Talk (Work in Progress): Christopher Menzel (Texas A&M University)
Modal Set Theory and Potential Hierarchies
As is well known, Russell's Paradox motivated the development of axiomatic set theory, and its natural “model” — viz., the cumulative hierarchy of sets — arguably provided a satisfying structural explanation of where the reasoning in the paradox goes wrong. There is thus an undeniably robust foundation for iterative set theoretic (IST) realism, that is, realism about the cumulative hierarchy. Two fundamental intuitions lie at the heart of IST realism. The first, of course, is that sets are “constructed” in stages such that, beginning (perhaps) with an initial plurality of urelements, each stage consists of all the sets that can be formed from the things in the preceding stage. Call this the iterative intuition. The second — call it the realist intuition — is that all the sets are there, indeed, necessarily so in the case of pure sets. But these two fundamental intuitions of IST realism themselves give rise to a paradox of their own. For if all the sets are there, then why does the iterative intuition not apply to them? Why does the hierarchy not continue on, starting with the sets there in fact are, into yet higher levels? On the face of it, the two intuitions cannot both be true.
A lot of interesting and important work, notably by Charles Parsons and more recently by Øystein Linnebo, has brought modal logic and set theory together into a framework that, by spelling out the idea that the cumulative hierarchy is merely potential, promises (i) to reconcile the apparently conflicting intuitions that underlie IST realism, and moreover (ii) to account for the initially compelling intuitions underlying the principles that lead to Russell's paradox. However, in this paper, I will first argue that, while modal set theory solves the realist's dilemma if the modality is taken to be genuinely metaphysical, such modal set theoretic (MST) realism suffers from a similar and equally serious problem about the metaphysics of sets, viz., the apparent modal capriciousness of the existence of sets. The problems with both IST and MST realism cast doubts on the viability of either brand of realism. I will argue that the best hope for a reasonably robust form of mathematical realism about set theory lies with aspects of the modal structuralist program first proposed by Hilary Putnam and developed in great detail by Geoff Hellman (a recent critique by Sam Roberts notwithstanding).
An important consequence of the modal structuralist gambit is that it appears to entail the falsity of necessitism, i.e., the view (recently defended at extraordinary length by Timothy Williamson) that, necessarily, everything exists necessarily. I will close by discussing this consequence as well as some other tensions between necessitism and most any form of set theoretic reaiism.