International Journal of Mathematics Trends and Technology
Volume 71 | Issue 8 | Year 2025 | Article Id. IJMTT-V71I8P106 | DOI : https://doi.org/10.14445/22315373/IJMTT-V71I8P106
| Received | Revised | Accepted | Published |
|---|---|---|---|
| 17 Jun 2025 | 26 Jul 2025 | 14 Aug 2025 | 30 Aug 2025 |
Frantz Olivier, "The Riemann Hypotheses: A Lesson in Universal Paradoxes that Lead to the Axiom of Choice – A Path of Unexpected Outcomes," International Journal of Mathematics Trends and Technology (IJMTT), vol. 71, no. 8, pp. 34-43, 2025. Crossref, https://doi.org/10.14445/22315373/IJMTT-V71I8P106
The Riemann Hypothesis, in the article “A view on how to solve it”, challenges the schema of Russell’s Paradoxes through its result. This paper presents, in three segments, how the paradoxes can be viewed. In each segment, paradoxes are introduced to force a deeper look into the area of discourse. This main issue embraces the entire essay as it progresses to different aspects of this work. The first segment outlines the original naïve set concept and schema. The second segment submits the naïve set in the context of functional reasoning, which is a derivation of common sense and reason. Common sense will show that the empty set is irrelevant when functional reasoning, in its core derivation, is considered. Core derivation imposes the Gödel arguments for the existence of God. Thus, these sets of axioms and theorems strengthen the axiom of choice set theory and expose mathematics as the transport mechanism to touch upon the state of consciousness in its understanding of the universe as an evolving entity. Furthermore, the eloquent design of the universe in the quantum sphere of interaction is partly intrinsic to internal awareness of the self. Notwithstanding, the external makeup of human beings is influenced deeply by the classical knowledge of the world. Finally, the third segment, the functional reasoning, introduces Artificial Intellig ence and sets the stage for the Peano Axiom-Theory. The latter sets the stage for limits and quantifiers, which will open the door for omega, the recursive schema associated with a limit point.
Riemann Hypothesis, Quantum, Artificial Intelligence, Universe, Axiom.
[1] Alfred V. Aho et al., Data Structures and Algorithms, 1983.
[Google Scholar] [Publisher Link]
[2] John C. Amazigo, Advanced Calculus and Its Applications to the Engineering and Physical Sciences, Wiley, 1st Edition, 1980.
[Google Scholar] [Publisher Link]
[3] Frank Ayres, and Elliot Mendelson, Schaum’s Outline of Calculus (Shaum’s Outlines), McGraw-Hill Education – Europe, 2000.
[Publisher Link]
[4] Thomas Becker, and Volker Weispfenning, “Gröbner Bases A Computational Approach to Commutative Algebra,” Graduate Texts in
Mathematics, vol. 141, 1993.
[CrossRef] [Google Scholar] [Publisher Link]
[5] Thomas Becker, Volker Weispfenning, and Heinz Kredel, Grobner Bases A Computational Approach to Commutative Algebra,
Springer, Corrected Edition, 1993.
[Publisher Link]
[6] Robert Blitzer, Thinking Mathematically, 4th Edition, 2018.
[Google Scholar]
[7] Folkmar Bornemann, “PRIMES is in P: Breakthrough for Everyman,” Notices of the American Mathematical Society, pp. 545-552,
2003.
[Google Scholar] [Publisher Link]
[8] Arne Broman, Introduction to Partial Differential Equations from Fourier Series to Boundary-Value Problems, 2012.
[Google Scholar] [Publisher Link]
[9] Samuel Buss, Bounded Arithmetic, Department of Mathematics, University of California, Berkley, 1997.
[Google Scholar]
[10] P.J. Cameron, and J.H. Van Lint, Designs, Graphs, Codes, and their Links, Cambridge, Great Britain, 1991.
