Reference in arithmetic

Review of Symbolic Logic 11 (3):573-603 (2018)
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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.

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Lavinia Maria Picollo
National University of Singapore

Citations of this work

Gödel’s Theorem and Direct Self-Reference.Saul A. Kripke - 2023 - Review of Symbolic Logic 16 (2):650-654.
Reference and Truth.Lavinia Picollo - 2020 - Journal of Philosophical Logic 49 (3):439-474.
Alethic Reference.Lavinia Picollo - 2020 - Journal of Philosophical Logic 49 (3):417-438.

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References found in this work

The semantic conception of truth and the foundations of semantics.Alfred Tarski - 1943 - Philosophy and Phenomenological Research 4 (3):341-376.
Paradox without Self-Reference.Stephen Yablo - 1993 - Analysis 53 (4):251-252.

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