instance
LO.FirstOrder.Theory.Alt.instSubtheorySyntacticFormulaThyAlt_incompleteness
{L : LO.FirstOrder.Language}
{T : LO.FirstOrder.Theory L}
{U : LO.FirstOrder.Theory L}
[s : T ≼ U]
:
T ≼ U.alt.thy
Equations
- LO.FirstOrder.Theory.Alt.instSubtheorySyntacticFormulaThyAlt_incompleteness = s
instance
LO.FirstOrder.Theory.Alt.instSubtheorySentenceAltOfSyntacticFormula_incompleteness
{L : LO.FirstOrder.Language}
{T : LO.FirstOrder.Theory L}
{U : LO.FirstOrder.Theory L}
[s : T ≼ U]
:
T.alt ≼ U.alt
Equations
- LO.FirstOrder.Theory.Alt.instSubtheorySentenceAltOfSyntacticFormula_incompleteness = { prf := fun {f : LO.FirstOrder.Sentence L} (b : T.alt ⊢ f) => LO.System.Subtheory.prf b }
structure
LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
(T₀ : LO.FirstOrder.Theory L)
(T : LO.FirstOrder.Theory L)
:
Type u_1
- prov : LO.FirstOrder.Semisentence L 1
- spec : ∀ {σ : LO.FirstOrder.Sentence L}, T ⊢!. σ → T₀ ⊢!. (LO.FirstOrder.Rew.substs ![⌜σ⌝]).hom self.prov
Instances For
theorem
LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate.spec
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
{T₀ : LO.FirstOrder.Theory L}
{T : LO.FirstOrder.Theory L}
(self : LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate T₀ T)
{σ : LO.FirstOrder.Sentence L}
:
T ⊢!. σ → T₀ ⊢!. (LO.FirstOrder.Rew.substs ![⌜σ⌝]).hom self.prov
def
LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate.pr
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
{T₀ : LO.FirstOrder.Theory L}
{T : LO.FirstOrder.Theory L}
(𝔅 : LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate T₀ T)
(σ : LO.FirstOrder.Sentence L)
:
Equations
- 𝔅.pr σ = (LO.FirstOrder.Rew.substs ![⌜σ⌝]).hom 𝔅.prov
Instances For
instance
LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate.instCoeFunForallSentence
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
{T₀ : LO.FirstOrder.Theory L}
{T : LO.FirstOrder.Theory L}
:
Equations
- LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate.instCoeFunForallSentence = { coe := LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate.pr }
def
LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate.con
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
{T₀ : LO.FirstOrder.Theory L}
{T : LO.FirstOrder.Theory L}
(𝔅 : LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate T₀ T)
:
Instances For
class
LO.FirstOrder.DerivabilityCondition.Diagonalization
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
(T : LO.FirstOrder.Theory L)
:
Type u_1
- fixpoint : LO.FirstOrder.Semisentence L 1 → LO.FirstOrder.Sentence L
- diag : ∀ (θ : LO.FirstOrder.Semisentence L 1), T ⊢!. LO.FirstOrder.DerivabilityCondition.Diagonalization.fixpoint T θ ⭤ (LO.FirstOrder.Rew.substs ![⌜LO.FirstOrder.DerivabilityCondition.Diagonalization.fixpoint T θ⌝]).hom θ
Instances
theorem
LO.FirstOrder.DerivabilityCondition.Diagonalization.diag
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
{T : LO.FirstOrder.Theory L}
[self : LO.FirstOrder.DerivabilityCondition.Diagonalization T]
(θ : LO.FirstOrder.Semisentence L 1)
:
class
LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate.HBL2
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
{T₀ : LO.FirstOrder.Theory L}
{T : LO.FirstOrder.Theory L}
(𝔅 : LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate T₀ T)
:
- D2 : ∀ {σ τ : LO.FirstOrder.Sentence L}, T₀ ⊢!. 𝔅.pr (σ ➝ τ) ➝ 𝔅.pr σ ➝ 𝔅.pr τ
Instances
theorem
LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate.HBL2.D2
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
{T₀ : LO.FirstOrder.Theory L}
{T : LO.FirstOrder.Theory L}
{𝔅 : LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate T₀ T}
[self : 𝔅.HBL2]
{σ : LO.FirstOrder.Sentence L}
{τ : LO.FirstOrder.Sentence L}
:
class
LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate.HBL3
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
{T₀ : LO.FirstOrder.Theory L}
{T : LO.FirstOrder.Theory L}
(𝔅 : LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate T₀ T)
:
- D3 : ∀ {σ : LO.FirstOrder.Sentence L}, T₀ ⊢!. 𝔅.pr σ ➝ 𝔅.pr (𝔅.pr σ)
Instances
theorem
LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate.HBL3.D3
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
{T₀ : LO.FirstOrder.Theory L}
{T : LO.FirstOrder.Theory L}
{𝔅 : LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate T₀ T}
[self : 𝔅.HBL3]
{σ : LO.FirstOrder.Sentence L}
:
class
LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate.HBL
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
{T₀ : LO.FirstOrder.Theory L}
{T : LO.FirstOrder.Theory L}
(𝔅 : LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate T₀ T)
extends
LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate.HBL2
,
LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate.HBL3
:
Instances
class
LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate.Loeb
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
{T₀ : LO.FirstOrder.Theory L}
{T : LO.FirstOrder.Theory L}
(𝔅 : LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate T₀ T)
:
- LT : ∀ {σ : LO.FirstOrder.Sentence L}, T ⊢!. 𝔅.pr σ ➝ σ → T ⊢!. σ
Instances
theorem
LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate.Loeb.LT
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
{T₀ : LO.FirstOrder.Theory L}
{T : LO.FirstOrder.Theory L}
{𝔅 : LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate T₀ T}
[self : 𝔅.Loeb]
{σ : LO.FirstOrder.Sentence L}
:
class
LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate.FormalizedLoeb
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
{T₀ : LO.FirstOrder.Theory L}
{T : LO.FirstOrder.Theory L}
(𝔅 : LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate T₀ T)
:
- FLT : ∀ {σ : LO.FirstOrder.Sentence L}, T₀ ⊢!. 𝔅.pr (𝔅.pr σ ➝ σ) ➝ 𝔅.pr σ
Instances
theorem
LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate.FormalizedLoeb.FLT
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
{T₀ : LO.FirstOrder.Theory L}
{T : LO.FirstOrder.Theory L}
{𝔅 : LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate T₀ T}
[self : 𝔅.FormalizedLoeb]
{σ : LO.FirstOrder.Sentence L}
:
class
LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate.Rosser
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
{T₀ : LO.FirstOrder.Theory L}
{T : LO.FirstOrder.Theory L}
(𝔅 : LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate T₀ T)
:
- Ro : ∀ {σ : LO.FirstOrder.Sentence L}, T ⊢!. ∼σ → T₀ ⊢!. ∼𝔅.pr σ
Instances
theorem
LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate.Rosser.Ro
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
{T₀ : LO.FirstOrder.Theory L}
{T : LO.FirstOrder.Theory L}
{𝔅 : LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate T₀ T}
[self : 𝔅.Rosser]
{σ : LO.FirstOrder.Sentence L}
:
@[reducible, inline]
abbrev
LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate.D1
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
{T₀ : LO.FirstOrder.Theory L}
{T : LO.FirstOrder.Theory L}
{𝔅 : LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate T₀ T}
{σ : LO.FirstOrder.Sentence L}
:
Equations
- ⋯ = ⋯
Instances For
theorem
LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate.D2
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
{T₀ : LO.FirstOrder.Theory L}
{T : LO.FirstOrder.Theory L}
{𝔅 : LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate T₀ T}
[self : 𝔅.HBL2]
{σ : LO.FirstOrder.Sentence L}
{τ : LO.FirstOrder.Sentence L}
:
Alias of LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate.HBL2.D2
.
