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Recently Toshiyasu Arai submitted "An ordinal analysis of $\Pi_{N}$-Collection" and Henry Towsner submitted "Proofs that Modify Proofs", both of which claim ordinal analysis of full second-order arithmetic. There has been discussion before on MO about this, see:

The consensus seemed to be that we were a long way away from proving anything like this. For such a big result, I am struggling to find much information on it. I think the two most obvious questions for a lay-person is:

  1. What is claimed to be proof-theoretic ordinal of $Z_2$?
  2. What does this mean for the ordinal analysis of $ZFC$?

Additionally what are the wider implications Proof Theory and Reverse Mathematics?

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    $\begingroup$ Note that Towsner has also claimed a similar analysis. That said, I think a significant reason for the lack of response to either paper is the sheer technical difficulty involved; I don't think many people are able to easily read/assess these papers, even at a very high level (certainly I'm not!). $\endgroup$
    – Noah Schweber
    Commented Oct 6 at 3:06
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    $\begingroup$ With a non-expert, but interested-amateur, competence, I'd strongly second @NoahSchweber's observation of the difficulty of reading or appraising... $\endgroup$
    – paul garrett
    Commented Oct 6 at 3:15
  • $\begingroup$ @NoahSchweber I was not aware of Towsner's claim! I've extended my question to also include it. It does seem that it'll take time for these questions to be answered. Additionally as these results are quite strong, there is good reason to be cautious. I am interested to hear others opinion on this matter however. $\endgroup$
    – solatia
    Commented Oct 6 at 3:35
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    $\begingroup$ Towsner is still working on calculating the proof-theoretic ordinal of Z2, and there is nothing much to be said about it so far. My expectation (and what Towsner told me) is that it will involve with 'collapsing $n$-ptykes' (or recursion on $n$-ptykes.) Also, note that Pakhomov claimed he did an ordinal analysis of Z2, but nothing has been published about it. $\endgroup$
    – Hanul Jeon
    Commented Oct 6 at 4:05
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    $\begingroup$ I visited UPenn last late spring to discuss my research topic with Towsner, and we agreed that even if we can do an ordinal analysis of Z2, there is a very long way to go in ZFC. $\endgroup$
    – Hanul Jeon
    Commented Oct 6 at 5:34

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