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.
Set-theoretic Pluralism, Symposium I, July 12-17, 2016, at the University of Aberdeen. My talk will be the final talk of the conference.
Abstract. I shall discuss several bits of pluralism-inspired mathematics, including especially an account of Toshimichi Usuba’s recent proof of the strong downward-directed grounds DDG hypothesis, which asserts that the collection of ground models of the set-theoretic universe is downward directed. This breakthrough settles several of what were the main open questions of set-theoretic geology. It implies, for example, that the mantle is a model of ZFC and is identical to the generic mantle and that it is therefore the largest forcing-invariant class. Usuba’s analysis also happens to show that the existence of certain very large cardinals outright implies that there is a smallest ground model of the universe, an unexpected connection between large cardinals and forcing. In addition to these results, I shall present several other instances of pluralism-inspired mathematics, including a few elementary but surprising results that I hope will be entertaining.
Slides | Set-theoretic Pluralism Network | Conference program