These equations don't make sense dimensionally. Are there supposed to be constants of proportionality that aren't being mentioned? Are they using the convention c=1? Well, I doubt it's relevant (scaling things shouldn't change the result), but...
Edit: Also, perhaps I just don't know enough differential equations, but it's not obvious to me that a curve such as he describes exists. I expect it does; it's easy enough to write down a differential equation for the height, which will give you a curve that makes sense when r>0, but it's not obvious to me that everything still works when we allow x=0.
These equations don't make sense dimensionally. Are there supposed to be constants of proportionality that aren't being mentioned?
That is my guess. The simplest way IMO would be replace the g in eq. 1 by a constant c with units of distance^(-1/2). The differential equation becomes r'' = g c r^1/2, which works dimensionally. The nontrivial solution (eq. 4) is correct with an added (c g)^2 in front.
it's not obvious to me that a curve such as he describes exists.
I'm not sure what you mean here. What could be wrong in principle with a curve h = c r^3/2 describing the shape of a dome, even at r = 0?
We've discussed Edward Nelson's beliefs and work before. Now, he claims to have a proof of a contradiction in Peano Arithmetic; which if correct is not that specific to PA but imports itself into much weaker systems. I'm skeptical of the proof but haven't had the time to look at it in detail. There seem to be two possible weakpoints in his approach. His approach is to construct a system Q_0^* which looks almost but not quite a fragment of PA and then show that PA both proves this system's consistency and proves its inconsistency.
First, he may be mis-applying the Hilbert-Ackermann theorem-when it applies is highly technical and can be subtle. I don't know enough to comment on that in detail. The second issue is that in trying to show that he can use finitary methods to show there's a contradiction in Q_0^* he may have proven something closer to Q_0^* being omega-inconsistent. Right now, I'm extremely skeptical of this result.
If anyone is going to find an actual contradiction in PA or ZFC it would probably be Nelson. There some clearly interesting material here such as using a formalization of the surprise examiation/unexpected hanging to get a new proof of of Godel's Second Incompleteness Theorem. The exact conditions which this version of Godel's theorem applies may be different from the conditions under which the standard theorem can be proven.