IndisputableMonolith.Foundation.BiconditionalSelfNegation
IndisputableMonolith/Foundation/BiconditionalSelfNegation.lean · 287 lines · 18 declarations
show as:
view math explainer →
1import Mathlib
2import IndisputableMonolith.Cost
3import IndisputableMonolith.Foundation.LawOfExistence
4import IndisputableMonolith.Foundation.OntologyPredicates
5
6/-!
7# Biconditional Self-Negation: No Real Configuration Satisfies `P ↔ ¬P`
8
9This module proves the classical-logic fact that no real-valued configuration can
10satisfy a biconditional of the form `(defect c = 0) ↔ ¬(defect c = 0)`, together
11with a few corollaries about stabilization status and the unique zero-defect
12existent at `x = 1`.
13
14## What this module actually proves
15
16For any real `c`, the proposition `(defect c = 0) ↔ ¬(defect c = 0)` is
17inhabited iff `False`. The proof is a two-line case split on excluded middle.
18
19The same fact holds for any predicate `P`: classical logic has no fixed point
20for negation. The phenomenon is propositional-logic content, not anything
21specific to Recognition Science.
22
23## What this module does NOT prove
24
25**It does not address Gödel's first incompleteness theorem.**
26
27Gödel sentences do not satisfy `P ↔ ¬P`. They satisfy `G ↔ ¬Prov_F(⌜G⌝)`, where
28`Prov_F(⌜G⌝)` is a syntactic predicate over Gödel numbers and `G` is a sentence
29in the language of `F`. These are distinct propositions; the biconditional is
30consistent; that is the entire point of Gödel I.
31
32A logic-trained reader who sees the structure here labeled as a "Gödel
33dissolution" will reject the framing immediately. The historical filename
34`GodelDissolution.lean` was misleading. The canonical home is this module;
35the old file remains as a backward-compatibility shim with the same theorem
36content under deprecated names.
37
38The categorical argument for why Gödel I has no target inside the Recognition
39Science forcing chain (T-1 → T0 → ... → T8 → constants) lives at the
40meta-level and is not a Lean theorem. See
41`papers/Godel_And_RS_Closure_Honest_Assessment_20260520.html` for the
42honest accounting.
43
44## Cross-references
45
46- `LawOfExistence.defect_at_one`, `LawOfExistence.defect_pos_of_ne_one`:
47 the substantive content about the cost functional.
48- `OntologyPredicates.rs_exists_unique`: the unique zero-defect existent.
49- `papers/godel_dissolution.tex`: the philosophical paper. The Lean module
50 here proves only the propositional logic; the categorical argument is in
51 the paper.
52-/
53
54namespace IndisputableMonolith
55namespace Foundation
56namespace BiconditionalSelfNegation
57
58open Real
59open LawOfExistence
60open OntologyPredicates
61
62/-! ## Stabilization predicates (preserved from earlier API)
63
64These predicates are about real-valued configurations and are correctly named.
65They are re-exported by the legacy `GodelDissolution` namespace.
66-/
67
68/-- A real configuration "stabilizes" iff its defect vanishes. -/
69def RSStab (c : ℝ) : Prop := defect c = 0
70
71/-- A real configuration "diverges" iff its defect exceeds every bound. -/
72def RSDiverge (c : ℝ) : Prop := ∀ C : ℝ, defect c > C
73
74/-- A real configuration is "outside the stabilization classification" iff it
75neither stabilizes nor diverges. By `diverge_impossible` below, this reduces
76to "non-stabilizing" for real-valued configurations. -/
77def RSOutside (c : ℝ) : Prop := ¬RSStab c ∧ ¬RSDiverge c
78
79/-- Decidability of stabilization status for real configurations. Classical. -/
80theorem stab_decidable (c : ℝ) : RSStab c ∨ ¬RSStab c :=
81 em (RSStab c)
82
83/-- Divergence in the sense of "exceeds every real bound" is vacuous for any
84real-valued defect: take the bound equal to the defect itself. -/
85theorem diverge_impossible (c : ℝ) : ¬RSDiverge c := by
86 intro h
87 have : defect c > defect c := h (defect c)
88 linarith
89
90/-- Every real configuration either stabilizes or fails to stabilize. The
91extra `RSOutside` clause is included for compatibility with the legacy API;
92by `diverge_impossible` it adds no content. -/
93theorem config_classification (c : ℝ) : RSStab c ∨ RSOutside c := by
94 by_cases hs : RSStab c
95 · exact Or.inl hs
96 · exact Or.inr ⟨hs, diverge_impossible c⟩
97
98/-! ## The biconditional-self-negation structures
99
100The first structure encodes a real configuration claiming
101`(defect c = 0) ↔ ¬(defect c = 0)`. The classical-logic fact below shows
102no such configuration can exist.
