January 23rd, 2016, 12:37 PM  #1 
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 125  The real numbers are countable
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. 
January 23rd, 2016, 12:48 PM  #2 
Global Moderator Joined: Dec 2006 Posts: 20,484 Thanks: 2041 
Most reals can't be expressed using n decimal places for any finite n. Hence it doesn't follow that the reals are countable.

January 23rd, 2016, 01:04 PM  #3 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,639 Thanks: 2624 Math Focus: Mainly analysis and algebra 
Why do you keep on posting this nonsense? It's not going to become true however many times you say it. Your counting scheme omits all infinite decimals.

January 25th, 2016, 06:18 AM  #4  
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 125  Quote:
Please note "for all n." Last edited by zylo; January 25th, 2016 at 07:13 AM. Reason: removed reference to "nonsense"  
January 25th, 2016, 08:40 AM  #5 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,639 Thanks: 2624 Math Focus: Mainly analysis and algebra 
No, it's your understanding that doesn't progress past the first step. ALL OF THE NATURAL NUMBERS ARE FINITE Your statement "for all n" includes only finite numbers. The natural numbers are defined inductively by $1 \in \mathbb N$ and $n \in \mathbb N \implies (n+1) \in \mathbb N$. Since $1$ is finite, if there were any infinite natural numbers there must be one of them that is equal to $(N+1)$ where $N$ is finite. Show me such an $N$. Last edited by v8archie; January 25th, 2016 at 08:49 AM. 
January 25th, 2016, 09:02 AM  #6 
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 125 
"Mathematical induction is a mathematical proof technique, most commonly used to establish a given statement for all natural numbers" from https://en.wikipedia.org/wiki/Mathematical_induction 
January 25th, 2016, 09:05 AM  #7 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,639 Thanks: 2624 Math Focus: Mainly analysis and algebra 
So what?

January 25th, 2016, 09:10 AM  #8 
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 125  
January 25th, 2016, 09:14 AM  #9 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,639 Thanks: 2624 Math Focus: Mainly analysis and algebra 
It still hasn't become true. ALL OF THE NATURAL NUMBERS ARE FINITE (That still hasn't become false). 
January 25th, 2016, 09:21 AM  #10 
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 125  

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