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One more question about the cantor set.by cragar
Tags: cantor 
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#1
Jan712, 06:49 PM

P: 2,466

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?



#2
Jan712, 09:32 PM

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P: 1,716

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
Jan712, 09:51 PM

P: 2,466

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
Jan812, 10:48 AM

Sci Advisor
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One more question about the cantor set.



#5
Jan812, 10:52 AM

P: 2,466

can I define it as 1/x and x goes to infinity?



#6
Jan812, 12:18 PM

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#7
Jan812, 12:50 PM

P: 2,466

close to zero but not zero. could I define it as multiplying 1/2 to itself forever.



#8
Jan812, 02:12 PM

P: 1,622

One way to get infinitesimal elements involves using the compactness theorem to construct a nonstandard model of the reals. I am not familiar with the measure theory of nonstandard models of R so I cannot give you any more information than this. 


#9
Jan812, 03:30 PM

P: 2,466

ok, thanks for your responses. so I can have stuff like (0)*(Infinity)=1



#10
Jan812, 03:53 PM

P: 1,622




#11
Jan812, 04:10 PM

P: 2,466

why couldn't I just have [itex] \frac{1}{2^x}(2^x) [/itex] and have x go to infinity



#12
Jan812, 04:15 PM

P: 1,622




#13
Jan912, 12:17 PM

P: 2,466

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. 


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