Set theory is taken to serve as a foundation for mathematics. But it is well-known that there are set-theoretic statements that cannot be settled by the standard axioms of set theory. The Zermelo-Fraenkel axioms, with the Axiom of Choice (ZFC), are incomplete. Further, with the development of alternative foundational frameworks the question of what constitutes a foundation has been called into question. The primary goals of these conferences/symposia are to explore the different approaches that one can take to the phenomenon of incompleteness, and assess what it is for a theory to serve as a foundation of mathematics.
One way to address the former issue is to maintain the traditional “universe” view and hold that there is a single, objective, determinate domain of sets. Accordingly, there is a single correct conception of set, and mathematical statements have a determinate meaning and truth-value according to this conception. We should therefore seek new axioms of set theory to extend the ZFC axioms and minimize incompleteness. It is then crucial to determine what justifies some new axioms over others.
Alternatively, one can argue that there are multiple conceptions of set, depending on how one settles particular undecided statements. These different conceptions give rise to parallel set-theoretic universes, collectively known as the “multiverse”. What mathematical statements are true can then shift from one universe to the next. From within the multiverse view, however, one could argue that some universes are more preferable than others.
There is also a question as to what constitutes a foundation. The traditional approach has been to found mathematics using ZFC set theory. However, recent years have seen a more structural development of foundations based on category theory and type theory, an approach which has reached maturity with the recent book on Univalent Foundations.
These different approaches to incompleteness and foundations have wider consequences for the concepts of meaning and truth in mathematics. The conferences will address these foundational issues at the intersection of philosophy and mathematics.
The primary goal of the series is to showcase contemporary philosophical research on different approaches to these phenomena. To accomplish this, the conferences have the following general aims and objectives:
To bring to a wider philosophical audience the different approaches that one can take to the foundations of mathematics.
To elucidate the pressing issues of meaning and truth that turn on these different approaches.
To address philosophical questions concerning the need for a foundation of mathematics, and whether or not set theory can provide the necessary foundation