Published June 3, 2026 | Version v1
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Atomic Set Theory: A Unified Mathematical Framework Based on Atomic Primitives, Symmetry Breaking, and Entropy Quantification

Authors/Creators

Description

本文提出并系统构建了一个全新的数学基础理论——原子集合论(Atomic Set Theory, AST)。

AST以唯一不可分解的原子集合A = {2}为终极基元,重新定义了集合论、算术、几何、解析等数学分支的统一逻辑起点。围绕“原子构造、对称破缺、熵增量化、边界约束”这一核心脉络,构建了一个基于构造主义观点的动态演化的数学本体论框架。

主要结果:

  1. 为算术、几何、集合、解析提供统一公理基础,确立原子元公理及改造后的ZF公理体系。  
  2. 通过纽结幂集 $P_T$、凯勒势等价及A = {2} 的唯一性,建立跨域统一衔接与双射同构。  
  3. 以构造历史非等价性与路径冗余为核心,建立逻辑熵量化体系,导出熵增不可逆。  
  4. 以实无穷为本体,通过无穷公理调和实无穷与潜无穷观点。  
  5. 接纳并重构哥德尔不完备性,将“真但不可证”量化为可证熵与边界残熵的分解。  
  6. 剖析解析对称性(如 L-函数临界线零点)与离散对称性(如素数分布)的逻辑根源。  
  7. 基于三互质素数(47,59,71)的非交换纽结乘积,给出魔群(196883)与月光现象(196884)的构造性解释。  
  8. 计算出 P = NP 问题类真子集总信息量为 1.1 × 10¹²²比特。

AST为数学基础、数学物理与理论计算机科学提供了全新的公理化平台与统一视角。

Abstract

This paper proposes and systematically develops a novel foundational theory of mathematics—Atomic Set Theory (AST).  

AST takes the unique indecomposable atomic set A = {2} as its ultimate primitive and redefines the unified logical starting point of various mathematical branches, including set theory, arithmetic, geometry, and analysis. Centered around the sixteen-character core thread—“atomic construction, symmetry breaking, entropy quantification, and boundary constraints”—it constructs a dynamically evolving ontological framework of mathematics from a constructivist perspective.

Main results: 

  1.  Provides a unified axiomatic foundation for arithmetic, geometry, set theory, and analysis, establishing the atomic primitive axiom and a modified ZF axiom system.  
  2.  Via the knot power set  $P_T$ , Kähler potential equivalence, and the uniqueness of  A = {2}, establishes cross-domain unified connections and bijective isomorphisms.  
  3. With construction-history non-equivalence and path redundancy at its core, develops a quantification system for pure mathematical logical entropy, deriving the irreversibility of entropy increase.  
  4. Takes actual infinity as the ontology and reconciles actual and potential infinity through the axiom of infinity.  
  5. Embraces and reconstructs Gödel incompleteness, quantifying “true but unprovable” as a decomposition of provable entropy and boundary residual entropy.  
  6.  Analyzes the logical roots of analytic symmetry (e.g., critical-line zeros of  L-functions) and discrete symmetry (e.g., prime distribution).  
  7. Based on the non-commutative knot product of three coprime primes (47, 59, 71), gives a constructive explanation of the Monster group (196883) and moonshine phenomena (196884).  
  8.  Computes the total information content of the proper subclass related to the  P = NP  problem as $1.1 \times 10^{122}$ bits.

AST provides a new axiomatic platform and unified perspective for the foundations of mathematics, mathematical physics, and theoretical computer science.

  • Note on Language:The author‘s limited command of English makes it impractical to present a theory of this scale in a non-native language. For now, only the original Chinese version of Atomic Set Theory (AST) is provided. An English translation will take time.

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Translated title
原子集合论