103
104The second structure encodes a more general predicate-level biconditional
105self-negation. The same classical fact applies.
106
107Neither structure corresponds to a Gödel sentence in any technical sense.
108Both are propositional-logic content.
109-/
110
111/-- A real configuration claiming the biconditional self-negation
112`(defect c = 0) ↔ ¬(defect c = 0)`.
113
114By classical logic this structure has no inhabitants. Despite the historical
115naming (`SelfRefQuery`), this is not a model of Gödel-style self-reference.
116A Gödel sentence is not `P ↔ ¬P`; it is `P ↔ ¬Q(⌜P⌝)` with `Q` a syntactic
117provability predicate, and that biconditional is consistent. -/
118structure SelfNegatingConfig where
119 /-- The underlying real configuration. -/
120 config : ℝ
121 /-- The biconditional self-negation. This field is inhabited iff `False`. -/
122 self_negation : (defect config = 0) ↔ ¬(defect config = 0)
123
124/-- A general predicate-level biconditional self-negation. The fields together
125encode `RSStab c ↔ asserts ↔ ¬RSStab c`, which collapses to `RSStab c ↔ ¬RSStab c`,
126which is `P ↔ ¬P` and has no inhabitant. -/
127structure GeneralSelfNegatingPredicate where
128 /-- The underlying real configuration. -/
129 config : ℝ
130 /-- An associated proposition. -/
131 asserts : Prop
132 /-- That proposition is the negation of the stabilization status. -/
133 encodes_negation : asserts ↔ ¬RSStab config
134 /-- The configuration's stabilization status agrees with the proposition. -/
135 correctness : RSStab config ↔ asserts
136
137/-! ## The main theorems
138
139These are classical-logic facts. They are correctly proved; they should not
140be cited as resolutions of Gödel's incompleteness theorem.
141-/
142
143/-- **Classical-logic fact.** No real configuration satisfies
144`(defect c = 0) ↔ ¬(defect c = 0)`. Two-line proof by excluded middle. -/
145theorem no_self_negating_config : ¬∃ q : SelfNegatingConfig, True := by
146 intro ⟨q, _⟩
147 have h := q.self_negation
148 by_cases hd : defect q.config = 0
149 · exact (h.mp hd) hd
150 · exact hd (h.mpr hd)
151
152/-- **Classical-logic fact.** No real configuration carries a general
153predicate-level biconditional self-negation. -/
154theorem no_general_self_negating_predicate :
155 ¬∃ q : GeneralSelfNegatingPredicate, True := by
156 intro ⟨q, _⟩
157 have h1 := q.correctness
158 have h2 := q.encodes_negation
159 have h : RSStab q.config ↔ ¬RSStab q.config := h1.trans h2
160 by_cases hs : RSStab q.config
161 · exact (h.mp hs) hs
162 · exact hs (h.mpr hs)
163
164/-- Pointwise classical version. For every real `c`,
165`(defect c = 0) ↔ ¬(defect c = 0)` is uninhabited. -/
166theorem no_self_negation_at_point (c : ℝ) :
167 ¬((defect c = 0) ↔ ¬(defect c = 0)) := by
168 intro h
169 by_cases hd : defect c = 0
170 · exact (h.mp hd) hd
171 · exact hd (h.mpr hd)
172
173/-- A redundant compatibility statement: if a real `c` admitted a
174biconditional self-negation, then `False`. Equivalent to
175`no_self_negation_at_point`; kept for legacy API. -/
176theorem self_negation_implies_false
177 (c : ℝ)
178 (_h_encodes : ∀ P : Prop, (P ↔ RSStab c) → (P ↔ ¬RSStab c) → False)
179 (h_correct : RSStab c ↔ ¬RSStab c) :
180 False := by
181 by_cases hs : RSStab c
182 · exact (h_correct.mp hs) hs
183 · exact hs (h_correct.mpr hs)
184
185/-! ## Bundled theorem (formerly `GodelDissolutionTheorem`)
186
187The bundle collects four genuine facts:
188
1891. No `SelfNegatingConfig` exists (classical logic).
1902. No `GeneralSelfNegatingPredicate` exists (classical logic).
1913. Every real configuration has definite stabilization status (classical
192 excluded middle).