theorem
LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate.D3
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
{T₀ : LO.FirstOrder.Theory L}
{T : LO.FirstOrder.Theory L}
{𝔅 : LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate T₀ T}
[self : 𝔅.HBL3]
{σ : LO.FirstOrder.Sentence L}
:
Alias of LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate.HBL3.D3
.
theorem
LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate.LT
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
{T₀ : LO.FirstOrder.Theory L}
{T : LO.FirstOrder.Theory L}
{𝔅 : LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate T₀ T}
[self : 𝔅.Loeb]
{σ : LO.FirstOrder.Sentence L}
:
Alias of LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate.Loeb.LT
.
theorem
LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate.FLT
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
{T₀ : LO.FirstOrder.Theory L}
{T : LO.FirstOrder.Theory L}
{𝔅 : LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate T₀ T}
[self : 𝔅.FormalizedLoeb]
{σ : LO.FirstOrder.Sentence L}
:
Alias of LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate.FormalizedLoeb.FLT
.
theorem
LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate.Ro
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
{T₀ : LO.FirstOrder.Theory L}
{T : LO.FirstOrder.Theory L}
{𝔅 : LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate T₀ T}
[self : 𝔅.Rosser]
{σ : LO.FirstOrder.Sentence L}
:
Alias of LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate.Rosser.Ro
.
def
LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate.D1_shift
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
{T₀ : LO.FirstOrder.Theory L}
{T : LO.FirstOrder.Theory L}
[T₀ ≼ T]
{𝔅 : LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate T₀ T}
{σ : LO.FirstOrder.Sentence L}
:
Equations
- ⋯ = ⋯
Instances For
def
LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate.D2_shift
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
{T₀ : LO.FirstOrder.Theory L}
{T : LO.FirstOrder.Theory L}
[T₀ ≼ T]
{𝔅 : LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate T₀ T}
{σ : LO.FirstOrder.Sentence L}
{τ : LO.FirstOrder.Sentence L}
[𝔅.HBL2]
:
Equations
- ⋯ = ⋯
Instances For
def
LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate.D3_shift
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
{T₀ : LO.FirstOrder.Theory L}
{T : LO.FirstOrder.Theory L}
[T₀ ≼ T]
{𝔅 : LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate T₀ T}
{σ : LO.FirstOrder.Sentence L}
[𝔅.HBL3]
:
Equations
- ⋯ = ⋯
Instances For
def
LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate.FLT_shift
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
{T₀ : LO.FirstOrder.Theory L}
{T : LO.FirstOrder.Theory L}
[T₀ ≼ T]
{𝔅 : LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate T₀ T}
{σ : LO.FirstOrder.Sentence L}
[𝔅.FormalizedLoeb]
:
Equations
- ⋯ = ⋯
Instances For
def
LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate.D2'
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
{T₀ : LO.FirstOrder.Theory L}
{T : LO.FirstOrder.Theory L}
{𝔅 : LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate T₀ T}
{σ : LO.FirstOrder.Sentence L}
{τ : LO.FirstOrder.Sentence L}
[𝔅.HBL2]
[LO.System.ModusPonens T]
:
Equations
- ⋯ = ⋯
Instances For
def
LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate.prov_distribute_imply
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
{T₀ : LO.FirstOrder.Theory L}
{T : LO.FirstOrder.Theory L}
{𝔅 : LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate T₀ T}
[𝔅.HBL]
{σ : LO.FirstOrder.Sentence L}
{τ : LO.FirstOrder.Sentence L}
(h : T ⊢!. σ ➝ τ)
