Number Theory

    

The Fundamental Circularity Theorem: Why Some Mathematical Behaviours Are Inherently Unprovable

Authors: Alister John Munday

This paper introduces the Fundamental Circularity Theorem (FCT), establishing that certain mathematical behaviours are inherently unprovable because they emerge directly from fundamental properties that cannot themselves be proven without circular reasoning. Using the Collatz conjecture as our primary example, we demonstrate how mathematical behaviours that arise purely from the interaction of fundamental properties resist formal proof not due to complexity or logical paradox, but because any proof would require proving unprovable fundamentals. This insight offers a new understanding of mathematical unprovability distinct from Gödel's incompleteness theorems or independence results.

Comments: 6 Pages.

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[v1] 2025-01-04 14:22:10

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