This paper establishes a multi-layer classification system for mathematical structures based on their axiomatic systems, operational and relational symbols, and categorical equivalences. The system employs successive refinement criteria including the base type of the structure (algebraic, topological, order-theoretic, categorical, etc.), cardinality features, arities of symbols and logical complexity of axioms, properties of substructures and quotient structures, model-theoretic stability, additional compatible structures, and concrete models. This yields an arbitrarily extensible classification tree. Each layer is equipped with rigorous axiomatic principles and fundamental theorems, ensuring that any mathematical structure (including sets, groups, rings, fields, lattices, topological spaces, measure spaces, manifolds, categories, toposes, and higher categories) can be uniquely placed into a specific node of the tree. Conversely, any parameter combination corresponding to a node can mechanically generate an axiom system and predict as-yet-unstudied mathematical structures. The system exhibits unity, completeness, and extensibility, analogous to the periodic table of chemical elements, and can be used to systematically discover and construct new mathematical objects. This paper provides formal definitions, construction methods, fundamental theorems, and multiple examples, and shows how classical branches of mathematics are embedded into the system, as well as how to build axiom systems and fundamental theorems for vacant parameter combinations. All theorems are given rigorous proofs (general theorems at least 4 steps, important theorems at least 8 steps), and all predictions are equipped with complete axiom systems and existence constructions, transforming the development of mathematics from sporadic discovery into a systematic fill-in-the-blank construction.
Liu S (Wed,) studied this question.
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