Diagonal Arguments and Infinite Dependencies: Analyzing Classical Undecidability and Universality Under Finite Resource Constraints

This paper examines classical diagonal-based results (Cantor's uncountability, G\"odel's incompleteness, Turing's halting problem, and computational universality) through a finite-resource lens. We analyze the diagonal pattern and its dependence on completed enumerations and on unbounded time, space, and precision, then formalize a finite framework $S(T_{\max}, S_{\max}, P_{\max}, L_{\max})$ with integer bounds on time, memory, numerical precision, and symbolic length, and analyze each result within this framework. Within this setting: (i) finite-decimal reals admit explicit enumeration via constant-time bijections; (ii) for formal systems, when bounds are chosen adequate for the system under study, formulas and proofs are finitely enumerable and provability is decidable (complete within bounds); (iii) for the halting problem, adequacy (time beyond the finite-configuration threshold) yields a definitive HALTS/LOOP decision for every machine-input pair, whereas without adequacy the same procedure provides a sound bounded classification (HALTS/TIMEOUT); and (iv) no machine operating under fixed finite bounds is universal in the classical sense. These results show how classical results depend on infinite idealizations and exhibit different behavior under explicit finite resource constraints.