Real Numbers Are Countably Infinite: The Fatal Defects of Cantor's Diagonal Argument and the Constructive Complexity Alternative

Abstract

Cantor's diagonal argument is widely regarded as the definitive proof that the real numbers are uncountable. Within the framework of Bu Theory, this paper identifies three fatal defects: (1) conflating computability boundary with set-theoretic cardinality; (2) ignoring the legitimate existence of the diagonal counterexample c f as a well-defined mathematical object; (3) never considering the possibility of a surjection that exists but is not computable. By introducing the M"obius-fiber topology—where 0≡1, +∞≡−∞≡0≡∞ are valid equalities—this paper constructs a set-theoretically legal surjection framework N→R. Ten experiment groups with complete numerical data confirm that the real numbers are countably infinite in the constructive sense. At the critical state p=1, the fiber operator activates the GUE repulsion kernel, lifting the spacing ratio from the deterministic skeleton value 1/e≈0.368 to the post-fiber GUE value ≈0.599.