A Mathematical Resolution of the Burali-Forti Paradox: Establishing Infinity as Real yet Unidentifiable

This paper presents a proof demonstrating the infeasibility of extending the definition of ordinal numbers beyond finite ordinals, thereby offering a foundational resolution to the Burali-Forti paradox. Complementing John von Neumann’s classical formulation—wherein an ordinal is defined as the set of all preceding ordinals—this proof evaluates the cumulative domain of all currently existing ordinals at stage , defined as , against the maximal-anchor condition: . Because is uniquely identified as the -maximal ordinal , the condition is satisfied at every finite stage. Under this complementary formulation, a domain of ordinals can be extensionally closed to form the next successor ordinal if and only if identifies a unique maximal anchor , confirming that the progression remains open to successor generation. Applying this criterion successively to the sequence of ordinals yields the following progression: When evaluated at the infinite limit collection , the condition fails because every element belongs to its successor . Because has no maximal element, the required witness cannot be identified; thus, while every finite stage is validated inductively as non-terminal, the infinite totality reaches an unanchored horizon whose terminal boundary remains formally unidentifiable as an ordinal. Consequently, can only be defined as the infinite domain of all finite ordinals; it cannot be closed extensionally as the first transfinite ordinal .