Large Cardinals and Determinacy

The developments of set theory in 1960’s led to an era of independence in which many of the central questions were shown to be unresolvable on the basis of the standard system of mathematics, ZFC. This is true of statements from areas as diverse as analysis (“Are all projective sets Lebesgue measurable?”), cardinal arithmetic (“Does Cantor’s Continuum Hypothesis hold?”), combinatorics (“Does Suslin’s Hypotheses hold?”), and group theory (“Is there a Whitehead group?”). These developments gave rise to two conflicting positions. The first position—which we shall call pluralism—maintains that the independence results largely undermine the enterprise of set theory as an objective enterprise. On this view, although there are practical reasons that one might give in favour of one set of axioms over another—say, that it is more useful for a given task—, there are no theoretical reasons that can be given; and, moreover, this either implies or is a consequence of the fact—depending on the variant of the view, in particular, whether it places realism before reason, or conversely—that there is no objective mathematical realm at this level. The second position—which we shall call non-pluralism—maintains that the independence results merely indicate the paucity of our standard resources for justifying mathematical statements. On this view, theoretical reasons can be given for new axioms and—again, depending on the variant of the view— this either implies or is a consequence of the fact that there is an objective mathematical realm at this level. The theoretical reasons for new axioms that the non-pluralist gives are quite different that the theoretical reasons that are customarily at play in mathematics, in part because, having stepped into the realm of independence, a more subtle form of justification is required, one that relies more heavily on sophisticated mathematical machinery. The dispute between the pluralist