alt Hacker NewsThe problem with that is the explanation of why each number is interesting becomes:
the smallest member of the original set of uninteresting numbers
the second smallest member of the original set of uninteresting numbers
the third ...
...
That version of "interesting" quickly becomes "not interesting". The concept simply defies mathematical logic.
Replies
This isn't how math works.
When you get to the point in a proof of the irrationality of root two where you've demonstrated that if it is expressible as a fraction p/q, then both p and q have to be even, you don't then need to proceed to prove that if they're both even, then they both have to be divisible by four, and then if they're both divisible by four, that means they're both divisible by eight...
I mean, you can, but you don't have to.
You can just say 'if it's a rational number then it has a reduced form where p and q have gcf of 1, so if p and q would both have to be even, that is a contradiction'.
Same with the 'set of uninteresting numbers'. If 'being uninteresting' is a property numbers can have, then the 'set of uninteresting numbers' exists, and it has a least member. Being the least member of the set of uninteresting numbers is interesting.
You don't have to infinitely regress from here and get tied up in knots saying that surely there is some 'first truly uninteresting number' to prove that the set is actually empty - you can just see that you must have gone wrong somewhere. Either:
1) Being the least member of the set of uninteresting numbers isn't as interesting as we assume.
or
2) 'Being uninteresting' is not a property numbers can have
I think actually of the two, 1) is more likely the case.
But that doesn't defy mathematical logic. It is a consequence of mathematical logic.
It reminds me about the logic puzzle of the criminal sentenced to death, where the judge says "you will be executed on or before Sunday, and you won't know what day it will be until we come for you."
The criminal knows it can't be Sunday, because he would wake up on Sunday and know he was going to be executed that day. But if Sunday isn't possible, on Saturday he would know he was being executed that day; so Saturday wasn't possible either. The same reasoning can be repeatedly applied to every day between now and Sunday.
It's obviously flawed reasoning (Surprise! they execute you on Thursday), but the flaw is difficult to articulate.