Is it? I don’t think the repetition aspect (can’t be hanged on Friday, therefore can’t be hanged on Thursday, …" etc.) is necessary to the paradox. The “paradox” is still there if the judge simply says “You will be hanged at noon today. It will come as a surprise to you”. Since it is not possible for the prisoner to believe the judge’s statements, he cannot conclude anything from them. Therefore it would indeed be a “surprise” to him if he were hanged at noon.

It’s just a more elaborate version of “this statement is false”, except that it’s “you do not believe this statement”, which makes it apply to you (the prisoner) only.

I agree with the OP, but I’m going to use slightly different reasoning.

Consider another example: What is the smallest number that has never been written out? It seems whichever number we identify, we’ll have to write out and thus it won’t qualify any more.
Therefore by similar logic to the “interesting number” example, all of the infinite number of integers, has been written out

This example has a similar fallacy as the “interesting number” example.
The fact that we cannot identify a solution, because it would invalidate that solution, does not rule out that at any time there is a correct solution.

Of course, this is a philosophical issue rather than a maths one. You can’t do maths with these concepts.

Let S be a subset of the natural numbers that contains every uninteresting natural number and no interesting natural numbers. Assume S is not empty. Since every subset of the natural numbers has a smallest element, let n be the smallest element of S. However, this means n is interesting and thus contradicts that n is in S. Therefore our assumption that S is not empty must be false so S is empty. Since S is empty every natural number is interesting.

Asking about the second smallest element of S at this point doesn’t make sense. We just proved that S is empty.

This reminds me of the famous anecdoteabout Hardy, Ramanujan and the number 1729:

If the concept of interesting is loose enough I am sure every single number has some interesting property, since the number of possible properties is pretty much infinite.

I don’t think this is materially very different from the original proof. The difference here is you use the falseness of one proposition to entail the falseness of another, which I don’t think works.

But more to the point, I think the original argument is a trick. It’s using the structure of a formal proof by contradiction on concepts too woolly to apply such logic. It’s like one of the proofs of the existence of god that begin with some bollocks like: Given that there can be no greater love than a parent to his children…

i was going to say something similar to what Lance Turbo said, but he beat me to it. i would point out, though, that the set of interesting numbers is specifically defined to include all numbers which are not a part of it. that’s the contradiction, and whence stems the paradox.

i disagree. if one considers the set of interesting numbers empty (i.e. that there are no interesting numbers; that all numbers are boring), then 0 is the first uninteresting natural number and no apparent paradox arises.

A lot of numbers are only interesting because they fit in with other numbers in a certain way. I don’t think that is fair. We should treat them all as individuals and not judge them as a group.

“Oh, there goes 3, 5, and 7. Those guys are so prime… What the hell happened to you 614?”. I don’t want to live in a society like that. 614 is probably unique in many other ways and that is what we should focus on. If it turns out that 614 is just a regular, perfectly ordinary number with no special properties, that is OK too. Not every number can be completely unique.

Coming to terms this fact is why math majors have to take liberal arts classes at most reputable universities.

Except it doesn’t work. You can’t do proof by contradiction where the crucial property we’re categorizing is changed by the process.

As I said earlier, you can use the same proof to show that every integer has been written out, or thought about, which is clearly false. This is known as a reductio ad absurdum.

Indeed. Integers, while truly amazing, are only used so much because they are useful at describing the real world. If we lived in another universe with completely different rules, we might still “discover” the integers but find them about as elegent-but-useful as the Mandelbrot Set.

It doesn’t matter what “interesting” means. All you need for the apparent paradox to get off the ground is a stipulation that

–to be un"interesting" (whatever that means) is itself “interesting” (whatever that means).

If this is true of the property “interesting” then nothing can be uninteresting. Ipso facto no number can be uninteresting. Every number is interesting.

From this we can see that you don’t even have to do the mathematical-induction-esque procedure described in the OP (going through the numbers one at a time and asking “is this interesting?”).

I think the answer is to deny that the above stipulation is true of the English predicate “interesting”. To be uninteresting is not to be interesting. My wall is uninteresting. That doesn’t make it interesting.

Well–it might, but it depends on what your interests are. Which reminds us of the important (and obvious) fact that “interestingness” is interest-relative.

It is probably true that

–to be uninteresting relative to interest X is to be interesting relative to some interest Y.

It is definitely false that

–to be uninteresting relative to interest X is to be interesting relative to some interest Y.

And it is also definitely false that

–there is something interesting relative to all interests.

Put that all together and the paradox is resolved.

I should have been clearer in the above but it is now time for me to play D&D with my six-year-old.

That’s far from a settled question. Actually, the way you’ve phrased it is settled - non-human animals demonstrate some counting ability, but I doubt that’s what you meant.

I’m fairly sure all numbers are unique. Certainly, any number we could ever refer to would be.

It’s 17. Jeez, I have to explain everything to you people.

Who needs 17? The hell with 17. It’s not a square or a cube, it’s not even, and it has no particularly noteworthy application in popular culture or law. It’s a prime, but it’s not low enough to be remarkable for that and not high enough that you can say “huh, how interesting that such a big number is a prime.”

Let’s assume, for the sake of argument, that the concept of interesting exists and that some numbers have interesting properties. The question then is whether all numbers have interesting properties or whether some numbers lack interesting properties.

A property may be interesting in some cases and not in others. For a non-numerical example, Franklin Delano Roosevelt can be considering interesting because he was the first President to appear on television. But Barack Obama is not considered interesting because he appeared on television.

Same thing with numbers. Four can be considered interesting because it’s the smallest square number. But 5476 isn’t interesting just because it’s a square number.

So the smallest number with no interesting properties might be considered interesting for that reason. But that doesn’t mean that every number with no interesting properties is equally interesting.

So let me just check: you really believe that every natural number, all infinity of them, has the property of being “interesting” for a meaningful definition of “interesting”?

And you believe this because of a property of the edge case: that, in essence, there cannot be a first uninteresting number, since we’re defining “interesting” to include not being the first member of this set?

You’re aware of course that there are lots of paradoxes concerned with edge cases. The paradox of the unexpected hanging, Sorites paradox etc. There’s still dispute about some of these paradoxes, but no-one I know bites the bullet and says you cannot make a heap, or the exam will never take place.

Here’s something to consider: what happens when we extend the paradox to integers? Well, in this case the set of uninteresting numbers may have no boundary on either side, so the proof doesn’t work. Indeed, I suspect that for any reasonable, formal definition of “interesting”, it would be provable that there are an infinite number of uninteresting integers.
A very different answer to the “solution” for natural numbers. Does this seem odd to you?

The set of uninteresting integers would still have an element that’s smallest in absolute value, though. To get the kind of “no smallest element” behavior that you’re looking for, you’d have to go to the rationals, or reals, or some other non-integer set.

Really, the problem is just that the term “interesting” isn’t defined, like I said earlier. Start with a definition of “interesting”, any definition at all, and the problem goes away.