:
Equations
- ⋯ = ⋯
Instances For
def
LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate.prov_distribute_iff
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
{T₀ : LO.FirstOrder.Theory L}
{T : LO.FirstOrder.Theory L}
{𝔅 : LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate T₀ T}
[𝔅.HBL]
{σ : LO.FirstOrder.Sentence L}
{τ : LO.FirstOrder.Sentence L}
(h : T ⊢!. σ ⭤ τ)
:
Equations
- ⋯ = ⋯
Instances For
def
LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate.prov_distribute_and
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
[DecidableEq (LO.FirstOrder.Sentence L)]
{T₀ : LO.FirstOrder.Theory L}
{T : LO.FirstOrder.Theory L}
{𝔅 : LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate T₀ T}
[𝔅.HBL]
{σ : LO.FirstOrder.Sentence L}
{τ : LO.FirstOrder.Sentence L}
:
Equations
- ⋯ = ⋯
Instances For
def
LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate.prov_distribute_and'
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
[DecidableEq (LO.FirstOrder.Sentence L)]
{T₀ : LO.FirstOrder.Theory L}
{T : LO.FirstOrder.Theory L}
{𝔅 : LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate T₀ T}
[𝔅.HBL]
{σ : LO.FirstOrder.Sentence L}
{τ : LO.FirstOrder.Sentence L}
:
Equations
- ⋯ = ⋯
Instances For
def
LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate.prov_collect_and
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
{T₀ : LO.FirstOrder.Theory L}
{T : LO.FirstOrder.Theory L}
{𝔅 : LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate T₀ T}
[𝔅.HBL]
{σ : LO.FirstOrder.Sentence L}
{τ : LO.FirstOrder.Sentence L}
:
Equations
- ⋯ = ⋯
Instances For
@[reducible, inline]
abbrev
LO.FirstOrder.DerivabilityCondition.goedel
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
(T₀ : LO.FirstOrder.Theory L)
(T : LO.FirstOrder.Theory L)
[LO.FirstOrder.DerivabilityCondition.Diagonalization T₀]
(𝔅 : LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate T₀ T)
:
Equations
Instances For
theorem
LO.FirstOrder.DerivabilityCondition.goedel_spec
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
{T₀ : LO.FirstOrder.Theory L}
{T : LO.FirstOrder.Theory L}
[LO.FirstOrder.DerivabilityCondition.Diagonalization T₀]
{𝔅 : LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate T₀ T}
:
T₀ ⊢!. LO.FirstOrder.DerivabilityCondition.goedel T₀ T 𝔅 ⭤ ∼𝔅.pr (LO.FirstOrder.DerivabilityCondition.goedel T₀ T 𝔅)
class
LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate.GoedelSound
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
{T₀ : LO.FirstOrder.Theory L}
{T : LO.FirstOrder.Theory L}
(𝔅 : LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate T₀ T)
[LO.FirstOrder.DerivabilityCondition.Diagonalization T₀]
:
- γ_sound : T ⊢!. 𝔅.pr (LO.FirstOrder.DerivabilityCondition.goedel T₀ T 𝔅) → T ⊢!. LO.FirstOrder.DerivabilityCondition.goedel T₀ T 𝔅
Instances
theorem
LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate.GoedelSound.γ_sound
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
{T₀ : LO.FirstOrder.Theory L}
{T : LO.FirstOrder.Theory L}
{𝔅 : LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate T₀ T}
[LO.FirstOrder.DerivabilityCondition.Diagonalization T₀]
[self : 𝔅.GoedelSound]
:
T ⊢!. 𝔅.pr (LO.FirstOrder.DerivabilityCondition.goedel T₀ T 𝔅) → T ⊢!. LO.FirstOrder.DerivabilityCondition.goedel T₀ T 𝔅
theorem
LO.FirstOrder.DerivabilityCondition.unprovable_goedel
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
{T₀ : LO.FirstOrder.Theory L}
{T : LO.FirstOrder.Theory L}
[T₀ ≼ T]
[LO.FirstOrder.DerivabilityCondition.Diagonalization T₀]
{𝔅 : LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate T₀ T}
[LO.System.Consistent T]
:
theorem
