Project-declaredLean 4.32.1 · mathlib@520045ab

True restricted Gödel

LO.FirstOrder.Arithmetic.true_restrictedGödel

formal logic metatheory proof theory restricted provability

Project documentation

Gödel sentence by restricted provability -/ noncomputable abbrev restrictedGödel (e : ℕ) (T : Theory L) [T.Δ₁] : ArithmeticSentence := fixedpoint (∼(T.restrictedProvable e)) private noncomputable abbrev restrictedGödel' (e : ℕ) (T : Theory L) [T.Δ₁] : ArithmeticSentence := ∼(T.restrictedProvable e)/[⌜restrictedGödel e T⌝] private lemma restrictedGödel'_si...

Exact Lean statement

theorem true_restrictedGödel : ℕ↓[ℒₒᵣ] ⊧ T.restrictedGödel e

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Full Lean sourceLean 4

theorem true_restrictedGödel : ℕ↓[ℒₒᵣ] ⊧ T.restrictedGödel e := by  by_contra hC;  obtain ⟨e, _, he⟩ := models_neg_restrictedGödel (e := e) |>.mp hC;  apply hC;  apply iff_true_restrictedGödel_true_restrictedGödel'.mpr;  apply ArithmeticTheory.soundOnHierarchy T _ _ ?_ T.restrictedGödel'_sigmaOne;  apply iff_provable_restrictedGödel_provable_restrictedGödel'.mp;  apply Arithmetic.Bootstrapping.provable_of_standard_proof (T := T) (V := ℕ) (n := e);  simpa using he;

Project: Foundation
License: Apache-2.0
Commit: 8dcdb3196454
Source: Foundation/FirstOrder/Incompleteness/RestrictedProvability.lean:77-85

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Source project: Foundation

Person-level attribution pending.

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Project-declaredLean 4.32.1

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Plain-language statement

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formal logicmetatheoryproof theory

Source project: Foundation

Person-level attribution pending.

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