July 5th, 2016, 01:55 PM  #11 
Global Moderator Joined: May 2007 Posts: 6,454 Thanks: 567 
I surrender. I was trying to present a simplified proof based on measure theory without invoking measure theory. The basic proof is simply the following: The unit interval has measure 1. Any countable set has measure 0. Therefore, the unit interval is not countable. 
July 5th, 2016, 02:58 PM  #12 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,236 Thanks: 2412 Math Focus: Mainly analysis and algebra 
Measure theory is not something I'm familiar with. That bald statement is very clear and obviously serves as a proof, but it hides all understanding of the issues. For someone not familiar with measure theory it raises more questions than it answers. Not that that is necessarily a bad thing if one has the time and the inclination to follow up on those questions. It's certainly something that has some interest for me, but it's not my top priority right now. 
July 5th, 2016, 04:01 PM  #13 
Senior Member Joined: Nov 2010 From: Berkeley, CA Posts: 174 Thanks: 35 Math Focus: Elementary Number Theory, Algebraic NT, Analytic NT 
@mathman, Your proof is valid. For example, see Proposition 2.2 (d), Proposition 2.3 and Corollary 2.4 in this paper. 
July 6th, 2016, 03:30 AM  #14 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,297 Thanks: 94 
You can't compare distances with number of points because points have no measure (width), so the OP is meaningless. You can't use number of points as a distance. There is a distance between points, but a countably infinite number of points between them. Therefore the OP and post #11 are incorrect. 
July 6th, 2016, 04:55 PM  #15  
Global Moderator Joined: May 2007 Posts: 6,454 Thanks: 567  Quote:
 
July 16th, 2016, 01:53 PM  #16  
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,297 Thanks: 94  Quote:
If the marbles have no width, the number of marbles is 1M/0, which is undefined and meaningless. To get a pure number (count) you have to have a dimension in numerator and denominator to cancel out. You don't need abstract mathematics to think, on the contrary.  
July 16th, 2016, 06:14 PM  #17  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,236 Thanks: 2412 Math Focus: Mainly analysis and algebra  Quote:
 
July 17th, 2016, 01:06 PM  #18  
Global Moderator Joined: May 2007 Posts: 6,454 Thanks: 567  Quote:
 
July 18th, 2016, 05:08 AM  #19 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,297 Thanks: 94  
July 18th, 2016, 08:28 AM  #20 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,236 Thanks: 2412 Math Focus: Mainly analysis and algebra  

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argument, cantor, diagonal, reals, uncountable 
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