LO.FirstOrder.DerivabilityCondition.unrefutable_goedel
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
[DecidableEq (LO.FirstOrder.Sentence L)]
{T₀ : LO.FirstOrder.Theory L}
{T : LO.FirstOrder.Theory L}
[T₀ ≼ T]
[LO.FirstOrder.DerivabilityCondition.Diagonalization T₀]
{𝔅 : LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate T₀ T}
[LO.System.Consistent T]
[𝔅.GoedelSound]
:
theorem
LO.FirstOrder.DerivabilityCondition.goedel_independent
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
[DecidableEq (LO.FirstOrder.Sentence L)]
{T₀ : LO.FirstOrder.Theory L}
{T : LO.FirstOrder.Theory L}
[T₀ ≼ T]
[LO.FirstOrder.DerivabilityCondition.Diagonalization T₀]
{𝔅 : LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate T₀ T}
[LO.System.Consistent T]
[𝔅.GoedelSound]
:
theorem
LO.FirstOrder.DerivabilityCondition.first_incompleteness
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
[DecidableEq (LO.FirstOrder.Sentence L)]
{T₀ : LO.FirstOrder.Theory L}
{T : LO.FirstOrder.Theory L}
[T₀ ≼ T]
[LO.FirstOrder.DerivabilityCondition.Diagonalization T₀]
{𝔅 : LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate T₀ T}
[LO.System.Consistent T]
[𝔅.GoedelSound]
:
theorem
LO.FirstOrder.DerivabilityCondition.formalized_consistent_of_existance_unprovable
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
[DecidableEq (LO.FirstOrder.Sentence L)]
{T₀ : LO.FirstOrder.Theory L}
{T : LO.FirstOrder.Theory L}
{𝔅 : LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate T₀ T}
{σ : LO.FirstOrder.Sentence L}
[𝔅.HBL]
:
theorem
LO.FirstOrder.DerivabilityCondition.formalized_unprovable_goedel
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
[DecidableEq (LO.FirstOrder.Sentence L)]
{T₀ : LO.FirstOrder.Theory L}
{T : LO.FirstOrder.Theory L}
[T₀ ≼ T]
[LO.FirstOrder.DerivabilityCondition.Diagonalization T₀]
{𝔅 : LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate T₀ T}
[𝔅.HBL]
:
T ⊢!. 𝔅.con ➝ ∼𝔅.pr (LO.FirstOrder.DerivabilityCondition.goedel T₀ T 𝔅)
Formalized First Incompleteness Theorem
theorem
LO.FirstOrder.DerivabilityCondition.iff_goedel_consistency
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
[DecidableEq (LO.FirstOrder.Sentence L)]
{T₀ : LO.FirstOrder.Theory L}
{T : LO.FirstOrder.Theory L}
[T₀ ≼ T]
[LO.FirstOrder.DerivabilityCondition.Diagonalization T₀]
{𝔅 : LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate T₀ T}
[𝔅.HBL]
:
T ⊢!. LO.FirstOrder.DerivabilityCondition.goedel T₀ T 𝔅 ⭤ 𝔅.con
theorem
LO.FirstOrder.DerivabilityCondition.unprovable_consistency
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
[DecidableEq (LO.FirstOrder.Sentence L)]
{T₀ : LO.FirstOrder.Theory L}
{T : LO.FirstOrder.Theory L}
[T₀ ≼ T]
[LO.FirstOrder.DerivabilityCondition.Diagonalization T₀]
{𝔅 : LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate T₀ T}
[𝔅.HBL]
[LO.System.Consistent T]
:
T ⊬. 𝔅.con
theorem
LO.FirstOrder.DerivabilityCondition.unrefutable_consistency
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
[DecidableEq (LO.FirstOrder.Sentence L)]
{T₀ : LO.FirstOrder.Theory L}
{T : LO.FirstOrder.Theory L}
[T₀ ≼ T]
[LO.FirstOrder.DerivabilityCondition.Diagonalization T₀]
{𝔅 : LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate T₀ T}
[𝔅.HBL]
[LO.System.Consistent T]
[𝔅.GoedelSound]
:
def
LO.FirstOrder.DerivabilityCondition.kreisel
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
(T₀ : LO.FirstOrder.Theory L)
(T : LO.FirstOrder.Theory L)
[LO.FirstOrder.DerivabilityCondition.Diagonalization T₀]
(𝔅 : LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate T₀ T)
[𝔅.HBL]
(σ : LO.FirstOrder.Sentence L)
:
Equations
- One or more equations did not get rendered due to their size.
