Research

My research is in the philosophy of logic and mathematics, with focus on issues concerning quantification and set theory. Additional interests include early analytic philosophy and the philosophy of Bertrand Russell. I am a member of the Bertrand Russell Society. A full statement of my research interests is available here.

According to the iterative conception of set, each set is a collection of sets formed prior to it. The notion of priority here plays an essential role in explanations of why contradiction-inducing sets, such as the Russell set, do not exist. Consequently, these explanations are successful only to the extent that a satisfactory priority relation is made out. I argue that attempts to do this have fallen short: understanding priority in a straightforwardly constructivist sense threatens the coherence of the empty set and raises serious epistemological concerns; but the leading realist interpretations---ontological and modal interpretations of priority---are deeply problematic as well. I conclude that the purported explanatory virtues of the iterative conception are, at present, unfounded.

"Explanation and Paradox"

I discuss the explanatory virtues of proofs that contradiction-inducing sets, such as the universal set, do not exist in standard ZFC set theory. Do such proofs explain why these sets don't exist? Do they help explain the consistency of ZFC? Finally, do the explanations we seek rely on the so-called 'iterative conception of set'? And what sorts of pressures do our explanatory demands place on this conception?

"A Formulation of Restrictivism"

Straightforward attempts to express restrictivism presuppose unrestricted quantification and are therefore self-defeating. I argue that the restrictivist avoids self-defeat if she expresses her view as the schematic claim that any domain of quantification has an expansion.

"Axiomatic Set Theory and the Iterative Conception of Set"

It is generally held (a) that axiomatic set theory expresses the iterative conception of set and (b) that the iterative conception of set explains the consistency of axiomatic set theory. I argue that a consistent naive set theory, based on a variant of the axiom of full comprehension, also expresses the iterative conception of set, which in turn explains the consistency of this naive set theory. Consequently, the epistemic allure of the iterative conception does not justify favoring axiomatic set theory over naive set theory