Zeilberger proposes that there are a finite number of integers and indeed that they wrap around like computer integers. Woodin objects to that as a limiting view, but then worries that it is not ‘correct’. I ask is it a view or is it a proposition? I agree with Wooden that infinity works well for math and I suspect better than some limit in the sense that useful theorems are shorter. The article here, however, is about physics. I also suspect that if a finite mathematics digs physics out of its hole then the new conceptual world will be subject to classical infinite math. At that point however the finite math do the job more simply. I would not bet much either way. That would not impact classic infinite math in my opinion. It would not discredit infinite math any more than Einstein made Newtonian physics useless.
Tegmark is quoted as saying that since we use computers to check our physics theories and computers are finite, there is no reason to suppose that nature is infinite. Like Zuse I think that the reals, with their infinite precision, are an approximation to the finite, and logically simpler as well. If they are only an approximation we must dig still deeper to find more and I suspect that that may be finite.
At the end Gefter quotes Woodin:
My extremely tentative exploration of what computer proofs about programs might look like did not require infinite integers. A particular proof would presume some point at which the integers wrapped but it was not clear that it was necessary to name this point. It perhaps would be necessary to assert that 2128 ≠ 0.