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Informational Cardinality: A Unifying Framework for Set Theory, Fractal Geometry, and Analytic Number Theory

Zhengqiang Li

Abstract

This paper investigates a class of deterministic fractals whose construction is governed by arithmetic sequences. We introduce the essential fractal prime set P_{ess} , a variant of the Cantor set constructed using the sequence of prime numbers modulo 4. We compute its Hausdorff dimension, \dim_H(P_{ess}) , and analyze its geometric complexity. In contrast to the classical middle-third Cantor set C_{1/3} , we demonstrate that while both sets are uncountable and share the same cardinality, their differing fractal dimensions (dim_H(C_{1/3}) versus the computed dimension of P_{ess}) reflect a fundamental difference in their geometric complexity. Furthermore, we propose a potential connection between the density of this prime-driven fractal and the distribution of zeros of the Riemann zeta function, formalized through the construction of a fractal zero set Z_F . This framework provides a novel geometric perspective on analytic number theory, illustrating how the fine-scale structure of primes can be encoded in deterministic fractal geometries.

Informational Cardinality: A Unifying Framework for Set Theory, Fractal Geometry, and Analytic Number Theory

Abstract

This paper investigates a class of deterministic fractals whose construction is governed by arithmetic sequences. We introduce the essential fractal prime set P_{ess} , a variant of the Cantor set constructed using the sequence of prime numbers modulo 4. We compute its Hausdorff dimension, \dim_H(P_{ess}) , and analyze its geometric complexity. In contrast to the classical middle-third Cantor set C_{1/3} , we demonstrate that while both sets are uncountable and share the same cardinality, their differing fractal dimensions (dim_H(C_{1/3}) versus the computed dimension of P_{ess}) reflect a fundamental difference in their geometric complexity. Furthermore, we propose a potential connection between the density of this prime-driven fractal and the distribution of zeros of the Riemann zeta function, formalized through the construction of a fractal zero set Z_F . This framework provides a novel geometric perspective on analytic number theory, illustrating how the fine-scale structure of primes can be encoded in deterministic fractal geometries.
Paper Structure (56 sections, 15 theorems, 20 equations, 2 tables)

This paper contains 56 sections, 15 theorems, 20 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Informational Cardinality: Definitions and Basic Properties
  3. Motivation and Historical Context
  4. Formal Definitions
  5. Philosophical Interpretation
  6. The Essential Fractal Prime Set $P_{\text{ess}}$
  7. Motivation and Construction
  8. Mathematical Properties
  9. Information Measure of $P_{\text{ess}}$
  10. The Generalized Cantor Set $C_{1/3}$
  11. Comparison of Informational Cardinalities
  12. Geometric Duality and Information Conservation
  13. The Fractal Zero Set $Z_F$
  14. Information Conservation Conjecture
  15. Geometric Riemann Hypothesis
  16. ...and 41 more sections

Key Result

Theorem 3.3

$P_{\text{ess}}$ is a compact, perfect, nowhere dense subset of $[0,1]$ with Lebesgue measure zero.

Theorems & Definitions (46)

  • Definition 2.1: Cardinality Indicator
  • Definition 2.2: Geometric Dimension
  • Definition 2.3: Information Measure - Preliminary
  • Remark 2.4: Distinction from Classical Information Theory
  • Definition 2.5: Informational Cardinality
  • Definition 2.6: Comparison Rule
  • Definition 3.1: Essential Fractal Prime Set
  • Remark 3.2: Idealization
  • Theorem 3.3: Basic Properties
  • proof
  • ...and 36 more