# Set-theoretic geology and the downward directed grounds hypothesis, Bonn, January 2017

This will be a talk for the University of Bonn Logic Seminar, Friday, January 13, 2017, at the Hausdorff Center for Mathematics.

Abstract. Set-theoretic geology is the study of the set-theoretic universe $V$ in the context of all its ground models and those of its forcing extensions. For example, a bedrock of the universe is a minimal ground model of it and the mantle is the intersection of all grounds. In this talk, I shall explain some recent advances, including especially the breakthrough result of Toshimichi Usuba, who proved the strong downward directed grounds hypothesis: for any set-indexed family of grounds, there is a deeper common ground below them all. This settles a large number of formerly open questions in set-theoretic geology, while also leading to new questions. It follows, for example, that the mantle is a model of ZFC and provably the largest forcing-invariant definable class. Strong downward directedness has also led to an unexpected connection between large cardinals and forcing: if there is a hyper-huge cardinal $\kappa$, then the universe indeed has a bedrock and all grounds use only $\kappa$-small forcing.

Slides

# The Ground Axiom

• J. D. Hamkins, “The Ground Axiom,” Mathematisches Forschungsinstitut Oberwolfach Report, vol. 55, pp. 3160-3162, 2005.
@ARTICLE{Hamkins2005:TheGroundAxiom,
AUTHOR = "Joel David Hamkins",
TITLE = "The {Ground Axiom}",
JOURNAL = "Mathematisches Forschungsinstitut Oberwolfach Report",
YEAR = "2005",
volume = "55",
number = "",
pages = "3160--3162",
month = "",
note = "",
abstract = "",
keywords = "",
source = "",
eprint = {1607.00723},
archivePrefix = {arXiv},
primaryClass = {math.LO},
url = {http://jdh.hamkins.org/thegroundaxiom/},
file = F,
}

This is an extended abstract for a talk I gave at the 2005 Workshop in Set Theory at the Mathematisches Forschungsinstitut Oberwolfach.