The proof I would like share with you amounts to a trivial extension of Kurt Gödels' and Alfred Tarski's respective Incompleteness and Undefinability Theorems, so those of you already familiar with them, and what they mean[t] for epistemelogy and philosophical discourse will likely find this paper to seem redundant/superfluous. The argument employs three neologisms of my own creation that it could technically do without, but they're included because a) the proof was originally elaborated in the context of this full paper, whose core message they enabled me to elaborate more succinctly and b) I just like them as words and secretly hope one or more of them catches on.

TL:DR this is a proof that it is not possible to construct a theory that explains both itself and everything else, because attempts to do so invariably generate a infinitely descending hierarchy of meta-level explanatory demands.

Definitions:

Let T denote a theory of reality.

Definition 1 (Autological Theory): A theory is autological if it contains an account of itself. In particular, it includes propositions concerning its own structure, operation, and explanatory claims.

Definition 2 (Autotelic Theory): A theory is autotelic if it requires no explanatory resources external to itself. All explanations necessary for its adequacy are contained within the theory.

Definition 3 (Autognostic Theory): A theory is autognostic if it is both autological and autotelic. An autognostic theory possesses complete self-knowledge and complete self-explanatory closure.

Definition 4 (Explanatory Completeness): A theory is explanatorily complete if it accountsfor all truths concerning reality.

Definition 5 (Final Theory): A final theory is a theory that is both explanatorily complete and autognostic.

Proof

Axiom 1 (Self-Explanatory Adequacy): A satisfactory theory of reality should be capable of expressing and establishing all truths relevant to its domain, including truths concerning its own adequacy, truthfulness, consistency, existence, and relation to reality.

Proposition 1: Any autological theory contains propositions concerning its own adequacy.

Proposition 2: Questions concerning the adequacy of a theory generate a meta-level description.

Proposition 3: Incorporating a meta-level account into a theory generates further meta-level demands.

Assumption: Assume- for contradiction- that an autognostic and explanatorily complete theory T exists.

(I) Because T is explanatorily complete, it must account for all truths concerning reality.

(II) Because T is autognostic, it must also contain a complete account of its own adequacy, truthfulness, consistency, existence, and explanatory status.

(III) This requirement generates a meta-level account concerning the adequacy of T.

(IV) To remain autognostic, the theory must incorporate this account.

(V) The enlarged theory now requires an account of the adequacy of the enlarged theory itself.

(VI) Repeating the process generates an indefinitely extending hierarchy of explanatory demands.

(VII) Gödel’s incompleteness theorems prevent sufficiently expressive systems from internally establishing all truths relevant to themselves.

(VIII) Tarski’s undefinability theorem prevents sufficiently expressive systems from internally providing a complete account of their own truth conditions.

(IX) Therefore no stage in the hierarchy achieves complete self-explanatory closure. Consequently, T cannot be both explanatorily complete and autognostic.

(X) This contradicts the original assumption.

(XI) Conclusion: Therefore no such theory exists.

Corollaries

Proposition 4: No final theory exists. Proof: Per our definition of Final Theories and (XI) Conclusion.

Proposition 5: Reality exceeds representation whenever satisfactory representation requires autognosis. Proof: If satisfactory representation requires explanatory completeness together with autognosis, and autognosis is impossible, then no representation can fully satisfy the criterion. Some explanatory residue necessarily remains beyond the representational boundary.