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.


The Ground Axiom

  • J. D. Hamkins, “The Ground Axiom,” Mathematisches Forschungsinstitut Oberwolfach Report, vol. 55, p. 3160–3162, 2005.
    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 = {},
    file = F,

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

Oberwolfach Research Report 55/2005 | Ground Axiom on Wikipedia