What question did this study set out to answer?

The central aim is to redefine numbers through a polar structure and explore the implications of incompleteness in mathematics.

June 19, 2026Open Access

A Number System Generated from the Divisions of Unity: An Axiomatic Polar Re-description of the Complex Field, with a Localisation of Incompleteness

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Foundational work explores the nature of numbers using polar form and axiomatic methods, suggesting implications for mathematical completeness.

The central aim is to redefine numbers through a polar structure and explore the implications of incompleteness in mathematics.
Introduced two divisions in a polar framework: magnitude and phase.
Utilized axioms to interpret essential mathematical constants and functions through a new lens.
Emphasized the reliance on existing mathematical structures and completeness in the formulation.
Identified the polar structure as inherent in existing axioms, leading to results reliant on classical mathematics.
Established a model for decidability of the complex field while highlighting incompleteness in foundational arithmetic.
Provided a framework that emphasizes generating processes within number theory, addressing the absence of fixed points.

Özgür Ünsal (2026) studied this question.

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Foundational work explores the nature of numbers using polar form and axiomatic methods, suggesting implications for mathematical completeness.