Exploring the Frontiers of Incompleteness
11 videos • 1,438 views • by Ivan Grozev 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.