Comment by swatow

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.