TY - JOUR
T1 - The downward directed grounds hypothesis and very large cardinals
AU - Usuba, Toshimichi
N1 - Funding Information:
We would like to thank Joel David Hamkins and Daisuke Ikegami for many fruitful discussions. Some of the ideas that came out of those discussions were used in this paper. We also thank Paul Larson and Yo Matsubara, they gave the author many corrections of the draft and insightful suggestions. Finally, we are grateful to David Aspero, Hiroshi Sakai, Kostantinos Tsaprounis, Yasuo Yoshinobu for their useful comments. This research was supported by KAKENHI Grant Nos. 15K17587 and 15K04984.
Publisher Copyright:
© 2017 World Scientific Publishing Company.
PY - 2017/12/1
Y1 - 2017/12/1
N2 - 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.
AB - 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.
KW - Forcing method
KW - downward directed grounds hypothesis
KW - generic multiverse
KW - large cardinal
KW - set-theoretic geology
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U2 - 10.1142/S021906131750009X
DO - 10.1142/S021906131750009X
M3 - Article
AN - SCOPUS:85031900598
SN - 0219-0613
VL - 17
JO - Journal of Mathematical Logic
JF - Journal of Mathematical Logic
IS - 2
M1 - 1750009
ER -