Some Naturalistic Reflections on Set Theoretic Method

Abstract

Set theory has been subject to sharp methodological disputes from its very beginnings, but over the years most of these - like those concerning impredicative definitions or the axiom of choice - have been resolved to the general satisfaction of the set theoretic community. So many themes have run through these debates, so many theses have been raised, challenged, and defended, that it's difficult to sort out which of these considerations actually brought about the stable outcome. Some observers think these matters were decided on philosophical grounds; some pessimists see only sociological forces at work. These questions are important for the practice of contemporary set theory, because methodological questions remain to this day: what is the status of independent questions like the Continuum Hypothesis? Should they be abandoned or pursued? If the latter, what means are appropriate? In particular, how are new axiom candidates to be evaluated? On these issues, I disagree with both the philosophers and the pessimists. In the now-settled methodological disputes, I think consensus was eventually reached for good mathematical reasons, that is, for reasons integrally connected to the mathematical goals the developing theory hoped to attain: for example, a classical theory of real numbers, in the case of impredicative defInitions; a staggeringly wide range of particular benefits in the case of the axiom of choice (a wellbehaved theory of infinite cardinals, to take just one example). In other words, I think one can argue for the rationality of these methodological decisions of the past and illuminate the underlying justificatory structure of debates of the present by strict attention to the detailed mathematical considerations in play.! This is what I call 'naturalized methodology', and I have tried to apply it in some particular cases.2 One factor often cited in philosophical and methodological discussions of mathematics is the widespread