“How did the serpent of inconsistency enter Frege’s paradise?”

The paper explores the alleged connection between indefinite extensibility and the classic paradoxes of Russell, Burali-Forti, and Cantor. It is argued that while indefinite extensibility is not per se a source of paradox, there is a degenerate subspecies—reflexive indefinite extensibility—which is. The result is a threefold distinction in the roles played by indefinite extensibility in generating paradoxes for the notions of ordinal number, cardinal number, and set respectively. Ordinal number, intuitively understood, is a reflexively indefinitely extensible concept. Cardinal number is not. And Set becomes so only in the setting of impredicative higher-order logic—so that Frege’s Basic Law V is guilty at worst of partnership in crime, rather than the sole offender.

Russell Reductio Redux

By far and away the strongest argument against there being an absolutely comprehensive domain of quantification comes from the set-theoretic paradoxes. The argument from indefinite extensibility can be rigorously regimented with the help of schematic or modal resources. After dispensing with the charge that the argument relies on an incoherent conception of set, this chapter offers a defence of its premisses. Advocates of the orthodox absolutist means to defend absolute generality have yet to give a non-ad-hoc response to the paradoxes. A heterodox absolutist view, which seeks to give an absolutist-friendly account of indefinite extensibility, leads to severe problems with impure set theory. The chapter closes by considering a revenge problem for hybrid relativists, who take modalized quantifiers to achieve absolute generality.

Russell, Zermelo, and Dummett

Concerns about generality in the context of set theory are not new. Russell seeks to resolve the set-theoretic antinomies by maintaining that we cannot legitimately speak of ‘all classes’. Zermelo attempts to avoid the paradoxes without ‘constriction and mutilation’ by adopting an open-ended conception of the cumulative hierarchy of sets. Dummett takes the indefinite extensibility of concepts such as set and ordinal to impugn absolutism about quantifiers. But not every paradox-inspired argument is an argument for relativism about quantifiers. This chapter aims to fill in the logical and philosophical background to the contemporary absolute generality debate, with an eye to disentangling my favoured indefinite-extensibility-based argument from others in its vicinity.

The Question of Platonism

This book defends the existence of abstract mathematical objects. Should this be regarded as a defense of Platonism? Platonism involves an analogy between mathematical and physical objects. Although mathematical objects are counterfactually independent of us, just like paradigmatic physical objects, there are other respects in which mathematical objects are strikingly different from physical objects: by giving rise to the phenomenon of indefinite extensibility and by having a shallow nature. The view here is therefore not a full-blown form of Platonism. However, the shallow nature of mathematical objects has the advantage of enabling an epistemology of mathematics where our mathematical beliefs are appropriately sensitive to the truth of these beliefs.