general_self_ref_impossible
General self-referential queries are impossible in Recognition Science because the correctness and negation-encoding conditions on a query q force RSStab(q.config) to be equivalent to its own negation. Researchers formalizing cost-minimization accounts of physical closure cite the result to reclassify Gödel sentences as non-configurations outside the ontology. The proof assumes existence of q, composes the two biconditionals into a single contradictory equivalence, and derives an immediate falsehood by case analysis on stabilization.
claimNo real number $c$ and proposition $A$ exist such that $A$ is equivalent to the negation of RSStab$(c)$ and RSStab$(c)$ is equivalent to $A$.
background
The Gödel Dissolution module translates self-reference into stabilization queries under Recognition Science dynamics. RSStab$(c)$ is the predicate that holds exactly when defect$(c) = 0$. A GeneralSelfRefQuery packages a configuration $c$ with a proposition $A$ that the query asserts, together with the two conditions that $A$ is equivalent to the negation of RSStab$(c)$ and that RSStab$(c)$ is equivalent to $A$. The module setting is that Gödel sentences become queries asserting non-stabilization, and the framework shows they cannot possess a fixed point under cost minimization.
proof idea
The term-mode proof assumes such a query $q$ exists. It extracts the correctness equivalence RSStab$(q.config) ↔ q.asserts$ and the encodes_negation equivalence $q.asserts ↔ ¬$RSStab$(q.config)$. Transitivity produces the contradictory biconditional RSStab$(q.config) ↔ ¬$RSStab$(q.config)$. Case analysis on whether RSStab$(q.config)$ holds then yields an immediate falsehood in each branch.
why it matters in Recognition Science
The theorem supplies the general-self-ref-impossible field of the GodelDissolutionTheorem structure, which records that self-referential stabilization queries are contradictory and therefore lie outside the RS ontology. It supports the paper's claim that Gödel's theorem does not obstruct physical closure because RS closure requires only a unique cost minimizer rather than arithmetic completeness. In the T0-T8 forcing chain it confirms that self-reference cannot produce a fixed point under the J-cost dynamics.
scope and limits
- Does not address self-reference inside standard formal arithmetic.
- Does not claim every proposition in RS receives a definite status.
- Does not construct non-self-referential configurations.
- Does not resolve independent statements such as the continuum hypothesis.
formal statement (Lean)
139theorem general_self_ref_impossible : ¬∃ q : GeneralSelfRefQuery, True := by
proof body
Term-mode proof.
140 intro ⟨q, _⟩
141 -- q.correctness: RSStab q.config ↔ q.asserts
142 -- q.encodes_negation: q.asserts ↔ ¬RSStab q.config
143 -- Combining: RSStab q.config ↔ ¬RSStab q.config
144 have h1 := q.correctness
145 have h2 := q.encodes_negation
146 -- RSStab ↔ asserts ↔ ¬RSStab
147 have h : RSStab q.config ↔ ¬RSStab q.config := h1.trans h2
148 by_cases hs : RSStab q.config
149 · exact (h.mp hs) hs
150 · exact hs (h.mpr hs)
151
152/-! ## The Three-Part Theorem from the Paper -/
153
154/-! For the paper's Theorem 4.1, we need to show that IF a self-referential
155query existed, it would be:
1561. Not RSTrue
1572. Not RSFalse
1583. Outside the ontology
159
160But we've already shown such queries can't exist consistently.
161We can still state what WOULD happen if we could define one: -/
162
163/-- If we could define a self-referential query c where:
164 - c is "correct" when c stabilizes iff c encodes truth
165 - c encodes "I don't stabilize"
166
167 Then c cannot be RSTrue. -/