February 5th, 2016, 08:30 AM  #31 
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 126  My assertion is the decimal places are countable, using the above standard definition for countable, which in turn leads to the reals are countable via my induction argument.
Last edited by zylo; February 5th, 2016 at 08:36 AM. Reason: add "...reals are countable..." 
February 5th, 2016, 08:33 AM  #32  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,683 Thanks: 2661 Math Focus: Mainly analysis and algebra  Quote:
Quote:
Last edited by v8archie; February 5th, 2016 at 09:15 AM.  
February 5th, 2016, 08:45 AM  #33 
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 126  
February 5th, 2016, 08:54 AM  #34  
Global Moderator Joined: Dec 2006 Posts: 20,968 Thanks: 2217  Quote:
Quote:
By using induction in the way you suggested, you can reach the conclusion that terminating decimals are countable. You were incorrectly concluding that the set of all reals is countable. It is correct that each real has a decimal representation the digits in which are countable, but that is not the same as the set of all reals being countable. For example, the decimal representation of $\pi$ has infinitely many digits, but the set {$\pi$} is finite because it has only one element. A property possessed by each real is not necessarily a property possessed by the set of all reals.  
February 5th, 2016, 09:16 AM  #35 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,683 Thanks: 2661 Math Focus: Mainly analysis and algebra  
February 5th, 2016, 09:30 AM  #36  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,683 Thanks: 2661 Math Focus: Mainly analysis and algebra  Quote:
Again, the key to understanding the flaw in your "proof" is that:
Last edited by skipjack; February 5th, 2016 at 10:35 AM.  
February 5th, 2016, 01:27 PM  #37 
Math Team Joined: Nov 2014 From: Australia Posts: 689 Thanks: 244  
February 8th, 2016, 08:31 AM  #38 
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 126  
February 8th, 2016, 08:50 AM  #39 
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 126 
"This post is meant to replace the OP of Counting the Irrational Numbers which is more intuitive than precise. It also provides a precise simple clear proof to comment on. I also include the proof that the reals are countable for easy reference and comparison.  I) The Reals can be placed in countable order Proof by induction: With 2 decimal places I can represent all 2place decimals between 0 and 1 in countable order: 0 .01 .02 . .99 1 If I have n+1 decimal places, I can arrange n of them in countable order. Then I can add an additional decimal place to divide each interval into ten countable intervals: 0 .001 .002 . .009 .010 .011 . .019 .020 .021 . . .999 By induction, all the real decimals can be placed in countable order. Including pi, skipjack's question. Any real number can be expressed as an integer (countable) and a decimal (countable).  II) A companion to this proof, "the reals are countable," is Any real number can be expressed as a decimal. The number of numbers that can be expressed by n decimal places is 10^n. 10^n is countable for all n. Proof: 10^2 is countable. 10^(n+1) = 10x10^n is countable. The reals are countable." The above is the OP of The reals can be placed in countable order It is the clarification of and supersedes the present OP, which was my initial, undeveloped response to a question of skipjack. It was closed by greg1313 and referred to this thread. I would appreciate it if this post was considered a replacement for OP and referred to as the basis for discussion. Thank you Last edited by skipjack; March 5th, 2016 at 11:40 AM. 
February 8th, 2016, 09:23 AM  #40 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 
You are still making the same mistake that everyone has been pointing out before. A "proof by induction" proves that some statement, P(n), depending on positive integer, n, is true for every positive integer. What you have proved is that the set of all real numbers, between 0 and 1, expressible in base 10 with "n" decimal places, where n can be any integer, is countable. That's obviously true but that is NOT "all real numbers". It is rather, a small subset of the set or rational numbers those fractions that, reduced to least terms, have only powers of 2 and 5 in the denominator. It does not, for example, include even the rational number 1/3.


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