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.