Infinity as an Undefined Concept: Structural Limits of Mathematical Formalization

This paper examines infinity not as a large or extreme quantity, but as an intrinsically undefined concept that resists formal objectification.We argue that many unresolved problems in mathematics and theoretical computer science do not arise from technical difficulty, but from a structural mismatch: mathematics operates on defined objects, while infinity represents the absence of a definable boundary. By carefully distinguishing infinity from both numerical zero and null, the paper reframes infinity as an undefined magnitude rather than a manipulable value. From this perspective, we show that mathematical frameworks remain coherent only when infinity is treated as a process, limit, or directional constraint, and not as a completed entity. Classical successes—such as limits, convergence, and differential calculus—are shown to rely on this implicit restriction. In contrast, many undecidable or intractable problems can be interpreted as structural failures caused by treating infinity as an object. Rather than proposing new axioms, this work clarifies the conceptual role of infinity and highlights how respecting its undefined nature sharpens the boundary between meaningful formalization and structurally ill-posed problems.