The downward directed grounds hypothesis and very large cardinals

Toshimichi Usuba

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arXiv · 1707.05132

The downward directed grounds hypothesis and very large cardinals

Toshimichi Usuba

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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.

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Toshimichi Usuba. 2017-07-17. The downward directed grounds hypothesis and very large cardinals. https://arxiv.org/abs/1707.05132

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