One more question about the cantor set.

In summary, the conversation discusses the concept of removing infinitesimal amounts from a line segment and its implications on the measure of the set. It also explores the idea of non-standard models of the real numbers and the possibility of having 0 * ∞ as a defined value. However, it is concluded that even in non-standard models, this operation is not defined.
  • #1
cragar
2,552
3
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?
 
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  • #2
cragar said:
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?
 
  • #3
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,
 
  • #4
cragar said:
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?
 
  • #5
can I define it as 1/x and x goes to infinity?
 
  • #6
cragar said:
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.
 
  • #7
close to zero but not zero. could I define it as multiplying 1/2 to itself forever.
 
  • #8
cragar said:
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.
 
  • #9
ok, thanks for your responses. so I can have stuff like (0)*(Infinity)=1
 
  • #10
cragar said:
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.
 
  • #11
why couldn't I just have [itex] \frac{1}{2^x}(2^x) [/itex] and have x go to infinity
 
  • #12
cragar said:
why couldn't I just have [itex] \frac{1}{2^x}(2^x) [/itex] 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.
 
  • #13
ok. I thought we could do some 0*infinity limits with L'Hôpital's rule,but maybe I am wrong.
And yes I do see something wrong with what you said.
 

1. What is the Cantor Set?

The Cantor Set is a mathematical set that is created by repeatedly removing the middle third of a line segment. The resulting set is a fractal with infinitely many points, but has a measure of zero.

2. How is the Cantor Set related to the concept of infinity?

The Cantor Set is related to the concept of infinity because it is a set that is made up of infinitely many points, yet has a finite length of zero. This concept challenges our understanding of infinity and shows that there are different levels or types of infinity.

3. What is the significance of the Cantor Set in mathematics?

The Cantor Set has significant implications in various areas of mathematics, including topology, analysis, and set theory. It has also been used to construct other fractals and to prove the existence of uncountably infinite sets.

4. Can the Cantor Set be visualized?

Yes, the Cantor Set can be visualized through a process called the Cantor dust construction, where each step of the construction is represented as a set of points. As the number of iterations increases, the resulting image becomes increasingly complex and resembles a self-similar pattern.

5. How does the Cantor Set relate to real-world applications?

The Cantor Set has been used in various real-world applications, such as in signal processing and image compression. It has also been used as a model for understanding the behavior of certain physical systems, such as fluid dynamics and turbulence.

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