On the Concept of Arithmetic Conseqeunce

TL;DR

This paper offers a proof-theoretic semantics perspective on G"odel's incompleteness theorem, showing that arithmetical theories can support their own consistency statements through semantic support, despite not proving them syntactically.

Contribution

It introduces a semantic notion of consequence based on inferential roles, revealing divergence from derivability and reframing incompleteness without relying on external models.

Findings

Arithmetical theories support their formalized consistency statements semantically.

Semantic support can diverge from syntactic derivability in arithmetic.

Reframes G"odel's incompleteness as a divergence between two notions of consequence.

Abstract

G\"odel's second incompleteness theorem is standardly understood as showing that no sufficiently strong, consistent theory of arithmetic can prove its own consistency, a result typically interpreted against a model-theoretic background in which arithmetical language is evaluated with respect to an independently given structure of natural numbers. This paper develops an alternative perspective grounded in proof-theoretic semantics. We distinguish between derivability and a semantic notion of consequence given by support, defined compositionally in terms of the inferential roles fixed by a theory. For suitable arithmetical theories A formulated in a finite signature (such as Robinson's Q and Peano Arithmetic), these two notions can diverge in a principled way: although A does not prove its own consistency, it nevertheless supports its formalized consistency statement, and more generally supports sentences not derivable within it. This does not conflict with G\"odel's incompleteness theorem, but instead reframes incompleteness as a divergence between two internally determined notions of consequence associated with a single theory, rather than as a gap between syntactic provability and truth in a mind-independent structure. The result clarifies the relationship between reflection, consistency, and inferentialist approaches to meaning, and shows how substantial semantic determinacy may arise from the inferential structure of arithmetic itself.