Thematically, formally and structurally, Wallace’s writing concerned itself with the infinite, from the antinomies of set theory and the obese Bombardini in The Broom of the System to the featureless horizon of Peoria in The Pale King, by way of the title of Infinite Jest and the brief and not wholly successful exploration of Cantorian mathematics in Everything and More, the idea of the infinite was never far from any of Wallace’s writing. Moreover, the structures of the writing continually reinscribe this obsession with infinity, with none of the novels conforming to a traditional boundaried structure and the collections of short fiction troubling the very concept of order in their use of pagination and enumeration. This chapter illuminates the importance of infinity to Wallace’s writing by exploring its formal and thematic development through his career, demonstrating that infinity worked as a conceptual counterpoint to solipsism, both an existential threat and a source of profound hope for the disassociated subject of contemporary culture.

Historians are constantly confronted with the twin problems of translating texts and interpreting their meanings. When mathematicians like Georg Cantor or Abraham Robinson demonstrate the consistency of concepts that, since the paradoxes of Zeno and Democritus, have been assumed to be paradoxical notions like infinitesimals or the actual infinite, how should the works of earlier mathematicians be regarded, who either used such concepts or believed they had proven their impossibility? Is it anachronistic to use nonstandard analysis or transfinite numbers to “rehabilitate” or explain the works of Leibniz, Euler, Cauchy, or Peirce, for example, as recent mathematicians, historians, and philosophers of mathematics have attempted? At the other extreme, chronologically, how may ideas readily accepted in the West – like incommensurable numbers, parallel lines, and similar triangles – but foreign to traditional Chinese mathematics have adversely affected the interpretations of ancient Chinese mathematical works?

This chapter is the first of two exploring the idea that the core thought of the naïve conception – that there is an intimate connection between sets and properties – can be preserved as long as we build into the conception the idea that certain properties are pathological and, for this reason, do not determine a set. The chapter first uses a result from Incurvati and Murzi (2017) to show that restricting attention to those properties that do not give rise to inconsistency will not do. It then focuses on the limitation of size conception of set, according to which the pathological properties are those that apply to too many things. Various versions of the doctrine are distinguished. The chapter also discusses what it calls the definite conception, according to which the pathological properties are the indefinitely extensible ones. It is argued that the limitation of size fails to provide a complete explanation of the set-theoretic paradoxes. The definite conception faces the same problem and, in addition, it is unclear whether it has the resources to develop a reasonable amount of set theory.