CHALLENGE THE INFINITE SPEAKERS

Geoffrey Hellman

Geoffrey Hellman is Professor of Philosophy at the University of Minnesota, having held appointments and visitorships at several Universities across the world, including being a Visiting Fellow at Wolfson College, Oxford, in 1983 and ’85,

He is widely regarded as one of the most important Philosophers of Mathematics and his vast and varied output includes developing a modal-structural approach in philosophy of mathematics, publishing Mathematics without Numbers in 1989 (Oxford), with many follow-up papers over the next two decades.

He has also worked on assessing the adequacy of constructivism, predicative foundations of arithmetic and, more recently, pluralism in mathematics and implications for foundations.

Title: Modal Set Theory and Modal-Structural Interpretation vis-à-vis Critics and Then Each Other

Abstract

Recent critiques claim that "potentialist" versions of set theory fail to improve on standard "actualist" or single-universe set theories, either in explaining why there cannot be, e.g., a set of all sets (Soysal) or because they are mutually interpretable (Button). Au contraire, we explain why both critiques miss their mark by failing to appreciate modal extendability principle, and how they transcend the resources of single-universe set theories.

We then consider the "pros and cons" of modal set theories vs. modal-structuralist interpretations of set theories, especially first- and second-order ZFC. While the former can claim greater simplicity and proximity to standard ZFC, theory and practice, the latter can avoid certain problematic issues confronting the former, can motivate small-large-cardinal axioms, and is compatible with a nominalist ontology of the actual world.