zlacker

>>jeremy+(OP)
Mathematician Edward Nelson has a very interesting take on "big" numbers. He claims that the exponential function is not necessarily total, and a number like 2^1000000000000 might not actually exist. The reason he singles out exponentiation is that the reasoning that exponential numbers are "real" numbers, is circular (impredicative). According to Nelson, speaking about such numbers might lead to condradictions, just like speaking about "the set of all sets that don't contain themselves" leads to a contradiction.

He also relates these issues to Christian philosophy, which I find very interesting. In particular, he claims that the a priori belief in the objects defined by Peano Arithmetic, is equivalent to worshipping numbers, as the Pythagoreans did.

I think this is the best starting point if you're interested in reading about his ideas: https://web.math.princeton.edu/~nelson/papers/warn.pdf

>>swatow+im
I'm definitely missing something about his position. I think I understand why he accepts addition and multiplication but not exponentiation. But I don't understand how he can illustrate that with 80^5000 or other concrete examples. While you can't guarantee to him that x ^ n is a countable number for arbitrary x and n, for any specific x and n, you can, right?

>>hyperp+iq
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.

>>swatow+Aq
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...?)