A Paradox on the Law of Excluded Middle in the framework of category of set

TL;DR

This paper questions whether the Law of Excluded Middle (LEM) is universally valid within classical mathematics by constructing a paradox in complex analysis. It defines $F_1$, $F_2$, $F_3$ and $f(z)=\Ln(F_1(z))+\Ln(F_2(z))+\Ln(F_3(z))-\Ln(z)+\Ln(2)-\frac{i\pi}{2}-i c$ on a connected open set $\mathcal{D}$ and shows $f$ is analytic, nonzero, yet has zeros that are not isolated. This contradicts the standard result that zeros of a non-identically-zero analytic function are isolated, linking the paradox to LEM's role in discovering relations rather than objects and suggesting epistemic uncertainty and undecidability in the logical framework. The authors advocate exploring alternative logical paradigms (e.g., intuitionistic or paraconsistent logics) to ground mathematics where LEM may fail.

Abstract

In this paper, we present a paradox arising from the acceptance of the Law of Excluded Middle (LEM) within classical mathematics. Specifically, we construct a nonzero analytic function on a connected open subset of the complex plane whose zeros are not isolated. This contradicts a fundamental theorem in complex analysis, thereby revealing an inconsistency tied to LEM. Unlike traditional critiques that reject LEM in favor of intuitionistic or constructive mathematics, we argue that LEM is instrumental in discovering \textbf{relations} between objects and facts rather than the objects themselves. Since we are not always in direct attachment with objects, this relational perspective introduces \textbf{inherent uncertainty} in mathematical reasoning. Consequently, we propose that the logical framework of the world is undecidable, making contradictions possible in more complex contexts. Our findings suggest that LEM, while powerful, may not be universally reliable in all mathematical frameworks. This work has implications for foundational mathematics, particularly in relation to the limits of classical logic and the necessity of alternative logical paradigms.

Figures (3)

• Figure 1: The semicircle of $|z-\frac{\mathbf{i}}{2}|=\frac{1}{2}$, $x\le 0,\ y\ge 0$.
• Figure 2: The segment $z=\mathbf{i}\frac{r}{r+1}$, $r\ge 0$.
• Figure 3: The region $\mathcal{D}$.

Theorems & Definitions (4)

• Remark 3.1
• Remark 3.2
• Remark 3.3
• Remark 3.4