Friday, November 05, 2004

Math was never this interesting

2 is an interesting number : Among other things it is the smallest prime. Ramanujam's number ,1729, is interesting, it is the smallest number that can be written as the sum of two cubes in two different ways. Perhaps the largest interesting number we know of is Graham's number.

Is it possible to define exactly what the set of 'interesting' numbers is? Not in any non-trivial way, if you're working with only the natural numbers. Suppose I is a set of of interesting numbers, such that I' is non-empty. Let d be the least element of I'. Since d is the least non-interesting number, it is therefore interesting. This contradicts the assumption that I' is non-empty. Therefore we have,

Theorem: There does not exist a set of dull natural numbers.

Corollary : Every natural number is interesting.

This argument can be extended to any countable set of numbers. How about the real numbers?

Can there exist a dull set of real numbers? Maybe, as long as they are uncountable and do not contain any natural numbers (or integers or rationals) , for then the least such natural number (or integer or rational) would be interesting.

It might seem that since a countable set can always be embedded into such a uncountable dull set of real numbers, its minimum element must always be interesting, but we have to be careful here. We cannot consider arbitrary orderings since they are not interesting.

Thus, if there exists only a countable number of interesting orderings, then there must exist a set of dull real numbers, because each interesting number can also be represented as the least element of an interesting ordering (interesting because it has an interesting element as its least element). Assuming the continuum hypothesis, the converse follows.

(Disclaimer: This article was written after drinking one too many tequila shots and should therefore be taken with a generous dose of salt, rather like the tequila.)

3 comments:

Jeff Erickson said...

Theorem: All positive intergers are boring.

Proof: Let n be the smallest interesting positive integer. So what?

Deepak said...

Trust a Computational Geometer to find the integers boring.

Dave said...

They were all boring before a constraint changed the point of view. If you're bored, find the constraint.