Instances For
theorem
LO.FirstOrder.DerivabilityCondition.kreisel_spec
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
{T₀ : LO.FirstOrder.Theory L}
{T : LO.FirstOrder.Theory L}
[LO.FirstOrder.DerivabilityCondition.Diagonalization T₀]
{𝔅 : LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate T₀ T}
[𝔅.HBL]
(σ : LO.FirstOrder.Sentence L)
:
T₀ ⊢!. LO.FirstOrder.DerivabilityCondition.kreisel T₀ T 𝔅 σ ⭤ (𝔅.pr (LO.FirstOrder.DerivabilityCondition.kreisel T₀ T 𝔅 σ) ➝ σ)
theorem
LO.FirstOrder.DerivabilityCondition.loeb_theorm
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
{T₀ : LO.FirstOrder.Theory L}
{T : LO.FirstOrder.Theory L}
[T₀ ≼ T]
[LO.FirstOrder.DerivabilityCondition.Diagonalization T₀]
{𝔅 : LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate T₀ T}
{σ : LO.FirstOrder.Sentence L}
[𝔅.HBL]
(H : T ⊢!. 𝔅.pr σ ➝ σ)
:
T ⊢!. σ
instance
LO.FirstOrder.DerivabilityCondition.instLoebOfHBL
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
{T₀ : LO.FirstOrder.Theory L}
{T : LO.FirstOrder.Theory L}
[T₀ ≼ T]
[LO.FirstOrder.DerivabilityCondition.Diagonalization T₀]
{𝔅 : LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate T₀ T}
[𝔅.HBL]
:
𝔅.Loeb
Equations
- LO.FirstOrder.DerivabilityCondition.instLoebOfHBL = { LT := ⋯ }
theorem
LO.FirstOrder.DerivabilityCondition.formalized_loeb_theorem
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
[DecidableEq (LO.FirstOrder.Sentence L)]
{T₀ : LO.FirstOrder.Theory L}
{T : LO.FirstOrder.Theory L}
[T₀ ≼ T]
[LO.FirstOrder.DerivabilityCondition.Diagonalization T₀]
{𝔅 : LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate T₀ T}
{σ : LO.FirstOrder.Sentence L}
[𝔅.HBL]
:
instance
LO.FirstOrder.DerivabilityCondition.instFormalizedLoebOfHBL
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
[DecidableEq (LO.FirstOrder.Sentence L)]
{T₀ : LO.FirstOrder.Theory L}
{T : LO.FirstOrder.Theory L}
[T₀ ≼ T]
[LO.FirstOrder.DerivabilityCondition.Diagonalization T₀]
{𝔅 : LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate T₀ T}
[𝔅.HBL]
:
𝔅.FormalizedLoeb
Equations
- LO.FirstOrder.DerivabilityCondition.instFormalizedLoebOfHBL = { FLT := ⋯ }
theorem
LO.FirstOrder.DerivabilityCondition.unprovable_consistency_via_loeb
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
{T₀ : LO.FirstOrder.Theory L}
{T : LO.FirstOrder.Theory L}
{𝔅 : LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate T₀ T}
[LO.System.Consistent T]
[𝔅.Loeb]
:
T ⊬. 𝔅.con
theorem
LO.FirstOrder.DerivabilityCondition.formalized_unprovable_not_consistency
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
[DecidableEq (LO.FirstOrder.Sentence L)]
{T₀ : LO.FirstOrder.Theory L}
{T : LO.FirstOrder.Theory L}
[T₀ ≼ T]
[LO.FirstOrder.DerivabilityCondition.Diagonalization T₀]
{𝔅 : LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate T₀ T}
[LO.System.Consistent T]
[𝔅.HBL]
[𝔅.GoedelSound]
:
theorem
LO.FirstOrder.DerivabilityCondition.formalized_unrefutable_goedel
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
