Proof of the Non-Existence of Infinity: Thermodynamic and Informational Constraints on Physical Reality

Abstract This paper provides a logical proof of the non-existence of “infinity” within the hierarchy of physical reality, grounded in the laws of thermodynamics and information theory. Key Highlights: Existence as Difference: Building upon the author’s previous definition (Saito, 2025), I demonstrate that maintaining any difference requires a minimum thermodynamic energy cost (Landauer’s Principle). Physical Upper Bound: Given the finite energy of the universe, the number of definable differences is necessarily capped; thus, infinity cannot manifest as an actual physical entity. Resolving the Continuum Hypothesis: This proof illustrates how the fundamental resolution of physical space (Planck scale) renders questions of infinite cardinality moot. Conclusion: I conclude by re-defining the universe as a dynamic computing system with finite resources, establishing the logical foundation for the "Law of Mutual Interdependence" and the "Principle of Mutual Optimization" detailed in subsequent works.