Documentation

Incompleteness.Arith.Second

def LO.Arith.Formalized.substNumerals {V : Type u_1} [ORingStruc V] [V ⊧ₘ* 𝐈𝚺₁] {k : } (φ : V) (v : Fin kV) :
V
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def LO.FirstOrder.Arith.ssnums {k : } :
𝚺₁.Semisentence (k + 2)
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@[simp]
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diagi(x):=(y)[(jssnums(yj,xj,x))θi(y)]

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@[reducible, inline]
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@[reducible, inline]
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theorem LO.FirstOrder.Arith.provableₐ_D1 {T : Theory ℒₒᵣ} [𝐈𝚺₁ T] {U : Theory ℒₒᵣ} [U.Delta1Definable] {σ : Sentence ℒₒᵣ} :
U ⊢!. σT ⊢!. U.bewₐ σ
theorem LO.FirstOrder.Arith.provableₐ_D2 {T : Theory ℒₒᵣ} [𝐈𝚺₁ T] {U : Theory ℒₒᵣ} [U.Delta1Definable] {σ π : Sentence ℒₒᵣ} :
T ⊢!. U.bewₐ (σ π) U.bewₐ σ U.bewₐ π
theorem LO.FirstOrder.Arith.provableₐ_sigma₁_complete {T : Theory ℒₒᵣ} [𝐈𝚺₁ T] {U : Theory ℒₒᵣ} [U.Delta1Definable] {σ : Sentence ℒₒᵣ} (hσ : Hierarchy 𝚺 1 σ) :
T ⊢!. σ U.bewₐ σ
theorem LO.FirstOrder.Arith.provableₐ_D3 {T : Theory ℒₒᵣ} [𝐈𝚺₁ T] {U : Theory ℒₒᵣ} [U.Delta1Definable] {σ : Sentence ℒₒᵣ} :
T ⊢!. U.bewₐ σ U.bewₐ (U.bewₐ σ)
theorem LO.FirstOrder.Arith.goedel_iff_unprovable_goedel {T : Theory ℒₒᵣ} [𝐈𝚺₁ T] {U : Theory ℒₒᵣ} [U.Delta1Definable] :
T ⊢!. U.goedelₐ U.bewₐ U.goedelₐ
theorem LO.FirstOrder.Arith.provableₐ_D2_context {T : Theory ℒₒᵣ} [𝐈𝚺₁ T] {U : Theory ℒₒᵣ} [U.Delta1Definable] {Γ : List (Sentence ℒₒᵣ)} {σ π : Sentence ℒₒᵣ} (hσπ : Γ ⊢[T.alt]! U.bewₐ (σ π)) (hσ : Γ ⊢[T.alt]! U.bewₐ σ) :
Γ ⊢[T.alt]! U.bewₐ π
theorem LO.FirstOrder.Arith.provableₐ_D3_context {T : Theory ℒₒᵣ} [𝐈𝚺₁ T] {U : Theory ℒₒᵣ} [U.Delta1Definable] {Γ : List (Sentence ℒₒᵣ)} {σ : Sentence ℒₒᵣ} (hσπ : Γ ⊢[T.alt]! U.bewₐ σ) :
Γ ⊢[T.alt]! U.bewₐ (U.bewₐ σ)
theorem LO.FirstOrder.Arith.provableₐ_sound {T U : Theory ℒₒᵣ} [U.Delta1Definable] [ ⊧ₘ* T] [𝐑₀ U] {σ : Sentence ℒₒᵣ} :
T ⊢!. U.bewₐ σU ⊢! σ
theorem LO.FirstOrder.Arith.provableₐ_complete {T : Theory ℒₒᵣ} [𝐈𝚺₁ T] {U : Theory ℒₒᵣ} [U.Delta1Definable] [ ⊧ₘ* T] [𝐑₀ U] {σ : Sentence ℒₒᵣ} :
U ⊢! σ T ⊢!. U.bewₐ σ
theorem LO.FirstOrder.Arith.goedel_unprovable (T : Theory ℒₒᵣ) [𝐈𝚺₁ T] [T.Delta1Definable] [System.Consistent T] :
T T.goedelₐ
theorem LO.FirstOrder.Arith.not_goedel_unprovable (T : Theory ℒₒᵣ) [𝐈𝚺₁ T] [T.Delta1Definable] [ ⊧ₘ* T] :
T T.goedelₐ
theorem LO.FirstOrder.Arith.consistent_iff_goedel (T : Theory ℒₒᵣ) [𝐈𝚺₁ T] [T.Delta1Definable] :
T ⊢! T.consistentₐ T.goedelₐ
theorem LO.FirstOrder.Arith.goedel_second_incompleteness (T : Theory ℒₒᵣ) [𝐈𝚺₁ T] [T.Delta1Definable] [System.Consistent T] :
T T.consistentₐ

Gödel's Second Incompleteness Theorem

theorem LO.FirstOrder.Arith.inconsistent_unprovable (T : Theory ℒₒᵣ) [𝐈𝚺₁ T] [T.Delta1Definable] [ ⊧ₘ* T] :
T T.consistentₐ