# O’Hara Professor of Philosophy and Mathematics, University of Notre Dame

I have now taken up a position at the University of Notre Dame as the O’Hara Professor of Philosophy and Mathematics, beginning January 2022.

My appointment is with the Department of Philosophy with an affiliation with the Department of Mathematics. I expect to be teaching and working with students both in philosophy and mathematics.

Notre Dame offers a unique joint PhD degree program between mathematics and philosophy, the program in logic and the foundations of mathematics. For Notre Dame undergraduates of any major, I encourage you to consider the mathematical philosophy minor.

Notre Dame has strong research groups in logic in both philosophy and mathematics. In philosophy, Notre Dame recently came out very well in the speciality PGR rankings in philosophy of mathematics (#2, tied with NYU, Princeton, behind Harvard), mathematical logic (#2 tied with CMU, behind Harvard), and philosophical logic (group 2). In mathematics, Notre Dame has a strong research group in mathematical logic.

# The pluralist perspective on the axiom of constructibility, MidWest PhilMath Workshop, Notre Dame, October 2014

This will be a featured talk at the Midwest PhilMath Workshop 15, held at Notre Dame University October 18-19, 2014.  W. Hugh Woodin and I will each give one-hour talks in a session on Perspectives on the foundations of set theory, followed by a one-hour discussion of our talks.

Abstract. I shall argue that the commonly held $V\neq L$ via maximize position, which rejects the axiom of constructibility V = L on the basis that it is restrictive, implicitly takes a stand in the pluralist debate in the philosophy of set theory by presuming an absolute background concept of ordinal. The argument appears to lose its force, in contrast, on an upwardly extensible concept of set, in light of the various facts showing that models of set theory generally have extensions to models of V = L inside larger set-theoretic universes.

Set-theorists often argue against the axiom of constructibility V=L on the grounds that it is restrictive, that we have no reason to suppose that every set should be constructible and that it places an artificial limitation on set-theoretic possibility to suppose that every set is constructible. Penelope Maddy, in her work on naturalism in mathematics, sought to explain this perspective by means of the MAXIMIZE principle, and further to give substance to the concept of what it means for a theory to be restrictive, as a purely formal property of the theory. In this talk, I shall criticize Maddy’s proposal, pointing out that neither the fairly-interpreted-in relation nor the (strongly) maximizes-over relation is transitive, and furthermore, the theory ZFC + `there is a proper class of inaccessible cardinals’ is formally restrictive on Maddy’s account, contrary to what had been desired. Ultimately, I shall argue that the V≠L via maximize position loses its force on a multiverse conception of set theory with an upwardly extensible concept of set, in light of the classical facts that models of set theory can generally be extended to models of V=L. I shall conclude the talk by explaining various senses in which V=L remains compatible with strength in set theory.

This talk will be based on my paper, A multiverse perspective on the axiom of constructibility.

Slides