The downward directed grounds hypothesis and very large cardinals

Toshimichi Usuba*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

15 Citations (Scopus)


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 ofV 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.

Original languageEnglish
Article number1750009
JournalJournal of Mathematical Logic
Issue number2
Publication statusPublished - 2017 Dec 1


  • Forcing method
  • downward directed grounds hypothesis
  • generic multiverse
  • large cardinal
  • set-theoretic geology

ASJC Scopus subject areas

  • Logic


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