I am currently writing a proof for the strength of an ordinal notation. I want to analyse it with second order arithmetic subsystems and I would like to get some advice from experts if possible. I understand that to find a proof-theoretic ordinal of a theory, one has to find the set of well-orderings provably recursive in that theory, but how exactly is that done? I have tried to read some papers by Michael Rathjen to undertand it but a lot of it goes over my head. I would like if someone could also explain how Cut Elimination is done for proofs and what Ordinal Diagrams actually are. Thanks.
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