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January 13th, 2016, 12:53 PM  #11  
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 125  Quote:
sqrt2=1.414.......... pi= 3.14159......... And I have shown that the number of decimals is countable. *A nest of rational numbers defines a unique real number. Between any two rational numbers there is a decimal number. The cut number and the decimal representation are the same.  
January 13th, 2016, 01:16 PM  #12 
Global Moderator Joined: Dec 2006 Posts: 20,389 Thanks: 2015 
Your purported proof relates to the countability of the rationals (which is true), but that doesn't imply that the cuts you mentioned are countable. The rationals are countable, but the subsets of the rationals (or of any countably infinite set) are not. That means that subsets of the rationals can be used to define uncountably many reals. The decimal representations you refer to (for √2 and $\pi$) have an infinite number of decimal places and are not "recurring decimals". You haven't shown that such decimal representations are countable. Your assertion that they are doesn't follow from your previous statements. Note that your amplification needed to refer to "nests" of rationals, but "nests" are subsets, which means they needn't be countable. Last edited by skipjack; February 29th, 2016 at 03:00 AM. 
January 13th, 2016, 01:27 PM  #13  
Banned Camp Joined: Jun 2014 From: Earth Posts: 945 Thanks: 191  Quote:
. Last edited by Math Message Board tutor; January 13th, 2016 at 01:32 PM.  
January 13th, 2016, 01:58 PM  #14 
Global Moderator Joined: Dec 2006 Posts: 20,389 Thanks: 2015 
I avoided that wording because zylo didn't claim to be using all the subsets. The purported proof fails because zylo didn't prove that only countably many subsets suffice for defining the reals.

January 13th, 2016, 02:24 PM  #15  
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 125  Quote:
The decimals are countable. What is wrong with my proof that the decimals are countable?  
January 13th, 2016, 03:30 PM  #16 
Global Moderator Joined: Dec 2006 Posts: 20,389 Thanks: 2015 
If you used only a list of decimals that correspond to rationals, you would be okay, as the rationals are countable. However, you are using "cuts" and haven't provided a way of listing all of them. To explain "cuts", you need to use subsets of the rationals, and there are uncountably many subsets of the rationals.

January 13th, 2016, 04:24 PM  #17 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,623 Thanks: 2611 Math Focus: Mainly analysis and algebra  Your proof that the decimals are countable deals only with finite decimals, not infinite ones. All irrationals are infinite decimals, therefore your proof ignores the irrationals.

January 13th, 2016, 04:49 PM  #18  
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 125  Quote:
2) The number of decimals is countable: I can count the number of decimal places and the number of numbers each count represents: 1 > 10, countable 2 > 100, countable ......... n > 10^n, countable ......... ......... n=1> infinity The number of decimals is countable because a countable number of countable numbers is countable. Perhaps the problem is you don't know what an infinite decimal is. Do you know what pi=3.1415..... means? Hint: It's an infinite decimal.  
January 13th, 2016, 05:11 PM  #19 
Global Moderator Joined: Dec 2006 Posts: 20,389 Thanks: 2015 
You listed some cases that are countable, but you haven't given a way of listing the infinite decimals. Which infinite decimal corresponding to an irrational number would be the first such decimal to appear in the list?

January 13th, 2016, 05:33 PM  #20  
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 125  Quote:
Every real number can be expressed as an infinite decimal. The number of infinitesimal decimals is countable: 1) Number of 1place decimals: 10 2) Number of 2place decimals: 10^2 .... n) Number of nplace decimals: 10^n ....... ....... n= 1 to infinity (all n) A countable number of countable numbers is countable. 10^n is countable because the set of natural numbers is closed under multiplication.  

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