A Pure Logical Proof of the Zhu-Liang Natural Number Structure Recursive Element Theorem

Within the axiomatic system of recursive elements, using set theory, recursion theory, and category theory, this paper presents a pure logical proof that the natural number structure $\mathbb{N}$ (the model of Peano arithmetic) is a fundamental recursive element of the mathematical universe. We first define the four axioms that a recursive element must satisfy: Existence (A1), Encoding Invariance (A2), Metabolic Conservation (A3), and Generativity (A4). Subsequently, we employ set theory and category theory to prove existence and generativity, model theory to prove encoding invariance, and recursion theory with category theory to prove metabolic conservation. Combining these four parts yields the Zhu-Liang Natural Number Structure Recursive Element Theorem, and we further prove at the meta-level the self-consistency of the recursively nested structure. The theorem reveals that the natural numbers are not only the foundation of mathematics but also a core recursive element that recursively generates the entire mathematical edifice; their truth originates from the recursive self-consistency requirement of the formal system itself.