[Google Scholar] [Publisher Link]
[11] Georg Cantor, Contributions to the Founding of the Theory of Transfinite Numbers, Dover Publications, 1955.
[Google Scholar] [Publisher Link]
[12] Judith N. Cederberg, A Course in Modern Geometries, Second Edition, 2001.
[Google Scholar] [Publisher Link]
[13] Harvey Cohn, Advanced Number Theory, Dover Publications, 1980.
[Google Scholar] [Publisher Link]
[14] Harvey Cohn, Conformal Mapping on Riemann Surfaces, Dover Publications, 1967.
[Google Scholar] [Publisher Link]
[15] Richard E. Crandall, Projects in Scientific Computation, Springer-Verlag New York, 1994.
[Google Scholar] [Publisher Link]
[16] Nell B. Dale, and Chip Weems, Introduction to PASCAL and Structured Design, D.C. Heath, 1983.
[Google Scholar] [Publisher Link]
[17] Rene Descartes, The Geometry of Rene Descartes, David Eugene Smith, 1925.
[Google Scholar] [Publisher Link]
[18] John Derbyshire, Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics, Plume, 2004.
[Google Scholar] [Publisher Link]
[19] Reinhard B. Diestel, Graph Decompositions A Study in Infinite Graph Theory, Clarendon Press, 1990.
[CrossRef] [Google Scholar] [Publisher Link]
[20] A. Evyatar, and P. Rosenbloom, Motivated Mathematics, Cambridge University Press, 1981.
[Google Scholar] [Publisher Link]
[21] Paul C. Evyatar, and Paul Rosenbloom, Motivated Mathematics, CUP Archive, 1981.
[Google Scholar] [Publisher Link]
[22] Gill Williamson, Combinatorics for Computer Science, 1985.
[Publisher Link]
[23] Frisk Gustafson, Elementary Geometry, Third Edition, 1991.
[Google Scholar] [Publisher Link]
[24] David W. Henderson, and Daina Taimina, Experiencing Geometry, Third Edition, 2005.
[Google Scholar]
[25] Klaus Jänich, Vector Analysis, Springer, 2000.
[CrossRef] [Google Scholar] [Publisher Link]
[26] John W. Keese, Elementary Abstract Algebra, DC Heath/Order HM College, 1965.
[Google Scholar] [Publisher Link]
[27] Donald E. Knuth, The Art of Computer Programming, Fundamental Algorithms, 2nd Edition, Addison-Wesley, 1973.
[Google Scholar]
[28] Neal Koblitz, A Course in Number Theory and Cryptography, Springer-Verlag, Second Edition, 1994.
[Google Scholar] [Publisher Link]
[29] Neal Koblitz, A Course in Number Theory and Cryptography, Springer, 1994.
[Google Scholar] [Publisher Link]
[30] Bernard Kolman, and Robert C. Busby, Discrete Mathematical Structures for Computer Sciences, 1984.
[Google Scholar]
[31] Roland E. Larson, Robert P. Hostetler, Bruce H. Edwards, Calculus, Sixth Edition, Boston, New York, 1998.
[Publisher Link]
[32] Jean Louis Lassez, and Gordon Plotkin, Computational Logic: Essays in Honor of Alan Ronbinson, The MIT Press, 1991.
[Google Scholar] [Publisher Link]
[33] Harry R. Lewis, and Christos H. Papadimitriou, Element of the Theory of Computation, Prentice-Hall, 2nd Edition, 1997.
[Google Scholar] [Publisher Link]
[34] Melvin J. Maron, and Robert J. Lopez, Numerical Analysis: A Practical Approach, International Thomson Publishing, 1991.
[Publisher Link]
[35] The American Mathematical Monthly, Mathematical Association of America, 1894. [Online]. Available:
https://maa.org/publication/the-american-mathematical-monthly/
[36] Walter Meyer, Geometry and its Applications, Academic Press, 1999.
[Google Scholar] [Publisher Link]
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