Questions about coverings and some odd question.

[matt grime]

yes, idefined it such that x1 is in(x1-e/2,x1+e/2) and so forth.
what's wrong with that?

Nothing - just tidy it up. I.e. put in proper subscripts, notation, and don't use 'and so forth' after *one* example.

and for the second question, only to count the number of strings that can be made out of ten symbols.
in order to choose the first symbol we have 6 options, and thus it follows up until the tenth symbol, overall we have 10*6!, so why do you say the factorial stuff is irrelevant?
the number of strings that can be made must include factorial, is it not?

No. Why would it? Let's say we can only use the symbols 0,1,2,..,9. Now, how many numbers can be made from strings of length n in those symbols. Let's say n=2, so we have 00,01,02,03,04,..,99. Hmm, does 100=2*10!, which is what you've said it does by the above reasoning?

 

[matt grime]

but the total length isn't e, so i could squeeze such that (x1-e/4,x1+e/4)
(x2-e/8,x2+e/8)...
so the total length is e/2+e/4+...=e/2(1/1-1/2)=e

No, the total length is *less* than e; the intervals are not necessarily disjoint.

but surely that's not your problem from what i wrote, is it?

What are you referring to with 'my problem'?

 

[EnumaElish]

I think what the 1st question is getting at Cantor's argument that the measure of rationals (on the real line) is zero. Imagine you can visualize each rational point on the real line (and there are infinitely many of them, to be sure). Suppose a tiny creature wants to walk on the real line from 0 to 1 in infinitely tiny steps, and that each point corresponding to a rational number holds a landmine. So your advice to the creature is to avoid those points, but to make extra sure that it doesn't blow off a leg, you decide to cover each rational point with a tiny piece of red paper. Although your Xmas gifts are wrapped in miles of red paper, what you really need is a tiny, tiny, piece of red paper to cover all the rationals. Why? Suppose you have a piece of red paper with length L (which is tiny, e.g., 1/"zillion"). You use half of it to cover the first rational point that you see (and to be sure, even the tiniest bit of paper will cover more than just the point with a landmine, i.e. it will also cover some safe [i.e. irrational] points which are very near the rational point you'd like to cover, but there is no way to avoid this, and that's okay anyway, as long as each landmine is covered in the end, along with some safe points). For the next rational point, you use half of the remaining half, or 1/4ths. Use 1/8ths for the next rational. And so on. Question: can you cover all the rationals this way? Why (or why not)? Supposing that you can, what is the total length of paper that you'll ever need to cover all the rationals?

 

[EnumaElish]

i think i solved question 1, bacause every x in A is in the interval (x-e/2^k+2,x+e/2^k+2) so the set A is covered by: U(x-e/2^k+2,x+e/2^k+2) where k is from 1 to infinity,

1. Why do you need the +2 after each 2^k?
2. Don't you need an argument or a sentence as to why these intervals would cover all the rationals?

 

[EnumaElish]

#33 is almost verbatim from

Everything and More: a Compact History of ∞