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Can transfinite induction be defined as axiom scheme in FOL on bin-tree structures?

Transfinite induction requires a second order induction hypothesis. So, that can not be defined as axiom scheme in FOL. However, if I look to Goodstein's theorem en the Hydra games, then they have to ...
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2answers
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Can Goodstein's theorem been proven with first order PA + Constructive Omega Rule?

I am trying to understand transfinite induction and Gentzen's theories. But I was wondering, if there is any connection with the Constructive Omega Rule (COR). With COR I mean that if you can proof: ...