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This "proof" seems to be popular, and it always bothers me that it's invalid. It uses self-reference in an invalid way. Allow me to formalize; we take as a rig

This "proof" seems to be popular, and it always bothers me that it's invalid. It uses self-reference in an invalid way. 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. [1] https://plato.stanford.edu/entries/logic-higher-order/ https://plato.stanford.edu/entries/logic-higher-order/

zodiac 10y ago
> Specifically, a number n is interesting if there is some predicate P(x) which is true only for n. 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
- gcr 10y ago
  What's not captured by this criteria? You can express whatever idea of "interestingness" you like in this framework by finding a predicate that expresses it.
Someone 10y ago
”Specifically, a number n is interesting if there is some predicate P(x) which is true only for n” 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?
gweinberg 10y ago
I think it was always intended to be sort of a joke. "Interesting" is subjective, but obviously any self-consistent criteria of interesting cannot allow something to be interesting by it's sheer uninterstingness. 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.
gcr 10y ago
Thank you. As someone who struggles expressing concepts formally, it was enlightening to see how this sort of proof could be modeled.

zodiac 10y ago

> Specifically, a number n is interesting if there is some predicate P(x) which is true only for n. 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

gcr 10y ago
What's not captured by this criteria? You can express whatever idea of "interestingness" you like in this framework by finding a predicate that expresses it.

Someone 10y ago

”Specifically, a number n is interesting if there is some predicate P(x) which is true only for n” 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?

gweinberg 10y ago

I think it was always intended to be sort of a joke. "Interesting" is subjective, but obviously any self-consistent criteria of interesting cannot allow something to be interesting by it's sheer uninterstingness. 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.

gcr 10y ago

Thank you. As someone who struggles expressing concepts formally, it was enlightening to see how this sort of proof could be modeled.