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 January 23rd, 2016, 01:37 PM #1 Banned Camp   Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 124 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, 01:48 PM #2 Global Moderator   Joined: Dec 2006 Posts: 20,278 Thanks: 1963 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, 02:04 PM #3 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,599 Thanks: 2587 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. Thanks from topsquark
January 25th, 2016, 07:18 AM   #4
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Quote:
 Originally Posted by zylo 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.
Looks like the previous 2 posts never went beyond the title of the OP

Last edited by zylo; January 25th, 2016 at 08:13 AM. Reason: removed reference to "nonsense"

 January 25th, 2016, 09:40 AM #5 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,599 Thanks: 2587 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 09:49 AM.
 January 25th, 2016, 10:02 AM #6 Banned Camp   Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 124 "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, 10:05 AM #7 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,599 Thanks: 2587 Math Focus: Mainly analysis and algebra So what?
January 25th, 2016, 10:10 AM   #8
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Quote:
 Originally Posted by v8archie So what?
OP:

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 25th, 2016, 10:14 AM #9 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,599 Thanks: 2587 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, 10:21 AM   #10
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Quote:
 Originally Posted by v8archie It still hasn't become true. ALL OF THE NATURAL NUMBERS ARE FINITE (That still hasn't become false).
I said "for all n."

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