Exploring the Frontiers of Incompleteness
The "Exploring the Frontiers of Incompleteness" project is made possible by the generous support of the John Templeton Foundation, through a grant given to Peter Koellner. The aim is to bring together some of the most prominent thinkers who have struggled with the following questions:
- (1) Do the questions that are independent of the standard axioms admit of determinate answers?
- (2) If so then what are those answers and how might we go about determining them?
We shall do this by actually engaging with the major figures in this foundational debate. Through generous external support we have managed to secure the funding necessary to make this possible. Over the two semesters Fall 2011 and Spring 2012 there were 12 workshops. Each workshop involved a presentation by one of the major figures in the debate. The speakers were (in order of scheduled workshop):
Fall Semester 2011
Spring Semester 2012
The speaker presentations occurred (roughly) every two weeks and during the intervening weeks background material was provided for the upcoming presentation. In addition, the paper of the presentation was made available in advance. Moreover, all 12 speakers were involved throughout the process—they too received the papers in advance and were given an opportunity to comment on it. At the end of the workshop series there was a master-workshop (something like a conference but more interactive) involving all workshop-speakers and all participants.
General Background Material
Workshop DiscussionDiscussion of the workshops took place on the Discussing the Frontiers page. Highlights of the discussion are posted here.
Hugh Woodin (September 21, 2011): Workshop Materials · Slides · Discussion · Video
Comments on Woodin's Realm of the Infinite (Peter Koellner). Main Paper: The Realm of the Infinite. Supplementary Material:
Solomon Feferman (October 5, 2011): Workshop Materials · Slides · Discussion · Video
Comments on Feferman's Is the Continuum Hypothesis a definite mathematical problem? (Peter Koellner). Main Paper: Is the Continuum Hypothesis a definite mathematical problem? Supplementary Material:
- Infinity in Mathematics: Is Cantor Necessary?
- Does mathematics need new axioms?
- The philosophy of mathematics. 5 questions (mainly the end part)
- Conceptual structuralism and the continuum (slides for talks in San Sebastian, UCLA and Stanford)
- What's definite? What's not? (slides for the Harvey Fest)
- Conceptions of the continuum
Joel Hamkins (October 19, 2011): Workshop Materials · Slides · Discussion · Video
Main Paper: The Set-theoretic Multiverse Title: The multiverse perspective on determinateness in set theory Abstract: 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. Supplementary Material:
Charles Parsons (November 2, 2011): Workshop Materials · Discussion · Video
Main Paper: Evidence and the hierarchy of mathematical theories Supplementary Material:
- William Tait (November 16, 2011): Workshop Materials · Discussion · Video
- Michael Rathjen will not be presenting a paper as part of the workshop.
- Philip Welch (February 15, 2012): Workshop Materials · Discussion · Video
Stevo Todorcevic: Workshop Materials
Main paper: The power-set of ω1 and the continuum problem
James Cummings (March 21, 2012): Workshop Materials · Slides · Discussion · Video
Main Paper: Some challenges for the philosophy of set theory (preliminary draft)
- John Steel (March 28, 2012): Workshop Materials · Slides · Discussion · Video
Tony Martin (April 4, 2012): Workshop Materials · Discussion · Video
Main Paper: Completeness or Incompleteness of Basic Mathematical Concepts Supplementary Material:
- Gödel's Conceptual Realism
- Mathematical Evidence, in Truth in Mathematics
- Multiple Universes of Sets and Indeterminate Truth Values
Menachem Magidor (April 18, 2012): Workshop Materials · Discussion · Video
Main Paper: Some Set Theories Are More Equal