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September 11th, 2012, 08:12 PM  #11  
Senior Member Joined: Aug 2010 From: Germany Posts: 132 Thanks: 1  Re: Dichotomy paradox reverse engineering Quote:
 
October 2nd, 2012, 12:31 PM  #12 
Senior Member Joined: Aug 2010 From: Germany Posts: 132 Thanks: 1  Re: Dichotomy paradox reverse engineering
If we allow only finitely many copies of a line segment s in [0, 1], where the size of s is an element of (0, 0.5], how can we allow infinitely many line segments { [0, 0.5], [0.5, 0.75], [0.75, 0.875], … } in [0, 1], where the size of each line segment is an element of (0, 0.5]? We wouldn’t allow an infinite number of copies of any element of { [0, 0.5], [0.5, 0.75], [0.75, 0.875], … } in [0, 1], right?

October 2nd, 2012, 04:07 PM  #13 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Dichotomy paradox reverse engineering
Huh?

October 2nd, 2012, 10:44 PM  #14  
Senior Member Joined: Aug 2010 From: Germany Posts: 132 Thanks: 1  Re: Dichotomy paradox reverse engineering Quote:
 
October 3rd, 2012, 11:33 AM  #15  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Dichotomy paradox reverse engineering Quote:
 
October 5th, 2012, 01:08 PM  #16  
Senior Member Joined: Aug 2010 From: Germany Posts: 132 Thanks: 1  Re: Dichotomy paradox reverse engineering Quote:
 
October 5th, 2012, 01:30 PM  #17  
Senior Member Joined: Aug 2012 Posts: 2,343 Thanks: 732  Re: Dichotomy paradox reverse engineering Quote:
Yes it makes perfect sense. The infinitely many intervals keep getting smaller and smaller. The sum of their lengths converges. You're asking if it makes sense that 1/2 + 1/4 + 1/8 + /16 + ... = 1 but 1 + 1 + 1 + 1 + ... is infinite. Well yes, it makes perfect sense. Doesn't it? In fact it's a necessary condition of a convergent infinite series that its terms go to zero. All you've done is illustrate that fact, which is proved in freshman calculus.  

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