Proof is by contradiction: Assume that not every number is mathematically interesting and let X be the first such number. However, the fact that X is the lowest such number is itself pretty special, right?
Allow me to formalize; we take as a rigorous definition of an "interesting number" that a number has a unique property. Specifically, a number n is interesting if there is some predicate P(x) which is true only for n. In formal first order logic, n is interesting if there exist a predicate P and a number n such that P(n) is true and if m != n then P(n) is false.
Let I, as a subset of the natural numbers N, be the set of interesting numbers. There are two cases: either N - I is empty, or it is not. If it is not, let n be the least element of N - I. n is therefore interesting, having a unique property in that it is the smallest integer not in I; however, this is a contradiction, because we defined I to include all interesting numbers, and so N - I must be empty; in other words, every number is interesting.
Edit: Actually, my definition of "interesting" seems to be in second order logic [1], since I'm using an existential quantifier for predicates. It doesn't seem possible to give a definition of this sense of "interesting" in first order logic.
That’s not that good a definition. Since the predicate “P(x) = x = 4578634986” is only true for x = 4578634986, would that imply that 4578634986 is interesting?
But even if you're willing to accept the paradox, it's still a bullshit proof. It's one thing to say a number is interesting because it really is the first number you can't think of anything interesting to say about it. It's something else to say to say it's interesting because its the first number you can't find anything interesting about, not counting all the others that were considered "interesting" for the same reason.
Well, if you read the "every number is interesting" "proof", this clearly doesn't capture the proof's criteria of interestingness.
I see it as analogous to Berry's paradox - the proof isn't "wrong" per se, but the relevant notion of interestingness is not well defined
You can express whatever idea of "interestingness" you like in this framework by finding a predicate that expresses it.
You might like the alternative (and very nerdy) phrase "on the gripping hand" as an alternative. It (usually?) connotes a third option that invalidates the other two in some way, as you've done here.
{x | ∃ n ∈ ℕ : 10^n = 1}
=
{1, 0.1, 0.01, 0.001, 0.0001, 0.00001, …}
It is impossible to adapt this proof because the set of reals is uncountable. It is doable for the rationals, though, as they _are_ countable.
Of course since OEIS is finite, there will always be numbers which don't make it. What's really interesting is that some numbers are disproportionately underrepresented. They appear much less often than other numbers of the same size. And if you plot each number by the number of times it occurs, it makes a really interesting pattern: https://www.youtube.com/watch?v=_YysNM2JoFo
Oh, and happy new year's!
18159 appears in OEIS, but its search interface isn’t perfect.
For example it appears in http://oeis.org/A000027: ”The positive integers. Also called the natural numbers, the whole numbers or the counting numbers, but these terms are ambiguous.”
I have a diploma in math (equiv. to master of science in math) and I'm interested in 18159. Does that count? (Moreover, I'm interested in 18159 for exactly the reason you stated.)
(Waiting for the new OEIS sequence of uninteresting numbers.)