Recent advances in set-theoretic geology, Harvard Logic Colloquium, October 2016

I will speak at the Harvard Logic Colloquium, October 20, 2016, 4-6 pm.

harvard

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

A conference in honor of W. Hugh Woodin’s 60th birthday, March 2015

I am pleased to announce the upcoming conference at Harvard celebrating the 60th birthday of W. Hugh Woodin.  See the conference web site for more information. Click on the image below for a large-format poster.

woodin_conference_poster

Exploring the Frontiers of Incompleteness, Harvard, August 2013

I will be participating in the culminating workshop of the Exploring the Frontiers of Incompleteness conference series at Harvard University, to take place August 31-September 1, 2013.  Rather than conference talks, the program will consist of extended discussion sessions by the participants of the year-long series, with the discussion framed by very brief summary presentations.  Peter Koellner asked me to prepare such a presentation on the multiverse conception, and you can see the slides in The multiverse perspective in set theory (Slides).

My previous EFI talk was The multiverse perspective on determinateness in set theory, based in part on my paper The set-theoretical multiverse.

The multiverse perspective on determinateness in set theory, Harvard, 2011

This talk, taking place October 19, 2011, is part of the year-long Exploring the Frontiers of Incompleteness (EFI) series at Harvard University, a workshop focused on the question of determinateness in set theory, a central question in the philosophy of set theory. JDH at Harvard Streaming video will be available on-line, and each talk will be associated with an on-line discussion forum, to which links will be made here later.

In this talk, I will discuss the multiverse perspective on determinateness in set theory.  The multiverse view in set theory is the view that there are many distinct concepts of set, each instantiated in a corresponding set-theoretic universe. The universe view, in contrast, asserts that there is an absolute background set concept, with a corresponding absolute set-theoretic universe in which every set-theoretic question has a definite answer.  The multiverse position, I argue, explains our experience with the enormous diversity of set-theoretic possibilities, a phenomenon that challenges the universe view.  In particular, I shall argue that the continuum hypothesis is settled on the multiverse view by our extensive knowledge about how it behaves in the multiverse, and as a result it can no longer be settled in the manner formerly hoped for.

Workshop materials | Article | Slides | EFI discussion forum | Video Stream