# One more question about the cantor set.

by cragar
Tags: cantor
 P: 2,464 Lets start with a line segment from zero to 1 and instead of removing like the middle 1/3 can we remove an infinitesimal amount, and then keep doing this forever. It seems like this set would still have measure 1. Unless I don't understand measure or infinitesimals. And if we looked at the line would it look like a line or dust on the line?
P: 1,716
 Quote by cragar Lets start with a line segment from zero to 1 and instead of removing like the middle 1/3 can we remove an infinitesimal amount, and then keep doing this forever. It seems like this set would still have measure 1. Unless I don't understand measure or infinitesimals. And if we looked at the line would it look like a line or dust on the line?
you need to better define what you mean by removing an infinitesimal amount.

If you mean what would be the limit of the sets obtained say by removing middle fifths then middle sevenths then middle 11'ths and so on it is clear that it will have measure one. What do you think the set would look like?
 P: 2,464 I wanted to remove an infinitesimal amount from the start, like as close as I can get to zero. But on your example, it seems like the set would look like scattered points,
P: 1,716
One more question about the cantor set.

 Quote by cragar I wanted to remove an infinitesimal amount from the start, like as close as I can get to zero. But on your example, it seems like the set would look like scattered points,
What do you mean by an infinitesimal amount?
 P: 2,464 can I define it as 1/x and x goes to infinity?
P: 1,716
 Quote by cragar can I define it as 1/x and x goes to infinity?
to me that is zero. I do not know what an infinitesimal amount is.
 P: 2,464 close to zero but not zero. could I define it as multiplying 1/2 to itself forever.
P: 1,622
 Quote by cragar close to zero but not zero. could I define it as multiplying 1/2 to itself forever.
Nope. Notice that limn→∞2-n = 0. In fact, since the ordinary real number system is archimedean, it has no non-zero infinitesimal elements.

One way to get infinitesimal elements involves using the compactness theorem to construct a non-standard model of the reals. I am not familiar with the measure theory of non-standard models of R so I cannot give you any more information than this.
 P: 2,464 ok, thanks for your responses. so I can have stuff like (0)*(Infinity)=1
P: 1,622
 Quote by cragar ok, thanks for your responses. so I can have stuff like (0)*(Infinity)=1
No. Even in non-standard models of the reals, you still do not have anything like 0 * ∞ = 1.
 P: 2,464 why couldn't I just have $\frac{1}{2^x}(2^x)$ and have x go to infinity
P: 1,622
 Quote by cragar why couldn't I just have $\frac{1}{2^x}(2^x)$ and have x go to infinity
Notice that limx→∞2-x2x+1 = 2. Do you see any problem with this? If you want to include the term +∞ in your number system, then you have to leave things like 0 * ∞ as undefined.
 P: 2,464 ok. I thought we could do some 0*infinity limits with L'Hôpital's rule,but maybe im wrong. And yes I do see something wrong with what you said.

 Related Discussions Calculus & Beyond Homework 1 Set Theory, Logic, Probability, Statistics 4 Calculus & Beyond Homework 0 Calculus 2 General Math 36