I read Jordan Ellenberg’s excellent Quomodocumque weblog. He referenced a paper by Edward Nelson. This paper made my head hurt.

This paper is not terribly technical, and could be understood by people who are reasonably comfortable with mathematics (no Ph.D. needed). This paper is 12 pages long, so many of you may not want to take the time to read it.

To summarize: Nelson writes about the foundations of mathematics–roughly, verifying that the basics of mathematics is correct (e.g. Does “1+1” equal “2?”). Nelson uses that fact that, as long as we are only considering natural numbers, multiplication is repeated addition, and that exponentiation is repeated multiplication.

The good news: under the basic rules of mathematics, it easy to show that the sum of two natural numbers is a natural number. Similarly, it is easy to show that the product of two natural numbers is a natural number.

The bad news: it is not at all obvious that exponentiation of natural numbers yields a natural number. This creates the unpleasant situation that a number such as 999^{123456789} may not be an integer. For all we know, this could be, say, a fraction.

I do not think that we are going to find natural numbers n and m such that n^{m} is not a natural number, but it *seems* like we should be able to easily prove this from the very basics. Not all is right with the universe.

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October 7, 2009 at 1:56 am |

Wow. When I saw the Facebook post with the question, I thought, why wouldn’t it be? But there’s no proof, eh? That is one of the weirder mathematical results I’ve ever heard of.

October 7, 2009 at 4:36 pm |

I’m not an expert in this, but Nelson seems to think there is no proof. This scares me.

October 8, 2009 at 1:57 am

Is he maybe limiting himself to some weaker proof system, like Peano arithmetic or something?

October 7, 2009 at 2:44 am |

didn’t read it but

where m<0

October 7, 2009 at 4:37 pm |

Re: didn’t read it but

Yeah, I gave up hope a long time ago that 2-1 is a natural number. Alas, we have a need for fractions…

October 7, 2009 at 7:10 pm

Re: didn’t read it but

uh wow, I didn’t eat my Wheaties. m<0 would make m a non-natural number.

maybe I should read more and type less.

October 13, 2009 at 7:25 pm

Re: didn’t read it but

No problem. I do stuff like this hourly.

October 3, 2010 at 5:17 pm |

Wait… really?

I mean, shouldn’t you just be able to prove it inductively, using the recursive definition of exponentiation?

n1 is a natural number, namely n.

If nm is a natural number, then nm+1 is a natural number, namely nm times n.

Why isn’t that good enough?

I guess there’s another possible interpretation, though. We define exponentiation in the natural numbers as repeated multiplication. Then we define exponentiation in (say) the rational numbers, somehow. Then our job is to prove that, if two rational numbers a and b happen to also be natural numbers, then the (rational) exponential ab agrees with the (natural) exponential ab.

It seems to me like this is what it would mean to know that the exponential of two natural numbers is natural. So maybe Nelson is saying we can’t do that?

Wacky! Bookmarked, and I’ll read it as soon as I have the chance.

October 12, 2010 at 1:58 pm |

Quite honestly, I do not remember enough about this article (and I do

not have enough time to re-read it right now beyond a mere skimming).

But here is what my hazy memory says:

First, the problem with a simple induction argument is that the

language of formal logic is not strong enough to say “is a natural

number.” See the second full paragraph on page 10.

Second, he seems to be saying that Goedel proved that we cannot even

prove (in the formal language) that, if x and y are natural numbers,

then x+y is a natural number…

…but (third) he has a workaround for addition and multiplication

(pages 10-12).

Fourth, but this workaround does not work for exponentiation.

I almost went into logic when I was in graduate school, but the last

time I seriously thought about logic was eight years ago. So I am

largely trusting him about the power of the language and Goedel’s

results.

I would love to here your interpretation once you read it!

Bret