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?
These are very difficult questions and there are many prominent philosophers and mathematicians who have given them a great deal of thought. There is a broad spectrum of views. For example, at one end there are people like Solomon Feferman who think that there are objective facts of the matter about questions pertaining to the natural numbers but think that most of the questions of set theory (most notably, the Continuum Hypothesis (CH)) are indeterminate since the underlying notions of set theory are inherently vague. At the other end of the spectrum there are people like Hugh Woodin who have provided serious arguments (based on a wealth of mathematical results) for thinking that questions like CH are determinate and who have advanced major programs (again based on a wealth of mathematical results) for determining those answers. There are many views in between and there are views which are entirely orthogonal to this ordering. The main purpose of the two-part series is to investigate these various positions, compare their strengths and weaknesses, and make steps forward in determining the answers to the two guiding questions.
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.