I didn't downvote, but if I had it would have been because you seem to be missing the point a bit.
It may well be the case that we live in a discrete universe without infinities (I assume that's what you meant by integer) but this is not the case for Newtonian Mechanics, which asserts that both space and time remain continuous at every scale. Thomas never claimed to have a paradox in actual physics, only in Newtonian mechanics, so what you are saying is irrelevant to his claim, you might was well bring up GR or QM as a solution.
Appealing to the third law also doesn't help, his whole point is that if you do the calculations one way you get one answer and if you do them another way you get another answer, hence the use of the word paradox.
Appealing to the third law also doesn't help, his whole point is that if you do the calculations one way you get one answer and if you do them another way you get another answer, hence the use of the word paradox.
His point is if you only do some of the calculations, you can come up with the wrong answer. I fail to see how that implies a paradox rather than just sloppiness.
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.