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It is not the case that Gentzen's original 1936 proof was withdrawn precisely because its well-foundedness argument presupposed consistency in a circular way.
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Reasons For
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Reason for
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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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Reasons Against
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Reason against
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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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