On the Philosophical Roots of the Naïve and Axiomatic Set Theories: Determinatio est Negatio

Authors : Osman Gazi Birgül

Pages : 73-83

Doi:10.26650/arcp.1559355

View : 12 | Download : 34

Publication Date : 2024-12-31

Article Type : Research Paper

Abstract :The principle determinatio est negatio—that determination is achieved through negation—has philosophical roots extending back to Plato and Aristotle, and it later influenced early modern thinkers such as Francisco Suárez and Spinoza. This paper has two aims. The first demonstrates how the principle of negation functions as a tool for conceptual determination across various philosophical frameworks, and the second demonstrates that the principle plays a key role in the analysis and resolution of the Burali-Forti paradox within the context of the naïve and axiomatic set theories. In the first section, the analysis focuses on the evolution of the principle from ancient philosophy to early modern metaphysics, examining Plato’s dialectics, Aristotle’s metaphysical distinctions, Suárez’s scholastic theories, and Spinoza’s monist metaphysics. The second section shifts to mathematics, where determinatio est negatio plays a key role in resolving set-theoretical paradoxes, particularly the Burali-Forti paradox. By exploring Cantor’s solution and its reliance on the distinction between consistent and inconsistent sets, this study demonstrates how this principle is essential for avoiding self-referential inconsistencies. The contributions of Zermelo and von Neumann, who developed axiomatic frameworks to address these paradoxes, further elaborate the philosophical foundations of set theory. Ultimately, the study reveals a deep connection between metaphysical principles and the mathematical treatment of paradoxes, emphasizing the ongoing relevance of determinatio est negatio in both domains.
Keywords : Determinatio est negatio, Burali-Forti paradoksu, Cantor, kümeler teorisi, öz-referans