On determinacy and arithmetic
: First steps towards an account of the determinacy of arithmetic

Student thesis: Master's ThesisMaster of Philosophy

Abstract

Philosophers and mathematicians typically assume that arithmetic is determinate, i.e., that all well-formed arithmetical statements have a determinate truth-value. By contrast, many believe that set theory could be indeterminate. This thesis intends to take the first steps towards grounding the determinacy of arithmetic thus understood and establishing the contrast with the case of set theory.

The first chapter of the thesis surveys extant material on the determinacy of arithmetic. Thus, it first introduces the notion of arithmetical determinacy and presents a challenge in the literature that urges us to account for the determinacy of arithmetic. Then, it outlines and assesses some of the most influential arguments against the determinacy of arithmetic, paying special attention to Putnam’s modeltheoretic arguments [Put80, Put81], and the responses that different philosophers have offered against these arguments.

The second chapter takes a step back, and notices that the nature of the project requires an understanding of the broader picture regarding determinacy as a feature of formal systems. Thus, it addresses the more general phenomenon of truththeoretic determinacy: it aims at laying the foundations for a theory of determinate truth that allows us to speak about determinacy from within the object mathematical theories themselves. To this extent, it draws on three theories that present a desirable trait for a theory of truth, namely supervaluational truth. These theories are: the van Fraassen-Kripke fixed-point semantics [Kri75], Stern’s supervaluationalstyle truth [Ste18], and McGee’s theory of definite truth [McG91]. In the chapter, special attention is given to McGee’s theory, for it displays an attractive feature: material adequacy for the truth predicate.

Finally, the last chapter advances a first defence of the determinacy of arithmetic, based on what is often known as Isaacson’s thesis [Isa87, Isa92]. According to this thesis, arithmetical truth coincides with provability in Peano Arithmetic; thus, if the thesis holds, arithmetical determinacy is guaranteed, for all there is to arithmetic is provability or refutability in Peano Arithmetic. The chapter advances a challenge to the thesis, according to which the latter could entail that Peano Arithmetic proves non-arithmetical truths. It then sets out to argue that the challenge can be withstood, since all seemingly non-arithmetical truths could, in fact, be shown to be arithmetical.
Date of Award1 Feb 2023
Original languageEnglish
Awarding Institution
  • King's College London
SupervisorCarlo Nicolai (Supervisor)

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