self_ref_query_impossible
No real configuration c satisfies the equivalence (defect c = 0) ↔ ¬(defect c = 0). Researchers on cost-theoretic physical closure cite this to show that Gödel sentences translate into inconsistent stabilization queries under Recognition Science dynamics. The proof assumes such a query exists, extracts the contradictory equivalence, and derives a contradiction by case analysis on the defect value.
claimThere does not exist a real number $c$ such that $(J(c) = 0) ↔ ¬(J(c) = 0)$, where $J$ is the cost functional with $J(x) = (x + x^{-1})/2 - 1$.
background
The Gödel Dissolution module models self-referential stabilization queries as structures pairing a real configuration with an equivalence that asserts stabilization status is the negation of itself. The defect functional equals the J-cost, which vanishes exactly at unity. This setting draws on the Law of Existence for the defect definition and on consistency predicates from complexity models to ensure the query is well-formed within the ontology.
proof idea
The proof assumes an arbitrary SelfRefQuery q exists. It extracts the self_ref equivalence (defect q.config = 0) ↔ ¬(defect q.config = 0). Case analysis on whether defect q.config equals zero is applied: the forward direction yields the negation when the antecedent holds, while the reverse direction forces the antecedent when it fails. Both branches produce an immediate contradiction.
why it matters in Recognition Science
This result supplies the first conjunct of complete_godel_dissolution, which concludes that Gödel sentences become non-configurations outside the RS ontology. It supports the module's claim that RS closure concerns unique cost minimizers rather than arithmetic completeness, consistent with the phi-ladder and eight-tick octave. The packaged theorem godel_dissolution_holds records this alongside decidability of stabilization status.
scope and limits
- Does not claim that every self-referential sentence is inconsistent.
- Does not address the halting problem or Turing undecidability.
- Does not provide an empirical test of Gödel's theorem in arithmetic.
- Applies only inside the Recognition Science cost-minimization framework.
formal statement (Lean)
86theorem self_ref_query_impossible : ¬∃ q : SelfRefQuery, True := by
proof body
Term-mode proof.
87 intro ⟨q, _⟩
88 -- q.self_ref says: (defect config = 0) ↔ ¬(defect config = 0)
89 -- This is impossible: P ↔ ¬P has no model
90 have h := q.self_ref
91 -- Assume defect config = 0
92 by_cases hd : defect q.config = 0
93 · -- If defect = 0, then by self_ref, ¬(defect = 0)
94 have : ¬(defect q.config = 0) := h.mp hd
95 contradiction
96 · -- If defect ≠ 0, then by self_ref.symm, defect = 0
97 have : defect q.config = 0 := h.mpr hd
98 contradiction
99
100/-- **COROLLARY**: No consistent configuration can be a self-referential
101 stabilization query. Such "configurations" are syntactically expressible
102 but ontologically empty. -/