arXiv2026
Informal statements of Gödel's incompleteness theorems often run: "any consistent formal system with arithmetic has true but unprovable statements." That reading assumes a classical notion of truth, under which a universal statement is true as soon as each of its numeral instances is. This paper presents reflective grounded arithmetic (RGA), in which truth is grounded in computation and universal quantification is grounded reflectively: a universal statement is true when the system's own proof search certifies its schematic instance, and false when the system refutes one numeral instance. RGA allows recursive definitions without restriction and proves addition, multiplication and Ackermann's function total as theorems quantified inside the system. RGA is consistent, and it is complete for its own, grounded notion of truth: every well-formed statement that evaluates to true is provable. Gödel's theorems still hold for RGA, as a companion paper shows. The statements that classical reasoning calls true but unprovable have every numeral instance provable but no certified schematic instance, so they are ungrounded in RGA; read instance by instance, in an $ω$-semantics for which RGA is also sound, they are true. All results are machine-checked in Isabelle/HOL: soundness and consistency; completeness for grounded truth, open statements included; $ω$-soundness and $ω$-consistency; that grounded truth is recursively enumerable and RGA represents exactly the recursively enumerable sets; and $ω$-incompleteness. The logic is neither classical nor intuitionistic: double negation is eliminable as an entailment, excluded middle fails wherever a statement has no boolean value, RGA has the disjunction and witness properties, and the deduction theorem holds exactly at provably decided hypotheses.