[Paper Review] On the Concept of Arithmetic Conseqeunce

Alexander V. Gheorghiu|arXiv (Cornell University)|Mar 10, 2026

Philosophy and Theoretical Science0 citations

TL;DR

The paper argues that in arithmetic, derivability and semantic support diverge under proof-theoretic semantics; a theory can semantically support its Con but not derivably prove it, reframing Gödel’s incompleteness.

ABSTRACT

Gödel'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 Gödel'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.

Motivation & Objective

• Introduce a proof-theoretic semantics for arithmetic that ties meaning to inferential roles rather than external truth conditions.
• Differentiate between syntactic derivability and semantic support within arithmetic theories.
• Demonstrate, for suitable finite-signature theories (e.g., Q, PA), that Con(A) is semantically supported but not derivable.
• Clarify how this separation informs reflection principles and the interpretation of Gödel’s theorems.
• Relate inferentialism to traditional notions of consistency and meaning in arithmetic.

Proposed method

• Define a base as a set of atomic inference rules capturing an agent’s commitments.
• Introduce derivability (⊢_B) from a base B and the semantic support relation (⊩_B) for formulas.
• Present Sandqvist’s proof-theoretic semantics with compositional clauses for logical constants.
• Prove a soundness/completeness link under the condition of having infinitely many constants (Theorem 8).
• Formalize Con(A) as the negation of Prov_A(⊥) and show A ⊩ Con(A) while not having A ⊢ Con(A).
• Discuss maxiconsistent bases and reflection principles as part of the interpretive framework.

Experimental results

Research questions

• RQ1 Can semantic support diverge from syntactic derivability within a fixed arithmetical theory A?
• RQ2 Under what conditions does A semantically support Con(A) without proving it?
• RQ3 How does the finitary (finite-signature) nature of arithmetic affect the relation between derivability and semantic consequence?
• RQ4 What is the role of reflection principles in connecting proof-theoretic semantics to Gödel’s incompleteness theorems?
• RQ5 How does this framework reinterpret meaning and consistency in arithmetic?

Key findings

• For suitable arithmetical theories A in a finite signature, semantic support yields A ⊩ Con(A) even though A ⊬ Con(A).
• Gödel’s second incompleteness theorem remains valid in derivability terms but does not preclude semantic support of consistency statements.
• The gap between derivability and support arises from the absence of a completeness condition that requires infinitely many unused constants in the language.
• The result reframes incompleteness as a divergence between two internally determined notions of consequence rather than a gap between provability and mind-independent truth.
• This approach clarifies the relationship between reflection principles, consistency, and inferentialism about mathematical meaning.
• The framework connects Dummett’s idea of indefinite extensibility to formal reflection notions via provability as a constructive extension.