Categorical structure in coherent theory of arithmetic

In this paper we provide a semantic and syntactic analysis of parametrised natural numbers object in coherent categories, or pr-coherent categories. Semantically, we show the definable functions in the initial pr-coherent category are exactly given by primitive recursive functions. We also show that any pr-coherent category supports the construction of bounded universal quantifications, which are absent in an arbitrary coherent category. Under these semantic consideration, we construct a coherent theory of arithmetic and we show its syntactic category is equivalent to the initial pr-coherent category. From a logical perspective, we also show that this theory can be identified as the Σ1-fragment of IΣ1. Thus as an application, we provide a structural proof of the classical result in proof theory that the strongly Σ1-representable functions in IΣ1 are exactly primitive recursive functions.

💡 Research Summary

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This paper investigates the categorical foundations of primitive recursive arithmetic within the framework of coherent categories equipped with a parametrised natural numbers object (PNO), which the author calls pr‑coherent categories. The work has three intertwined strands: a semantic analysis of definable functions, a syntactic construction of a coherent arithmetic theory, and an application to proof‑theoretic characterisation of Σ₁‑representable functions in the fragment IΣ₁.

Semantic part.
A pr‑coherent category is a coherent category that possesses a PNO satisfying induction for all coherent formulas (not merely equations). The author constructs the initial such category C and shows that any function definable in C – i.e. any morphism from the PNO to itself – is exactly a primitive recursive function. The key technical device is the auxiliary category PriM, whose objects are recursively enumerable sets and whose morphisms are primitive recursive functions between them. PriM is shown to be pr‑coherent, and the canonical functor C → Set factors through PriM by initiality. Consequently every definable morphism in C lands in PriM and therefore is primitive recursive (Theorem 5.10). This mirrors Román’s earlier result for Cartesian categories but demonstrates that the additional logical structure of coherent categories (disjunction, existential quantification) does not enlarge the class of definable functions.

Syntactic part.
Coherent logic lacks a universal quantifier, yet the author proves that any pr‑coherent category internally supports bounded universal quantification (∀ z < t). Using this, a coherent arithmetic theory T is introduced. T contains constant 0, all primitive recursive function symbols (zero, successor, projections, composition, primitive recursion), a binary relation < defined via addition, a bounded universal quantifier rule, and an induction rule for the PNO. Although T extends coherent logic, all its formulas are Σ₁‑formulas (built from atomic formulas using ⊤, ∧, ⊥, ∨, ∃, and bounded ∀). The syntactic category C