Set Theory and Logic

2412 Submissions

[3] viXra:2412.0126 [pdf] submitted on 2024-12-22 04:26:36

Infinite Numbers in Mathematics

Authors: Weike Xu
Comments: 49 Pages. The sequence of sections in this article is carefully arranged for correct understandings. It is recommended for professionals to read in order.

A method named `motional construction' is introduced in this article. Questions involving continuum hypothesis and incompleteness theorems of formal systems are answered, but a major concern of this article is a nature of infinity: Infinity implies paradoxes.Many conclusions contradicting orthodox mathematics are proved, such as: a set of real numbers does not exist, Lebesgue measure of any set is zero, ZF axioms are not logically consistent, etc. Unreliable results are common in sub-fields of mathematics where infinite sets are used intensively.A valid mathematical conclusion describes finiteness in essence.When infiniteness is clear, the third mathematical crisis shall be over.
Category: Set Theory and Logic

[2] viXra:2412.0121 [pdf] submitted on 2024-12-20 00:42:02

A Progress on the P vs NP Problem

Authors: Theophilus Agama
Comments: 13 Pages.

This paper presents a significant advancement in understanding the P vs NP problem through the lens of problem theory. Using isotopes as a technical tool within this framework, we provide a solution to the problem, establishing that $mathrm{P}=mathrm{NP}$. The results demonstrate the effectiveness of the proposed theoretical framework in addressing fundamental problems in computational complexity.
Category: Set Theory and Logic

[1] viXra:2412.0005 [pdf] submitted on 2024-12-02 21:40:58

On the Principle of Structural Dependency and Applications

Authors: Theophilus Agama
Comments: 3 Pages. This is a short note containing a novel principle for reconstructing proofs.

This note formalizes and applies the emph{Principle of Structural Dependency}, which asserts that if the foundation of a mathematical structure ( B ) consists of another structure ( A ), then ( A ) cannot exhibit a property distinct from ( B ), while ( B ) may possess properties not shared by ( A ). We verify this principle and apply it systematically to reconstruct concise proofs of several classical theorems, including Cantor's theorem, the Fundamental Theorem of Algebra, the Jordan Curve Theorem, the Monotone Convergence Theorem, and the Pythagorean Theorem. These reconstructions emphasize the structural underpinnings of these results, offering a novel perspective and demonstrating how foundational relationships can simplify complicated proofs.
Category: Set Theory and Logic