The Term Model of ρ-Arithmetic / ρ-算术的项模型

Paper 7 of the ZFCρ series. Completes Milestone M3: the formal theory Tρ is defined as a multi-sorted first-order theory with three sorts (H for history terms, N for natural numbers, W for cost values), and its term model is constructed inside ZFC.

The axiom sources are entirely standard: H-sort uses a structural induction schema (four constructors, paralleling PA induction); N-sort uses PA axiom schema; W-sort uses Presburger arithmetic signature (addition, order, induction, no multiplication). The standalone pure W-language subsystem is Presburger arithmetic (complete, decidable), though the full theory Tρ is not. val and ρ are introduced via cross-sort recursive equations as axiom equations of Tρ.

The term model carries H = TΣ (free ΣH-term algebra, encoded as finite labeled trees in ZFC), N = ℕ, W = ℕ. val and ρ are the two canonical projections from TΣ as ΣH-initial algebra to the standard natural number structure and the cost algebra respectively — both uniquely determined by the universal property of initial algebras.

Key results: (1) Term-model existence theorem for any positive integer parameter triple. (2) Hist(n) is countably infinite (explicit injection constructed). (3) Hist*(n) is finite (strong induction proof). (4) Compactification lemma via six cross-sort semantic reduction rules with context lemmas for val-preservation and ρ-strict-decrease, termination by well-foundedness of ℕ. (5) ρE and ρval well-defined as term-model objects. All axiom satisfaction proofs stated and proved in metatheory ZFC. Bilingual CN-EN.

Resource type: Preprint

License: CC BY 4.0

Language: English, Chinese

Keywords: ρ-arithmetic, term model, multi-sorted first-order theory, initial algebra, Presburger arithmetic, structural induction, compactification lemma, free term algebra, Σ-homomorphism, foundations of mathematics

Related identifiers:

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