Proof-Theoretic Equivalence and Conservative Extension of ρ-Arithmetic / ρ-算术的证明论等价与保守扩张

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Paper 8 of the ZFCρ series. Completes Milestone M4: establishing the precise proof-theoretic position of ρ-arithmetic.

Core Theorem 1 (Equiconsistency): Con(PA) ⟺ Con(Tρ). The consistency strength of Tρ is exactly PA. Introducing H-sort (history terms) and W-sort (cost values) costs no additional consistency.

Core Theorem 2 (N-sort Conservative Extension): Tρ is a conservative extension of PA over the pure ΣN-language. Every pure natural-number sentence provable in Tρ is already provable in PA.

Corollary (W-sort Conservative Extension): Tρ is a conservative extension of Presburger arithmetic over the pure ΣW-language.

Method: a primitive recursive translation operator τ from Tρ-formulas to PA-formulas, with a guarded translation theorem (global H-variable Term guards), closed-sentence corollary, and guard elimination via UG+UI+MP. The proof covers the complete Hilbert calculus: propositional tautology schema, universal instantiation schema (with Term closure lemmas for general H-terms), quantifier distribution axiom schema (with propositional premise-swap bridge), equality axiom schema, modus ponens, and universal generalization. G5 structural induction translates to strong induction on Gödel codes. The argument is formalizable in PRA without ZFC.

ρ-arithmetic is a non-destructive upgrade of classical arithmetic: the extensional layer (N-sort) is fully conserved, the cost layer (W-sort) is fully conserved, and the operative layer (H-sort) adds higher-dimensional structure without altering any truth about natural numbers. Bilingual CN-EN.

Resource type: Preprint

License: CC BY 4.0

Language: English, Chinese

Keywords: ρ-arithmetic, equiconsistency, conservative extension, proof translation, Gödel encoding, primitive recursive, Hilbert calculus, Presburger arithmetic, proof-theoretic ordinal, foundations of mathematics

Related identifiers:

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