One I remember hearing is a proof that there is no first uninteresting number: If there were, it would be an interesting number, as the first of any kind of number would be interesting!
What are "interesting numbers"?
It’s your opinion if a number is interesting or not.
You have the proof right but the theorem wrong. What you're proving is that all numbers are interesting. Proof: If not, there must be a smallest uninteresting number... etc.
I think I heard the proof specifically for what I said, but one could also use it to prove what you said.
(can i necro?)
I think its from 3b1b.
Trivial as it may seem, I disagree with the alleged evidence. Suppose one calls all numbers between, and with the exception of, 0 and 1 uninteresting - one may legitimately do so since there is no universally accepted definition. What would then be the smallest of these "uninteresting" numbers? There is no such smallest number between 0 and 1; for example, any candidate would be defeated by its half.
It's all just a joke (including my rebuttal).
Yeah I think they have in mind nonnegative integers.
Of course there are other ways of doing it, like starting with 1/2 and halving it on either side each step, which gets all rationals with a power of 2 as a denominator. Similarly, you can do that for other fractions like 1/3, 1/5, 1/7, etc. And if you make all such 1/p-where-p-is-prime trees starting at each number reachable from one such tree, and call that tree-ing twice, then if you tree whichever infinity is appropriate for this ($$\infty$$ maybe? $$\omega$$?) times, I think you get all rationals in {0<x<1}.
Of course that doesn't get the rest of the reals in that range. Also, to better mathematicians than me: is there some way of getting all reals in some way like that?
no
The rules of this game say that a value is in an infinite sequence only if it's reachable in some finite number of TAILs. Note that doesn't say "all values are reachable in..."; clearly it takes infinite time to reach all of them. But any particular value has to be reachable in finite time.
So you can't append two infinite streams and say that the result has all the values from both streams; only the values of the first stream are effectively in the result. But what you can do is merge the two streams (take an item from A, then an item from B, etc). In fact you can do an infinite number of merges, and still any specific value will be reachable in finite time. (Merge A with B, then merge AB with C, then ABC with D, etc.)
It seems counterintuitive that you can make a stream of all the rationals because the rationals are dense on the number line, which means that between any two rationals you can always find another rational (namely their average). And so you can't make a stream in which all the rationals appear in order the way you can for the positive integers. (You can't make a stream of all the signed integers in order either, but that's not because the integers are dense, which they aren't; it's because there's no smallest integer to start with.)
But once you understand that you can make an infinite sequence of rational numbers if you're willing to have them out of order, it then becomes even more surprising that you can't do it for the reals. (Georg Cantor, who discovered the theory of infinite sets, famously said "I proved it, but I don't believe it" about this.) But any continuous set of reals (more broadly, any closed set, i.e., one in which the limit point of every sequence in the set is also in the set) can't be enumerated. Of course the proof has to be done by contradiction; you present a supposed enumeration and then show how to construct a real number that isn't in the list.
(why is this in italics?open set covering theorem?)
No, it's to indicate that what follows is the definition of a technical term. That's a pretty common convention in math books. The definition is usually given more tersely, as "A set is closed if every Cauchy sequence in the set has a limit in the set," but then I would have had to define Cauchy sequence.
Oh.ok