THE BOUNDARY LAW OF ANADIHILO: A FORMAL RESOLUTION TO RUSSELL’S PARADOX AND SET-THEORETIC RECURSION

Published: February 16, 2026 Author: Nitin Dagar
Title: The Boundary Law of Anadihilo: Resolving Russell's Paradox. Description: A formal mathematical resolution to Russell's Paradox and Set-Theoretic recursion using the Anadihilo Framework by Nitin Dagar. Keywords: Russell's Paradox, Set Theory, Anadihilo, Nitin Dagar, GSO Reset, Set-Theoretic Recursion, Formal Logic. Author: Nitin Dagar. DOI: 10.5281/zenodo.18265144
Formal Logic & Set Theory

The Boundary Law of Anadihilo: A Formal Resolution to Russell's Paradox

Researcher: Nitin Dagar  |  Release: January 2026  |  DOI: 10.5281/zenodo.18265144

Abstract

Traditional set-theoretic frameworks, such as Zermelo-Fraenkel (ZFC), fail to account for a primordial systemic boundary, leading to recursive loops. This article formalizes the Boundary Law of Anadihilo ($\anh$) to resolve structural paradoxes. By defining the Absolute Void preceding the functional Zero, we prove that self-referential paradoxes trigger a Global Systemic Overwrite (GSO) rather than logical collapse.

1. The Failure of Locality in Set Theory

Russell's Paradox ($S = \{x | x \notin x\}$) arises when a set is defined by its own non-membership, creating an infinite recursive loop. Standard mathematics lacks a "ceiling" to terminate this calculation. The Anadihilo framework introduces the Axiom of Normalization to establish this missing boundary.

$\anh + n = 0$ (for $n \ge 0$)

The Absorption interaction between the Absolute Void and systemic magnitude.

2. The Three Systemic Laws of Bounded Logic

To ensure mathematical consistency, every Anadihilo Group ($\delta_n^p$) is governed by these three laws:

I. Law of Identity Shift

$U_1 \cup U_2 \Rightarrow \gamma_{(n1+n2)}^{(p1+p2)}$

Interaction results in a re-initialization of total magnitude into a new base power.

II. Law of Non-Recursion

$\anh \notin \gamma_n^p$

The Absolute Void ($\anh$) is the field in which sets exist; it is ontologically external to the sets it initializes.

3. The GSO Reset: Resolving the Paradox

Russell's Paradox is resolved through the Global Systemic Overwrite (GSO) mechanism. When a collection attempts to include "All Sets," it reaches the absolute boundary ($\anh$).

Step Process Logic Outcome
1. Reach Limit Magnitude $n$ hits boundary $\anh$ Critical Saturation
2. Absorption Axiom of Normalization triggers Data Neutralization
3. Transformation Set is transformed into new Base GSO Reset Complete

4. Computational Validation

Comparing two systems with identical internal perceptions but different foundational bases reveals the scaling identity:

  • System A ($\gamma_{10}^5$): Total Magnitude $M_A = 150$
  • System B ($\gamma_5^5$): Total Magnitude $M_B = 75$

Finding: Identity of each set is anchored to its specific base ($n$). This differentiation ensures that elements from different scales cannot clash in a self-referential loop.

Systemic Conclusion

"Paradoxes are not errors, but signals for systemic re-initialization. All manifest existence is thus a sustained observation of the void within a specific magnitude of scale."

Primary References

  1. Dagar, N. (2026). ANADIHILO ($\anh$): The Ontological Primacy of the Absolute Void. DOI: 10.5281/zenodo.18194393
  2. Cantor, G. (1874). On a Property of the Collection of All Real Algebraic Numbers.
  3. Russell, B. (1903). The Principles of Mathematics. Cambridge University Press.

Peer Discussion