January 13th, 2016, 01:45 PM  #41  
Senior Member Joined: Aug 2012 Posts: 2,311 Thanks: 706  Quote:
I do confess I have not read through the entire thread. And my comment was not meant to be exactly on subject; but was rather a little bit of counterspin on the entire question. We take it as obvious that some but not all disjoint unions of a collections of dimensionless and lengthless points can have dimension and length. This is an assumption that underlies many if not all of the profound mysteries of the real numbers. There's no real explanation for it in our math or logic. It's a question that so far has eluded our rationality. So I would say that my comment was not so much offtopic, as it was a digression into the philosophical underpinnings of the entire subject. That's my story and I'm stickin' to it!! Last edited by skipjack; February 29th, 2016 at 02:23 AM.  
January 13th, 2016, 02:00 PM  #42  
Senior Member Joined: Dec 2015 From: France Posts: 103 Thanks: 1  Quote:
1/2+1/4+...=/=1 but >1  
January 13th, 2016, 02:57 PM  #43  
Senior Member Joined: Jun 2015 From: England Posts: 915 Thanks: 271  Quote:
An arrow will never reach his target because he is breaking down the path into lots of zeros instead of summing a null sequence of times. @pengkuan This is the trick to resolve all of Zeno type paradoxes. Transform the argument (problem) to a domain when you are summing a null sequence. Last edited by studiot; January 13th, 2016 at 03:04 PM.  
January 13th, 2016, 04:09 PM  #44 
Senior Member Joined: Aug 2012 Posts: 2,311 Thanks: 706  Which interval among [0,1/2], [1/2, 3/4], [3/4], [7/8], ... do you claim is zero? That's Zeno's argument. None of these intervals are zero.

January 14th, 2016, 09:30 AM  #45  
Senior Member Joined: Jun 2015 From: England Posts: 915 Thanks: 271  Quote:
$\displaystyle \left[ {0,\frac{1}{2}} \right[;\quad \left[ {\frac{1}{2},\frac{3}{4}} \right[;\quad \left[ {\frac{3}{4},\frac{7}{8}} \right[;........\quad $ and then noting that we must use halfopen intervals, closed at the lefthand end. This is because if we use closed intervals as you show then the 1/2, 3/4, etc. will be counted twice or you are saying that the arrow visits these points twice on its journey? We start at the lefthand end because the arrow must start at 0. Now the complete Zeno argument runs that in order to cover the whole distance the arrow must first cover half the distance and when it is half way it must cover half the distance remaining, i.e. one quarter the whole distance and so on. So however close the arrow gets to its target there is always a bit left to go. The fallacy is in the statement always a bit left to go. Always is a measure of time; and the times to traverse each successive interval forms a null sequence $\displaystyle {s_n} = 1,\frac{1}{2},\frac{1}{4},\frac{1}{8}...........$ Which does not sum to always (ie is not divergent). Actually, Zeno went on with his arrow to a more difficult paradox that does represent the partition into sets of single numbers of zero measure. He observed that the first half distance can itself be split down into a half distance and the argument can be applied recursively, with the conclusion that the arrow can never move. Last edited by skipjack; February 29th, 2016 at 02:28 AM.  
January 14th, 2016, 11:28 AM  #46  
Senior Member Joined: Aug 2012 Posts: 2,311 Thanks: 706  No typo, my notation was deliberate. A countable collection of singletons has measure zero and makes no possible difference to this discussion. The endpoints do not matter. The length of the open interval (a,b) is exactly the same as the length of the closed or halfopen intervals with the same endpoints. You went to great lengths to avoid answering the question I asked you, which was: Which interval has zero size? Clearly none of them do but you claimed some or all of them do earlier. Quote:
Earlier you wrote: I am challenging you to tell me which of the intervals listed, whether we take them to be open, closed, or halfopen/halfclosed, are zero. Last edited by Maschke; January 14th, 2016 at 11:34 AM.  
January 14th, 2016, 12:24 PM  #47  
Senior Member Joined: Jun 2015 From: England Posts: 915 Thanks: 271  Quote:
Quote:
 
January 14th, 2016, 12:56 PM  #48 
Senior Member Joined: Aug 2012 Posts: 2,311 Thanks: 706  Actually as I understand it Zeno proposed four separate paradoxes, and when people say "Zeno's paradox" without carefully stating which one they're talking about, there's some ambiguity. I'm not personally up on the specifics of which paradox is which. If you can remind me exactly what we're talking about, that will probably be a help to everyone. As I said earlier I jumped into the middle of a thread without reading it first, so whatever point you think I'm arguing, I'm probably not. You said something about zeros, maybe you can just say what that was in reference to. 

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cardinality, infinity, irrational, numbers 
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