In what number of steps will the recursion terminate? Why, in somewhat more that 2^b steps. The circularity in the argument is glaringly obvious.
This argument applies just as well to addition and multiplication.
I think I can explain that. Nelson doesn't believe in any kind of Platonic number, but he does believe in formal number systems and that they can sometimes accurately describe the world. We have essentially two number systems: the tally-mark system (e.g. SSSSS0 and so on), and the positional system (e.g. 2011 and so on). Both of these are successful at modeling aspects of the world.
Most people regard these two number systems as equivalent in some sense. Specifically, there is a procedure for rewriting a positional number as a tally number, and most peo...
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.