The Connection Between Fractal Spacetime Theory and G¨odel’s Incompleteness Theorem: Mathematical Foundations and Physical Realizations

This paper systematically explores the profound connection between fractalspacetime theory and G¨odel’s incompleteness theorem. Fractal spacetime theoryintroduces a dimension-parameterized mathematical framework, whose dynamicand relative characteristics reveal the evolution of mathematical laws as spacetimedimensions change. The study is conducted across three levels:1. Dimensional Relativity of Formal Systems: By constructing dimension?parameterized formal systems, we prove G¨odel incompleteness across differentdimensions and uncover the possibility of proposition migration between di?mensions.2. Enhanced Undecidability in Dynamic Mathematical Laws: We ana?lyze the recursive unsolvability of cross-dimensional propositions and its im?pact on the incompleteness of fractal arithmetic.3. G¨odelian Barriers in Physical Realizations: We explore the theoreticallimits in fractal quantum computing and experimental validations, highlight?ing undecidability obstacles in quantum gravity research.This study demonstrates that fractal spacetime theory not only extends the appli?cability of G¨odel’s incompleteness theorem but also provides a new mathematicalframework and research boundaries for quantum gravity theory.