Reference in Arithmetic

Abstract

Abstract. Self-reference has played a prominent role in the development of metamathematics in the past century, starting with Gödel's first incompleteness theorem. Given the nature of this and other results in the area, the informal understanding of self-reference in arithmetic has sufficed so far. Recently, however, it has been argued that for other related issues in metamathematics and philosophical logic a precise notion of self-reference and, more generally, reference, is actually required. These notions have been so far elusive and are surrounded by an aura of scepticism that has kept most philosophers away. In this paper I suggest we shouldn't give up all hope. First, I introduce the reader to these issues. Second, I discuss the conditions a good notion of reference in arithmetic must satisfy. Accordingly, I then introduce adequate notions of reference for the language of first-order arithmetic, which I show to be fruitful for addressing the aforementioned issues in metamathematics. §1. To prove his famous first incompleteness result for arithmetic, Gödel [5] developed a technique called "arithmetization" or "gödelization". It consists in codifying the expressions of the language of arithmetic with numbers, so that the language can 'talk' about its own formulae. Then, he constructed a sentence in the language that he described as stating its own unprovability in a system satisfying certain conditions, 1 and showed this sentence to be undecidable in the system. His method led to enormous progress in metamathematics and computer science, but also in philosophical logic and other areas of philosophy where formal methods became popular. Let's take a closer look.Let L be the language of first-order Peano arithmetic (PA). L contains =, ¬, ∧, ∨, →, ∀, and ∃ as logical constants, 0 as the only individual constant, S as a monadic function symbol, + and × as dyadic function symbols, and a stock of extra function symbols for recursive functions to be specified. All other logical and non-logical symbols are taken to be the usual abbreviations. We assume PA I am deeply indebted to Volker Halbach, with whom I had countless fruitful discussions on reference and self-reference over the last five years. I would also like to particularly thank Thomas Schindler, for great suggestions and encouragement, especially when it came to proofs.

Alethic Reference

“…A similar result can be found in Picollo [21], together with a proof. We are finally in a position to provide an adequate and precise definition of reference by quantification.…”

“…14 See Halbach & Visser [7,8]. 15 This is why extending the notions of reference for the pure language of arithmetic I put forward in Picollo [21] to L T is not viable, as anticipated in footnote 9. However, they can serve a heuristic role in the formulation of the alethic notions.…”

Reference and Truth

“…for ϕ ∈ L and ξ < 0 . 26 This means PA can prove that all ordinals below 0 are well ordered. Let L <0 be L and, for every α such that 0 < α ≤ 0 , let L <α be the result of extending L with monadic predicates T β , for each β < α. PAT <α consists of the axioms of PA formulated in L <α , with the induction schema extended to the whole language.…”

“…See Picollo[26] for an overview of the state of the matter. 2 See Herzberger[15], Visser[34], and Yablo[35,36].…”

“…8] 25. More comprehensive ordinal notation systems are also possible, but this is enough for our purposes 26. See Pohlers[28, chap.…”