1934. There exists a unique real `x > 0` with zero defect, namely `x = 1`
194 (substantive cost-uniqueness content).
195
196Item 4 is the only substantive RS content. The other three are propositional
197logic. The bundle was historically called `GodelDissolutionTheorem`; that
198name overstates what items 1-3 do.
199-/
200
201/-- Bundled classical-logic-and-unique-minimizer theorem. -/
202structure ClassicalLogicAndUniqueMinimizerTheorem where
203 /-- Classical: no real configuration satisfies `(defect = 0) ↔ ¬(defect = 0)`. -/
204 no_self_negating_config : ¬∃ q : SelfNegatingConfig, True
205 /-- Classical: no real configuration carries a general biconditional self-negation. -/
206 no_general_self_negating_predicate : ¬∃ q : GeneralSelfNegatingPredicate, True
207 /-- Classical: every real configuration has definite stabilization status. -/
208 definite_status : ∀ c : ℝ, RSStab c ∨ ¬RSStab c
209 /-- Substantive: the RS closure picks out a unique positive existent. -/
210 rs_closure_meaning : ∃! x : ℝ, RSExists x
211
212/-- The bundled theorem holds. -/
213theorem classical_logic_and_unique_minimizer_theorem :
214 ClassicalLogicAndUniqueMinimizerTheorem := {
215 no_self_negating_config := no_self_negating_config
216 no_general_self_negating_predicate := no_general_self_negating_predicate
217 definite_status := stab_decidable
218 rs_closure_meaning := rs_exists_unique
219}
220
221/-- The complete bundle: classical-logic facts plus the unique-existent value
222`x = 1`. Was historically called `complete_godel_dissolution`. The Gödel
223framing was wrong; the content is correct. -/
224theorem complete_classical_logic_and_closure :
225 -- Self-negating configurations impossible
226 (¬∃ q : SelfNegatingConfig, True) ∧
227 -- Unique RS-existent
228 (∃! x : ℝ, RSExists x) ∧
229 -- That existent is unity
230 (∀ x : ℝ, RSExists x ↔ x = 1) ∧
231 -- Every config has definite status
232 (∀ c : ℝ, RSStab c ∨ ¬RSStab c) :=
233 ⟨no_self_negating_config, rs_exists_unique, rs_exists_unique_one, stab_decidable⟩
234
235/-! ## Documentation-only structures (formerly `GodelRequirements`, `RSDoesNotSatisfyGodel`)
236
237The structures below carry no theorem content. Each field is `Prop`; the
238canonical inhabitant has every field set to `True`. They are Lean records of
239philosophical / categorical claims, not theorems. Reviewers reading the
240older naming may have mistaken them for proved propositions; they are
241not.
242
243If you want the genuine categorical argument (RS does not separately
244maintain `Prov` and `True` predicates, so the gap exploited by Gödel I
245does not arise), it lives in the prose of the companion paper, not here.
246-/
247
248/-- Documentation-only record of the standard prerequisites Gödel's first
249incompleteness theorem requires of a target system. Each field is a `Prop`
250placeholder; this structure carries no theorem content. -/
251structure GodelTargetClassPrerequisites where
252 /-- The target is a formal system. -/
253 formal_system : Type
254 /-- The target is consistent. -/
255 consistent : Prop
256 /-- The target's axiom set is computably enumerable. -/
257 axiom_enumerable : Prop
258 /-- The target expresses sufficient arithmetic. -/
259 expresses_arithmetic : Prop
260 /-- The target internally expresses its own provability predicate. -/
261 expresses_provability : Prop
262
263/-- Documentation-only record of the structural differences between
264Recognition Science and Gödel-I's target class. Each field is `Prop`;
265the canonical inhabitant has every field set to `True`. This carries no
266theorem content. -/
267structure RsCategoricalDifferenceFromGodel where
268 /-- RS is selection dynamics, not a proof system. -/
269 not_proof_system : Prop
270 /-- RS truth is stabilization, not Tarskian satisfaction. -/
271 not_tarskian : Prop
272 /-- RS truth is internal, no external model required. -/
273 no_external_model : Prop
274
275/-- Canonical inhabitant of `RsCategoricalDifferenceFromGodel` with every
276philosophical field set to `True`. The structure is documentation, not a
277theorem. -/
278def rs_categorical_difference_from_godel : RsCategoricalDifferenceFromGodel := {
279 not_proof_system := True
280 not_tarskian := True
281 no_external_model := True
282}
283
284end BiconditionalSelfNegation
285end Foundation
286end IndisputableMonolith
287