Self-Referential Logic and AI Consciousness: A Gödelian Approach

Cogito (“I think, therefore I am”) demonstrates a different logical ontology than Tarski’s Undefinability Theorem, prohibiting deterministic Tarskian AI from natively digitizing consciousness. Addressing this gap, we first establish consistent self-reference in naïve set theory showing that self-validating set SR = {x|P (x)} ∧ ∀x(x ∈ SR → P(x)) avoids Russell’s Paradox as R would fail x ∈ SR → P(x), i.e., if P(x) = {x|x ̸∈ x} then R ∈ R ↛ R ̸∈ R. Thus, R is never included in R. Next, we formalize and simulate Gödel’s meta-logic via set SR4 = {x|P(x)} ∧ ∀x(x ∈ SR4 → P(x)); where P(x) = [P1(x) ∨ [[¬(Prov(x) = true) ∧ ¬(Prov(x) = false)] ∧ x ≡ ¬P1(x)]] ∧ x ≡ x. To test G, we set P1(x) = Prov(x) and simulate G by G=“I am not provable.” SR4’s innovation is expressing G’s undecidability as ¬P1(G) syntax. Thus, SR4 determines G ≡ ¬Prov(G) as “Gödelian true” at SR4’s object level. Whereas Gödelian numbers aren’t expressive enough to objectify ¬P1(G) (stuck in undecidability until meta-logic), S4 permits reification of this negative syntactic property. Thus, SR4 algorithmically processes the G ≡ ¬Prov(G) truth of the First Theorem with consistency. With regards to the Liar’s Paradox and SR4, if we set P1(x) = True; ¬P1(x) = Not(True) and x1 = “I am not true” and x2 = “I am false” we see different results between x1 and x2. This is because [¬(P rov(x) = true) ∧ ¬(Prov(x) = false)] agrees with x1 and contra- dicts x2, meaning we fundamentally split negation, showing “not true” is not equal to “false” with regards to proofs. SR4 thus demonstrates consistent self-reference, matching the logical structure potentially required to express Cogito. We also explore the possibility of a logical ontology of Indeterminism, which can be split into Determined Indeterminism like G and Indeterminate Indeterminism, such as randomness.