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January 23rd, 2016, 12:37 PM   #1
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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.
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January 23rd, 2016, 12:48 PM   #2
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Most reals can't be expressed using n decimal places for any finite n. Hence it doesn't follow that the reals are countable.
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January 23rd, 2016, 01:04 PM   #3
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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.
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January 25th, 2016, 06:18 AM   #4
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Quote:
Originally Posted by zylo View Post
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

Please note "for all n."

Last edited by zylo; January 25th, 2016 at 07:13 AM. Reason: removed reference to "nonsense"
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January 25th, 2016, 08:40 AM   #5
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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.
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January 25th, 2016, 09:02 AM   #6
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"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
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January 25th, 2016, 09:05 AM   #7
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So what?
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January 25th, 2016, 09:10 AM   #8
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Quote:
Originally Posted by v8archie View Post
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.
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January 25th, 2016, 09:14 AM   #9
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It still hasn't become true.

ALL OF THE NATURAL NUMBERS ARE FINITE

(That still hasn't become false).
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January 25th, 2016, 09:21 AM   #10
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Quote:
Originally Posted by v8archie View Post
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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