GeneralSelfRefQuery
General self-referential queries pair a real configuration c with a proposition that asserts its own non-stabilization under RS dynamics. Researchers formalizing cost-theoretic resolutions to Gödel incompleteness cite this structure when translating self-reference into stabilization predicates. The definition assembles four fields that directly encode the contradictory biconditionals RSStab(c) ↔ P ↔ ¬RSStab(c).
claimA general self-referential query consists of a real number $c$ and a proposition $P$ such that $P$ holds if and only if $c$ does not stabilize, i.e., $P$ holds precisely when the defect of $c$ is nonzero, together with the biconditional that $c$ stabilizes if and only if $P$ holds.
background
The Gödel Dissolution module reinterprets self-reference inside Recognition Science by replacing formal provability with cost-minimization dynamics. RSStab(c) is defined as the predicate defect c = 0, marking a configuration as RS-true when its defect vanishes. The module contrasts this with classical Gödel sentences, showing that self-reference becomes a stabilization query asserting its own divergence rather than a truth gap inside arithmetic.
proof idea
This declaration is a structure definition that directly assembles the four fields config, asserts, encodes_negation, and correctness. No lemmas or tactics are invoked; the structure itself supplies the type used by the downstream impossibility theorems.
why it matters in Recognition Science
The structure supplies the general form of self-referential stabilization queries that feed into GodelDissolutionTheorem, which concludes such queries are non-configurations outside the RS ontology. It realizes the module's central claim that Gödel phenomena dissolve under cost-theoretic closure rather than obstructing it, aligning with the framework's emphasis on unique cost minimizers and the phi-ladder fixed-point condition.
scope and limits
- Does not prove non-existence of such queries; that step occurs in the downstream theorem.
- Does not define or compute the defect function.
- Does not address empirical programs or simplicial structures.
- Does not extend beyond real-number configurations.
formal statement (Lean)
124structure GeneralSelfRefQuery where
125 config : ℝ
126 /-- c "asserts" a proposition -/
127 asserts : Prop
128 /-- That proposition is ¬RSStab(c) -/
129 encodes_negation : asserts ↔ ¬RSStab config
130 /-- c is "correct" if what it asserts matches its stabilization status -/
131 correctness : RSStab config ↔ asserts
132
133/-- **THEOREM 2**: General self-referential queries are contradictory.
134
135 The correctness condition and encoding condition together imply:
136 RSStab(c) ↔ asserts ↔ ¬RSStab(c)
137
138 This is P ↔ ¬P, which is impossible. -/