Comment by swatow
11 years ago
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
Nelson withdrew the claim in this paper based on a blog interaction with Terry Tao.
http://m-phi.blogspot.com/2011/10/nelson-withdraws-his-claim...
The whole episode: - Reputed Princeton mathematician publishes paper questioning the foundations of math - Terry Tao 'disproves' him in a series of blog comments is one of the more interesting recent happenings in the world of math
You've misunderstood the situation.
I was referring to earlier work by Nelson in which he explains why he thinks Peano Arithmetic may be inconsistent.
Later, in the incident you referred to, he claimed to have an actual proof. Terry Tao showed the proof was wrong, and Nelson accepted this.
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?
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...?)
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I'll read the paper, it sounds interersting. But isn't all Math "circular" -- Goedels Second Incompleteness Theorem yeah?
http://en.wikipedia.org/wiki/G%C3%B6del's_incompleteness_the...
Goedel's incompleteness theorem uses Peano Arithmetic, which Nelson claims is (possibly) inconsistent. He develops his own "predicative" arithmetic, to which Goedel's theorem does not apply.
(large pdf) https://web.math.princeton.edu/~nelson/books/pa.pdf
I'll read it, thanks. I always (layman) understood Goedel's theorem as meaning you can be complete or consistent but not both. So you can have a language that can describe everything but you'll have paradoxes, or you'll have no paradoxes but not be able to describe everything (all possibilities) in your system
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Even if an axiom system could prove its own consistency, that wouldn't be any less circular - we could believe T because T proves that T is consistent - but if T were inconsistent then it might still prove that T was consistent.
Most of us take some axioms and believe them, ultimately because they correspond to physical experience - if you take a pile of n pebbles and add another pebble to the pile, you get a pile of n+1 pebbles, and n+1 is always a new number that doesn't =0 (that is, our pile looks different from a pile of 0 pebbles). Maybe there's some (very large) special n where this wouldn't happen - where you'd add 1 and get a pile of 0 pebbles, or where you can have n pebbles but you can't have n sheep. But so far we haven't found that. So far the universe remains simple, and the axioms of arithmetic are simpler than conceivable alternatives.