The downward directed grounds hypothesis and very large cardinals
Logic
2018-07-23 v2
Abstract
A transitive model of ZFC is called a ground if the universe is a set forcing extension of . We show that the grounds of 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) 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.
Cite
@article{arxiv.1707.05132,
title = {The downward directed grounds hypothesis and very large cardinals},
author = {Toshimichi Usuba},
journal= {arXiv preprint arXiv:1707.05132},
year = {2018}
}