Abstract.
We show that the consistency of the first order arithmetic \(PA\) follows from the pointwise induction up to the Howard ordinal. Our proof differs from U. Schmerl [Sc]: We do not need Girard's Hierarchy Comparison Theorem. A modification on the ordinal assignment to proofs by Gentzen and Takeuti [T] is made so that one step reduction on proofs exactly corresponds to the stepping down \(\alpha\mapsto\alpha [1]\) in ordinals. Also a generalization to theories \(ID_q\) of finitely iterated inductive definitions is proved.
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Received May 30, 1996
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Arai, T. Consistency proof via pointwise induction. Arch Math Logic 37, 149–165 (1998). https://doi.org/10.1007/s001530050089
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DOI: https://doi.org/10.1007/s001530050089