On the Inadmissibility of Infinity as an Operator

This paper establishes that infinity, when treated as an operator that licenses existence, totality, or completion, is inadmissible in principle. Appeals to infinity are shown to function not as neutral abstractions, but as powerful foundational operators that silently bypass admissibility constraints. The analysis proceeds by exhaustion across the major ways infinity is deployed: actual infinities, completed infinite totalities, unbounded quantification, infinite constructions, and limit-based completion principles. In each case, the use of infinity presupposes global scope, unrestricted domains, or meta-level existence assumptions that violate standing conservation and collapse admissibility. Infinity is therefore reclassified as a representational shorthand, not a source of foundational legitimacy. While infinite expressions may serve as internal bookkeeping devices within already-licensed frameworks, they cannot be used to ground existence, justify universality, or complete partial structures. Once stripped of operator status, no admissible form of infinity remains that can do foundational work. The result is negative and non-constructive. The paper does not advocate finitism, constructivism, or any alternative foundational program. Instead, it establishes a strict boundary: infinity cannot function as a generative or justificatory principle under admissible reasoning. This impossibility result closes a long-standing foundational escape route and forms a central component of a broader closure program that replaces appeals to infinity with admissibility- and standing-preserving constraints.