated this axiom, and [36] established its consistency with V = HOD. [32] then extended the investigations into "set-theoretic geology", digging into the remains of a model of set theory once the layers created by forcing are removed. On his side, Woodin [122, §8] used ground model definability to formalize a conception of the "generic-multiverse"; the analysis here dates back to 2004.…”