[DecidableEq (LO.FirstOrder.Sentence L)]
{T₀ : LO.FirstOrder.Theory L}
{T : LO.FirstOrder.Theory L}
[T₀ ≼ T]
[LO.FirstOrder.DerivabilityCondition.Diagonalization T₀]
{𝔅 : LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate T₀ T}
[LO.System.Consistent T]
[𝔅.HBL]
[𝔅.GoedelSound]
:
T ⊬. 𝔅.con ➝ ∼𝔅.pr (∼LO.FirstOrder.DerivabilityCondition.goedel T₀ T 𝔅)
@[reducible, inline]
abbrev
LO.FirstOrder.DerivabilityCondition.rosser
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
(T₀ : LO.FirstOrder.Theory L)
(T : LO.FirstOrder.Theory L)
[LO.FirstOrder.DerivabilityCondition.Diagonalization T₀]
(𝔅 : LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate T₀ T)
[𝔅.Rosser]
:
Equations
Instances For
theorem
LO.FirstOrder.DerivabilityCondition.rosser_spec
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
{T₀ : LO.FirstOrder.Theory L}
{T : LO.FirstOrder.Theory L}
[LO.FirstOrder.DerivabilityCondition.Diagonalization T₀]
{𝔅 : LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate T₀ T}
[𝔅.Rosser]
:
T₀ ⊢!. LO.FirstOrder.DerivabilityCondition.rosser T₀ T 𝔅 ⭤ ∼𝔅.pr (LO.FirstOrder.DerivabilityCondition.rosser T₀ T 𝔅)
theorem
LO.FirstOrder.DerivabilityCondition.unprovable_rosser
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
{T₀ : LO.FirstOrder.Theory L}
{T : LO.FirstOrder.Theory L}
[T₀ ≼ T]
[LO.FirstOrder.DerivabilityCondition.Diagonalization T₀]
{𝔅 : LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate T₀ T}
[LO.System.Consistent T]
[𝔅.Rosser]
:
theorem
LO.FirstOrder.DerivabilityCondition.unrefutable_rosser
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
{T₀ : LO.FirstOrder.Theory L}
{T : LO.FirstOrder.Theory L}
[T₀ ≼ T]
[LO.FirstOrder.DerivabilityCondition.Diagonalization T₀]
{𝔅 : LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate T₀ T}
[LO.System.Consistent T]
[𝔅.Rosser]
:
theorem
LO.FirstOrder.DerivabilityCondition.rosser_independent
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
{T₀ : LO.FirstOrder.Theory L}
{T : LO.FirstOrder.Theory L}
[T₀ ≼ T]
[LO.FirstOrder.DerivabilityCondition.Diagonalization T₀]
{𝔅 : LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate T₀ T}
[LO.System.Consistent T]
[𝔅.Rosser]
:
theorem
LO.FirstOrder.DerivabilityCondition.rosser_first_incompleteness
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
{T₀ : LO.FirstOrder.Theory L}
{T : LO.FirstOrder.Theory L}
[T₀ ≼ T]
[LO.FirstOrder.DerivabilityCondition.Diagonalization T₀]
{𝔅 : LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate T₀ T}
[LO.System.Consistent T]
[𝔅.Rosser]
:
theorem
LO.FirstOrder.DerivabilityCondition.kriesel_remark
{L : LO.FirstOrder.Language}
[LO.FirstOrder.Semiterm.Operator.GoedelNumber L (LO.FirstOrder.Sentence L)]
{T₀ : LO.FirstOrder.Theory L}
{T : LO.FirstOrder.Theory L}
[T₀ ≼ T]
{𝔅 : LO.FirstOrder.DerivabilityCondition.ProvabilityPredicate T₀ T}
[𝔅.Rosser]
:
T ⊢!. 𝔅.con
If 𝔅
satisfies Rosser provability condition, then 𝔅.con
is provable in T
.