Comment by swatow
11 years ago
No, and that's the key point. His claim is that if you tried to express 80^5000 as an ordinary number, you would just end up going in circles. Your calculation would never terminate. Now we can't in practice compute 80^5000 directly, in the sense of writing down this many 1's, so it's a good example to use. If the example where 2^10, we could indeed demonstrate that 2^10 is a real number.
I don't think that makes sense. We can certainly find the numeral for 80 ^ 5000 (open a command prompt and your numerical tool of choice if you doubt me). If your point is that we can't write out enough tally marks, then you're correct, but
1) Not unique to exponentiation, it's also true of 281474976710656 + 79766443076872509863361.
2) This is a vague notion and you're going to have to deal with the problem of the smallest number that isn't expressible (I believe it might be the same as "feasible numbers", and there's a whole theory here)
3) the whole tenor of his critique suggests something a bit more foundational (predicativity, I guess...?)
When I said Now we can't in practice compute 80^5000 directly, in the sense of writing down this many 1's I meant waht you referred to as "tally marks". I agree "1's" isn't quite the right way to put it.
As you say, the notion of a "real number" is vague without the details (which I'm not really able to describe properly, because I'm only summarizing something I vaguely understand. I'm not an expert on mathematical logic). The best source is Nelson's book "Predicative Arithmetic", https://web.math.princeton.edu/~nelson/books/pa.pdf In the book, he defines what he means by a real number, and shows why exponentation doesn't satisfy the property that if a^X is a real number, then a^(X+1) is.