The downward directed grounds hypothesis and very large cardinals

TL;DR

This paper explores the structure of grounds in set theory, demonstrating that the collection of grounds is downward set-directed and analyzing implications for the mantle and large cardinals.

Contribution

It proves that grounds are downward set-directed and establishes key theorems relating the mantle, grounds, and large cardinals in set-theoretic geology.

Findings

The grounds of V are downward set-directed.

The mantle is a model of ZFC.

If V has large cardinals, the mantle is a ground.

Abstract

A transitive model $M$ of ZFC is called a ground if the universe $V$ is a set forcing extension of $M$. We show that the grounds of $V$ are downward set-directed. Consequently, we establish some fundamental theorems on the forcing method and the set-theoretic geology. For instance, (1) the mantle, the intersection of all grounds, must be a model of ZFC. (2) $V$ has only set many grounds if and only if the mantle is a ground. We also show that if the universe has some very large cardinal, then the mantle must be a ground.