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sniffnoySo, as a special case of the theorems in Diana Schmidt's habilitation thesis, one gets that the maximum extending ordinal of the set of finite plane trees is at most the small Veblen ordinal.
It turns out that in 2005, Herman Jervell published a paper describing an order on the set of finite plane trees, extending the partial order, and with order type the small Veblen ordinal.
Therefore, the maximum extending ordinal of the set of finite plane trees is equal to the small Veblen ordinal.
What's annoying, though, is that Jervell didn't state his result in that form -- presumably because he wasn't aware of the theory of maximum extending ordinals. This is really annoying, because it means that the result stated in that form probably won't appear in the literature for quite some time. It's such an immediate consequence of Schmidt's upper bound and Jervell's implicit lower bound that it's not remotely separately publishable. Blargh.
Well, I wanted to publicize this fact here, at least. :P The maximum extending ordinal of the set of finite plane trees is equal to the small Veblen ordinal.
Edit: I guess actually there is a well-known connection between Kruskal's tree theorem and the small Veblen ordinal, the latter being the proof-theoretic ordinal of the former. Are these statements equivalent, meaning actually this has long been known in a different formalism? Um. I have no idea! I just care about order, I don't care about proof theory. But if maximum extending ordinals are actually the same as proof-theoretic ordinals of well-quasi-ordering statements, then I guess there's more of them out there than I thought? IDK, I am a bit skeptical of this. But logic -- like, proper logic, as opposed to order theory like I like to think about -- makes my head hurt... I'll probably just not look into this for now...
Edit again: Hold on, shouldn't the proof-theoretic ordinal be *greater* than the maximum extending ordinal? Apparently not, but... blargh. I'm going to ask math.SE about this.
Edit a third time:: Ha! So one of the people who proved the proof-theory statement mentioned above was... Andreas Weiermann! He's certainly familiar with the theory of maximum extending ordinals. The paper even cites Schmidt! It *doesn't* mention a connection between the two notions, though, leading me to suspect that none is known (although it's from 1993).
You know what? I'll skip asking math.SE (or really, MathOverflow, this is seeming more suited to there) for now; I'll just ask Weiermann himself in July...
It turns out that in 2005, Herman Jervell published a paper describing an order on the set of finite plane trees, extending the partial order, and with order type the small Veblen ordinal.
Therefore, the maximum extending ordinal of the set of finite plane trees is equal to the small Veblen ordinal.
What's annoying, though, is that Jervell didn't state his result in that form -- presumably because he wasn't aware of the theory of maximum extending ordinals. This is really annoying, because it means that the result stated in that form probably won't appear in the literature for quite some time. It's such an immediate consequence of Schmidt's upper bound and Jervell's implicit lower bound that it's not remotely separately publishable. Blargh.
Well, I wanted to publicize this fact here, at least. :P The maximum extending ordinal of the set of finite plane trees is equal to the small Veblen ordinal.
Edit: I guess actually there is a well-known connection between Kruskal's tree theorem and the small Veblen ordinal, the latter being the proof-theoretic ordinal of the former. Are these statements equivalent, meaning actually this has long been known in a different formalism? Um. I have no idea! I just care about order, I don't care about proof theory. But if maximum extending ordinals are actually the same as proof-theoretic ordinals of well-quasi-ordering statements, then I guess there's more of them out there than I thought? IDK, I am a bit skeptical of this. But logic -- like, proper logic, as opposed to order theory like I like to think about -- makes my head hurt... I'll probably just not look into this for now...
Edit again: Hold on, shouldn't the proof-theoretic ordinal be *greater* than the maximum extending ordinal? Apparently not, but... blargh. I'm going to ask math.SE about this.
Edit a third time:: Ha! So one of the people who proved the proof-theory statement mentioned above was... Andreas Weiermann! He's certainly familiar with the theory of maximum extending ordinals. The paper even cites Schmidt! It *doesn't* mention a connection between the two notions, though, leading me to suspect that none is known (although it's from 1993).
You know what? I'll skip asking math.SE (or really, MathOverflow, this is seeming more suited to there) for now; I'll just ask Weiermann himself in July...
