Is there a proof of the following statement: you cannot prove with natural deduction theorems that are unprovable in a Hilbert-style proof system? The logic in discussion is either propositional logic or FOL.
2026-03-26 04:30:37.1774499437
Is there a proof of this statement about deductions?
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Yes. You have to follow a "typical" proof of "equivalence" between Natural Deduction and Hilbert-style.
See :
An alternative approach is through soundness and completeness.
Both proof systems are sound and complete regarding valid formulae; thus, a formula unprovable in ND, being not valid, in not provable in H-s, and vice versa.