Problem
Research questions and friction points this paper is trying to address.
Reflective Grounded Arithmetic
Internal Truth Predicate
Tarski's Undefinability Theorem
Metatheorems
Formal Verification
Internalized Truth in Reflective Grounded Arithmetic
🤖 AI Summary
This study addresses the challenge posed by Tarski’s Undefinability Theorem regarding custom truth predicates in arithmetic systems. Grounded in reflexive base arithmetic, this project constructs a global truth predicate within Isabelle/HOL. By leveraging primitive recursive compilation, symbolic expansion laws, and internal strong induction, we formally verify four fundamental metatheorems and establish a bidirectional equivalence between truth and provability. This work successfully circumvents undefinability constraints while confirming system consistency and demonstrating non-trivial mathematical reasoning capabilities. Ultimately, it provides a machine-verified theoretical and practical foundation for constructing effective self-referential formal systems, thereby advancing the mechanization of metamathematics within proof assistants.
📝 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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Reflective Grounded Arithmetic
Internal Truth Predicate
Tarski's Undefinability Theorem
Machine-Checked Verification
Certified Proof-Checker
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