Independence and Ignorance: How Agnotology Informs Set-Theoretic Pluralism

Abstract

IntroductionMuch of the discussion of set-theoretic independence, and whether or not we could legitimately expand our foundational theory, concerns how we could possibly come to know the truth value of independent sentences. This paper pursues a slightly different tack, examining how we are ignorant of issues surrounding their truth. We argue that a study of how we are ignorant reveals a need for an understanding of set-theoretic explanation and motivates a pluralism concerning the adoption of foundational theory.Materials and MethodsOur strategy is as follows. First ("Varieties of Independence"), we note two different kinds of independence, those that are sensitive to large cardinal axioms, and those that are not. We pick two well-studied examples from the literature: {Cantor's Continuum Hypothesis} (henceforth CH and {Projective Determinacy} (henceforth ‘PD’). We then ("Multiversism and Pluralism") present two views concerning set-theoretic ontology (namely Multiversism and Universism), and explain how each might be linked to the acceptance or rejection of a Pluralism in set-theoretic foundations. Next ("Varieties of Ignorance"), we change tack and exposit some literature on the study of ignorance (often called agnotology). We then ("What is our Ignorance of Independent Sentences Like?"), examine how various positions might regard our ignorance of CH and PD. Finally ("How Ignorance Affects Pluralism"), we argue that certain views of our ignorance of independent sentences motivate pluralism concerning the study of set-theoretic foundations, even on a Universist picture.ConclusionWe conclude that despite the prima facie tension between Universism and Pluralism, the character of our ignorance suggests a fusion of the two positions.