### Abstract

A transitive model (Formula presented.) of ZFC is called a ground if the universe (Formula presented.) is a set forcing extension of (Formula presented.). We show that the grounds of(Formula presented.)(Formula presented.)(Formula presented.) 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) (Formula presented.) 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 language | English |
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Journal | Journal of Mathematical Logic |

DOIs | |

Publication status | Accepted/In press - 2017 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Logic

### Cite this

**The downward directed grounds hypothesis and very large cardinals.** / Usuba, Toshimichi.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - The downward directed grounds hypothesis and very large cardinals

AU - Usuba, Toshimichi

PY - 2017

Y1 - 2017

N2 - A transitive model (Formula presented.) of ZFC is called a ground if the universe (Formula presented.) is a set forcing extension of (Formula presented.). We show that the grounds of(Formula presented.)(Formula presented.)(Formula presented.) 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) (Formula presented.) 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 (Formula presented.) of ZFC is called a ground if the universe (Formula presented.) is a set forcing extension of (Formula presented.). We show that the grounds of(Formula presented.)(Formula presented.)(Formula presented.) 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) (Formula presented.) 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 - downward directed grounds hypothesis

KW - Forcing method

KW - generic multiverse

KW - large cardinal

KW - set-theoretic geology

UR - http://www.scopus.com/inward/record.url?scp=85031900598&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85031900598&partnerID=8YFLogxK

U2 - 10.1142/S021906131750009X

DO - 10.1142/S021906131750009X

M3 - Article

JO - Journal of Mathematical Logic

JF - Journal of Mathematical Logic

SN - 0219-0613

ER -