Use DerivabilityCondition for simple proof
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SnO2WMaN
commented
Aug 25, 2024
| @@ -133,6 +146,7 @@ lemma provableₐ_D2_context {Γ σ π} (hσπ : Γ ⊢[T]! U.bewₐ (σ ⟶ π) | |||
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| lemma provableₐ_D3_context {Γ σ} (hσπ : Γ ⊢[T]! U.bewₐ σ) : Γ ⊢[T]! U.bewₐ (U.bewₐ σ) := of'! provableₐ_D3 ⨀ hσπ | |||
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| instance : U.provₐ.GoedelSound T U := ⟨by sorry⟩ | |||
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これさえ証明が通れば残りは直ちに従う。競合を防ぐため自分はこのPRに今後触れないので、適当に書き直してマージするなりどうぞ。
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もちろんそうなのだけれど，現状 |
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素人考えでは
Logic.FirstOrder.Incompleteness.DerivabilityConditionを使えば証明はほとんど削減できるのではと思うのだが、どうなのだろうか？一応Diagonalizaation,D1,D2,D3は通っているので、GoedelSound（特にゲーデル文に対しての健全性）だけあとは証明できれば第二不完全性定理およびLoebの定理が直ちに従うはず。追伸: ArithmetizationのリポジトリにブランチをpushしてPRを作る権限がないので渡してもらえると若干助かる。