An Interdisciplinary Interpretation of G¨odel’s Incompleteness Theorem: Fractal Metaphors and Rigorous Exploration of System Openness

G¨odel’s incompleteness theorem reveals the intrinsic limitationsof formal systems, revolutionizing foundational mathematics whileoffering a unique perspective on the hierarchical openness of com?plex systems. This paper integrates perspectives from mathematics,physics, and cognitive science to explore the conceptual resonance be?tween fractal theory and G¨odel’s theorem under strict boundaries ofmetaphorical applicability and empirical prioritization. Specifically:1. Mathematical Perspective: A rigorous analysis of the dis?crete self-similarity in the recursive structures of formal systemsusing recursive function theory and fractal geometry.2. Physical Perspective: An exploration of the recursive featuresof physical systems and their potential mapping to formal sys?tems through loop quantum gravity, renormalization groups, andtopological order models.3. Cognitive Perspective: Insights into the fractal-inspired mech?anisms of self-referential cognitive processes, combining compu?tational complexity theory with neuroscience experiments.Through mathematical modeling, physical system analogies, and cog?nitive experimental design, this paper delineates the boundaries ofinterdisciplinary metaphors and constructs a testable experimentalframework.