Computable Quantification in Reflective Grounded Arithmetic

2026-07-28

AI Summary

This work addresses the limitations of Gödel’s incompleteness theorems in classical logic, which rely on specific foundational assumptions that hinder the simultaneous achievement of consistency and expressive power. The paper proposes Reflective Grounded Arithmetic (RGA), a novel framework grounded in computational semantics for truth, wherein universal quantification is interpreted reflectively via internal proof search, and unrestricted recursive definitions are supported. RGA operates within a non-classical, non-intuitionistic substructural logical setting, enabling it to precisely characterize recursively enumerable sets while preserving consistency. Using formal verification in Isabelle/HOL, the study establishes RGA’s consistency, openness completeness, N-soundness, Church–Turing expressiveness, and ω-incompleteness, thereby revealing its distinctive logical properties.

Abstract

Informal statements of Gödel's incompleteness theorems often run: "no consistent formal system with arithmetic can be complete" - omitting the fact that the theorems as proved assume classical logic. This paper presents reflective grounded arithmetic (RGA), a paracomplete arithmetic in which truth is grounded in computation rather than assumed by classical fiat, and in which universal quantification is grounded reflectively: a universal statement is true when the system's own proof search certifies its schematic instance, and false when it refutes a particular numeral instance. RGA permits unconstrained recursive definitions, proves the totality of addition and multiplication as internally quantified theorems, and represents exactly the recursively enumerable sets - the ingredient list of the folklore Gödel statement - while remaining consistent. This work proves, with all results machine-checked in Isabelle/HOL: soundness and consistency; open completeness - provability coincides with grounded truth on well-formed statements; N-soundness - every provable totality claim is backed by an actual value; a Church-Turing characterization of RGA's expressive power; and $ω$-incompleteness - grounded truth is recursively enumerable, and therefore some family of statements has every numeric instance provable while its universal closure is not merely unprovable but semantically ungrounded. The resulting logic occupies a Markov-flavored, substructural corner distinct from both classical and intuitionistic arithmetic: double-negation elimination holds, quantified excluded middle fails, refuted universals yield explicit counterexample witnesses, and the deduction theorem's abstraction direction fails precisely at ungrounded hypotheses.

Problem

• Gödel incompleteness
• grounded truth
• paracomplete arithmetic
• recursive enumerability
• ω-incompleteness

Innovation

• reflective grounded arithmetic
• paracomplete logic
• computable truth
• ω-incompleteness
• machine-checked verification