
#1
Jan712, 06:49 PM

P: 2,453

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

Sci Advisor
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: 2,453

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
P: 1,716

One more question about the cantor set. 



#5
Jan812, 10:52 AM

P: 2,453

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




#6
Jan812, 12:18 PM

Sci Advisor
P: 1,716





#7
Jan812, 12:50 PM

P: 2,453

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




#8
Jan812, 02:12 PM

P: 1,623

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: 2,453

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




#10
Jan812, 03:53 PM

P: 1,623





#11
Jan812, 04:10 PM

P: 2,453

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: 1,623





#13
Jan912, 12:17 PM

P: 2,453

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. 


Register to reply 
Related Discussions  
Cantor Set Question  Calculus & Beyond Homework  1  
Question about Cantor Set  Set Theory, Logic, Probability, Statistics  4  
Fat Cantor Set question  Calculus & Beyond Homework  0  
Fat Cantor Set question  Calculus  2  
A question on Cantor's second diagonalization argument  General Math  36 