Categorical characterizations of the natural numbers require primitive recursion

Simpson and the second author asked whether there exists a characterization of the natural numbers by a second-order sentence which is provably categorical in the theory RCA$^*_0$. We answer in the negative, showing that for any characterization of the natural numbers which is provably true in WKL$^*_0$, the categoricity theorem implies $\Sigma^0_1$ induction. On the other hand, we show that RCA$^*_0$ does make it possible to characterize the natural numbers categorically by means of a set of second-order sentences. We also show that a certain $\Pi^1_2$-conservative extension of RCA$^*_0$ admits a provably categorical single-sentence characterization of the naturals, but each such characterization has to be inconsistent with WKL$^*_0$+superexp.Comment: 17 page

Categoricity

After a short preface, the first of the three sections of this paper is devoted to historical and philosophic aspects of categoricity. The second section is a self-contained exposition, including detailed definitions, of a proof that every mathematical system whose domain is the closure of its set of distinguished individuals under its distinguished functions is categorically characterized by its induction principle together with its true atoms (atomic sentences and negations of atomic sentences). The third section deals with applications especially those involving the distinction between characterizing a system and axiomatizing the truths of a syste

Topics in arithmetic and determinacy

This thesis is about Arithmetical Determinacy. Loosely, this is the problem of whether every question in arithmetic has a determinate answer. In this work I discuss how to exactly understand the concept of determinacy, I criticise arguments for and against the claim that arithmetic is determinate, and examine how questions about determinacy may be applied to other debates in the philosophy of mathematics. Chapter 1 isolates different ways of understanding the problem of arithmetical determinacy. Chapter 2 turns to mathematical structuralism and explains how popular computability constraints thought to determine the reference of our arithmetical vocabulary are actually unsuccessful in securing determinacy. Chapter 3 criticises an interesting idea for securing determinacy via our experience with supertasks. Chapter 4 explores the phenomenon of mutually inconsistent satisfaction classes and motivates a new account of determinacy in terms of sentences possessing non-classical truth-values. Chapter 5 defends strict finitism, framing some objections against the view in terms of the concept of arithmetical determinacy