My Math Forum Reals are uncountable - without Cantor diagonal argument

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 July 5th, 2016, 01:55 PM #11 Global Moderator   Joined: May 2007 Posts: 6,510 Thanks: 584 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: 7,305 Thanks: 2443 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. Thanks from manus
 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: 1,363 Thanks: 100 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. Thanks from manus
July 6th, 2016, 04:55 PM   #15
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 Originally Posted by v8archie 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.
That was the point of my original statement. I was trying to present a simplified version, so one would not have to learn measure theory to understand the proof.

July 16th, 2016, 01:53 PM   #16
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 Originally Posted by mathman That was the point of my original statement. I was trying to present a simplified version, so one would not have to learn measure theory to understand the proof.
You don't need measure theory. If you have a 1 M long line and marbles that are .2 meters in dia, you can fit 1M/(.2M/marble) = 5 marbles in the line.

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
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 Originally Posted by zylo You don't need measure theory. If you have a 1 M long line and marbles that are .2 meters in dia, you can fit 1M/(.2M/marble) = 5 marbles in the line. 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.
This is just a statement that you don't understand the subject.

July 17th, 2016, 01:06 PM   #18
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 Originally Posted by zylo You don't need measure theory. If you have a 1 M long line and marbles that are .2 meters in dia, you can fit 1M/(.2M/marble) = 5 marbles in the line. 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.
If the number of marbles is countable, you can sum the zeros to get zero. That is why the number cannot be countable.

July 18th, 2016, 05:08 AM   #19
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 Originally Posted by mathman If the number of marbles is countable, you can sum the zeros to get zero. That is why the number cannot be countable.
Makes no sense. What is the size of a marble? If I have n marbles that fit into a unit length, the size of the marble is 1/n.

July 18th, 2016, 08:28 AM   #20
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 Originally Posted by zylo Makes no sense.
The fact that you are not able to understand something is a sign of a flaw in your own intelligence, not in that of other people.

 Tags argument, cantor, diagonal, reals, uncountable

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