Internalized Truth in Reflective Grounded Arithmetic

Bryan Ford

arXiv:2608.16140v1 Primary: math.LO Secondary: cs.LO, cs.PL

Abstract

By Tarski's undefinability theorem, no consistent classical formal system that includes arithmetic can define its own truth predicate. Reflective Grounded Arithmetic (RGA) is a powerful arithmetic whose universal quantifier is grounded in its own reflected proof search, and whose paracompleteness circumvents Tarski's theorem. This paper presents a machine-checked Isabelle/HOL development that defines a truth predicate for RGA's full language, quantifiers included, as an internal term of RGA itself. This term is compiled from a primitive-recursive decider for its operational semantics, and proven adequate in both directions. Around this predicate the development closes a square of metatheorems: for every formula RGA proves, RGA derives the formula's internal truth; every grounded-true formula is internally provable; internal truth implies internal provability; and the consistency of RGA follows. The two directions run on disjoint internal machines---a certified decider and a certified proof-checker, both RGA terms. Reaching these results involved substantial ordinary reasoning carried out within RGA: coded syntax and substitution, compiled primitive-recursive functions with symbolic unfolding laws, internal strong induction, and a verified proof-checker for the system written in the system's own formal language. The development thus demonstrates along the way that RGA is a workable formal system supporting nontrivial mathematical reasoning.

Disclosure

“on 8 reports as a methodology result in its own right. AI disclosure: both the machine-checked proofs and the first drafts of this paper’s text were produced with sub- stantial assistance from artificial intelligence (Claude Fable from Anthropic), working under the direction and review of the human author, as detailed later in Section 8. Section 2 develops the background sketched above; Section 3 builds the substrate and the two machines; Section 4 assembles the adequacy squar”
PDF page 3

Classification: Substantial proof generation
Multiplier: 10
Verified: 2026-08-23T05:19:03.650694Z

Structural counts

Pages 22 pdf

Theorems 16 source

Lemmas 0 source

Propositions 0 source

Corollaries 0 source

Definitions 0 source

Displayed equations 6 source

Bibliography entries 149 source

Appendix pages 0 estimated

Count notes

Source counts use the expanded primary TeX file main.tex.
Appendix pages include the first PDF page with an explicit Appendix heading through the final page.