All Numbers Are Interesting
By way of contraction, suppose there exists some collection of non-interesting numbers. According to the well-ordering principle of nonnegative integers, this set has a minimal element, x. However this makes x interesting: a contradiction. Therefore, all numbers are interesting. Q.E.D.
Doesn’t this mean that all non-negative numbers are interesting?
You could do the same with negative numbers, but the greatest element is interesting. And then zero is the only non negative and non positive number, so it’s interesting as well. The proof is incomplete, but easily finished.
My issue is less with the integers, but more with non-ordered fields. I think this implies that there are non-interesting complex numbers, for example.
Yes I was of course not being completely technical. The idea was to not make it overly complicated and use “number” to mean what most people typically think of when they talk about interesting numbers, their favorite number, number theory, etc: these being non-negative interegers or natural numbers.
However, you raise a good point when thinking about the extension of this to the integers, real numbers, complex numbers, etc. As @the-poetic-mathematician mentioned, it is easily extended to integers. But, we run into issues with the reals; there is no well-ordering of the real and even worse, for complex numbers, there is no usual ordering at all (Complex numbers are not comparable in a less then, greater than way. At best, they can be partially ordered via complex modulus, i.e., absolute value.)
At the same time, just because we cannot use well-ordering doesn’t mean we can’t fix the proof for reals, complex, etc.
is there a practical use for this
@canyoufaceme Of course! It is a great topic of discussion at dinner parties!