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I was working through the intuitionistic soundness theorem and came up with a proof, but had a critical insight when I trying to understand the mechanics of completeness. It looks like the soundness theorem works equally well for terms that denote false – for which the only satisfying value will be ⊥. That's fine and even expected for soundness – we can prove false things by putting falsehoods into Γ, and indeed any proof of a tautology of the form ~A will do exactly this.
But for valid_of_provable it seems like we need something stronger – that ∅ ⊨ A is non-bottom for any valuation, in some Heyting algebra H (perhaps every non-trivial Heyting algebra H). We'll need something similar for completeness as well.
The text was updated successfully, but these errors were encountered:
I was working through the intuitionistic soundness theorem and came up with a proof, but had a critical insight when I trying to understand the mechanics of completeness. It looks like the soundness theorem works equally well for terms that denote false – for which the only satisfying value will be ⊥. That's fine and even expected for soundness – we can prove false things by putting falsehoods into Γ, and indeed any proof of a tautology of the form ~A will do exactly this.
But for valid_of_provable it seems like we need something stronger – that ∅ ⊨ A is non-bottom for any valuation, in some Heyting algebra H (perhaps every non-trivial Heyting algebra H). We'll need something similar for completeness as well.
The text was updated successfully, but these errors were encountered: