Reading the chapter on Criticisms against the singularity, the strongest one so far, the one to which Kurzweil makes a really weak argument using the example of the “Busy Beaver” problem is putting Godel’s incompleteness theorems and Turings and Church’s findings of the limitations of logic and math to solve problems.
Kurzweil basically has no counter argument against this, yet he tries to distract the reader with the Tibor Rado’s “Busy Beaver” problem. He says they managed to compute the Busy Beaver function for a number of N’s that would be impossible for any human to calculate, however, this is merely an example of how a computer can handle more numbers better than humans, yet, this is no example of how logic could be different in a turing machine than from the human brain.
Given enough time (probably till the end of it), human beings could in theory compute those results, given the computers that can do it were programmed with human logic. This also means, that if there are unsolvable problems (on which immortality could well be one of them), a turing machine (a computer) would not be able to solve given the limitations of logic and math.