Dynamics of Metarepresentation and Structural Limits of Self Containment: A Diagonalization and Transfinite Cardinal Approach

We formalize the problem of epistemic self-containment in reflective agents usingstate-space representation and operator theory. By defining a metarepresentation operator Φover a family of cognitive frontiers F ⊆ P(U), we demonstrate that complete self-containmentis strictly impossible for any internal representation framework. Utilizing a Tarski-Cantor diag-onal argument, we prove that no finite internal frame can contain its own evaluation predicate,forcing a strictly ascending chain of representation frontiers F0 ⊊ F1 ⊊ F2 . . . . Furthermore, byanalyzing exhaustive metarepresentation through power-set dynamics, we map this expansionto the transfinite hierarchy of Beth numbers (ℶn). We conclude that the reflexive zero-pointlimit Fω is structurally inaccessible in finite steps, establishing a general barrier theorem forself-modeling systems with applications to recursive AI self-improvement, logical proof barriers,and physical observer bounds.