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  • $\begingroup$ I feel like the proof is simply a definition of a WQO, rather than a proof by induction on a WQO. The proof itself is usually in the form of a "minimal bad sequence" argument. I'd also note that it's significantly easier to define and use WQOs and well-founded orders than ordinals, so it might not be quite in the spirit of the question (or maybe it suggests that ordinals are not such useful objects after all...) $\endgroup$ Commented Dec 26, 2019 at 22:45
  • $\begingroup$ Are this proof of Kruskal's tree theorem uses such a wqo, or it just proves that the trees forms a wqo? And I don't think that means ordinals are not such useful. $\endgroup$ Commented Dec 27, 2019 at 1:52
  • $\begingroup$ I don't fully understand the distinction that you're trying to draw. The conclusion of Kruskal's tree theorem is that something is a wqo. The "minimal bad sequence" argument is what I'd call an induction argument. I'd call this proof by induction on a wqo, but maybe you reserve that term for a proof that starts with something that is already known to be a wqo, and uses that fact to prove some statement about the wqo other than "this is a wqo"? Anyway you can read the proof and decide for yourself. $\endgroup$ Commented Dec 27, 2019 at 4:11
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    $\begingroup$ The proof of the Kruskal's Tree Theorem itself needn't involve such an induction, but arguably Schmidt's proof that its type (maximum linearization) is $\Gamma_0$ does? This relates to another question I've been wondering about -- I'm a little unclear here on the connection between type on the one hand & proof-theoretic ordinal on the other; I'm not sure how well that's understood actually... but I guess that's not something to start a discussion on in the comments; maybe I should ask that as a separate question here...) $\endgroup$ Commented Dec 27, 2019 at 5:40
  • $\begingroup$ Hold on, wait -- I just realized we've been talking about the wrong ordinal, right? Kruskal's tree theorem is small Veblen ordinal, not $\Gamma_0$, right? May have to go look up this Gallier paper... $\endgroup$ Commented Dec 28, 2019 at 22:33