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    Gentzen's original 1936 proof was withdrawn precisely bec... — Carmelics
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    Challenges→A proof of a sequent in first-order arithmetic gives rise to a well-founded reduction tree

    Gentzen's original 1936 proof was withdrawn precisely because its well-foundedness argument presupposed consistency in a circular way.

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    1 reason for
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    Reasons For

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    Reason for
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    • 1.Gentzen's 1936 transfinite induction on ordinals presupposes that the ordering principle itself is well-founded, which assumes consistency.
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    • 2.Gentzen explicitly withdrew the proof and later developed cut-elimination, suggesting he recognized a foundational problem in his original approach.
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    • 3.Using proof-theoretic methods to establish consistency while relying on ordinal recursion risks petitio principii if ordinals encode the very system being justified.
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    Reasons Against

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    Reason against
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    • 1.Gentzen withdrew the proof for technical reasons (finite axiomatization), not because circularity was discovered—documented correspondence shows this.
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    • 2.Well-foundedness of ε₀ can be established proof-theoretically without assuming PA's consistency; it requires only elementary recursive reasoning about ordinals.
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    • 3.The 1936 proof's argument doesn't claim to prove consistency *from within* PA but uses finitary methods external to the system, avoiding direct circularity.
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    Proof of definition segments1 linkedTruth & Knowledge1 linked

    Related

    A proof of a sequent in first-order arithmetic gives rise to a well-founded redu...Gentzen explicitly withdrew the proof and later developed cut-elimination, sugge...Gentzen withdrew the proof for technical reasons (finite axiomatization), not be...Gentzen's 1936 transfinite induction on ordinals presupposes that the ordering p...
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    The 1936 proof's argument doesn't claim to prove consistency *from within* PA bu...Using proof-theoretic methods to establish consistency while relying on ordinal ...Well-foundedness of ε₀ can be established proof-theoretically without assuming P...

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    claim
    Perspectives
    2 (1 for, 1 against)
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    1 edit