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July 1st, 2016, 11:49 AM   #1
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Power Set of the Natural Numbers is Countable

If Sn = {1,2,....n}
P(Sn) has (2^n)-1 members, no matter what n is.
Therefore the power set of the natural numbers is countable.


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P(A) is the set of all subsets of A. If A has n members, the number of subsets of A is nC1+nC2+...nCn=(2^n)-1.
(1+1)^n=nC0+nC1+nC2+...nCn.

Also, n elements are countable if n is a natural number.

Last edited by skipjack; July 2nd, 2016 at 02:52 PM.
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July 1st, 2016, 12:17 PM   #2
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July 1st, 2016, 02:51 PM   #3
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It is spam. Zylo is apparently incapable of learning anything about infinite sets.
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July 1st, 2016, 07:24 PM   #4
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Equivalently: 8 is greater than 7. Therefore the power set of the natural numbers is countable.

Or: today is Friday. Therefore the power set of the natural numbers is countable.
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July 1st, 2016, 09:00 PM   #5
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https://en.wikipedia.org/wiki/Power_set

P(N) = P({1,2,…}) is countable if P({1,2…,n}) is countable for any natural number n.
P({1,2..,n}) is countable because the 2^n subsets (including empty set) of {1,2,..,n} are unique and so correspond 1:1 with the natural numbers (up to 2^n).

Are there any mathematical objections?

Last edited by skipjack; July 2nd, 2016 at 03:05 PM.
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July 2nd, 2016, 02:30 AM   #6
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As n is a natural number, it is finite. Hence it doesn't follow that the power set of a countably infinite set, such as the set of all the natural numbers, is countable.
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July 2nd, 2016, 05:16 AM   #7
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Quote:
Originally Posted by skipjack View Post
As n is a natural number, it is finite. Hence it doesn't follow that the power set of a countably infinite set, such as the set of all the natural numbers, is countable.
There is no such thing as n=infinity. You show something is true for ALL n. All n and any n are the same thing.
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July 2nd, 2016, 05:50 AM   #8
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This demonstrates that your reasoning applies only to finite sets. It says nothing about infinite sets.

Learn some mathematics.
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July 2nd, 2016, 05:58 AM   #9
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x^n=xxxx...x, n times, for any/all n.

Any person with a licence can drive is the same as all persons with a licence can drive. The semantics is just a way to distract from truth of OP.

None of the replies so far invalidates the OP, on the contrary.
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July 2nd, 2016, 07:22 AM   #10
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Quote:
Originally Posted by zylo View Post
Any person with a licence can drive is the same as all persons with a licence can drive.
And your claim is that a person of infinite size can drive.

Whereas the truth is that there are no persons of infinite size.

In this case, there are sets of infinite size, but your deductions do not apply to sets of infinite size because you have specified the size of the set to be $n$. And, as you say
Quote:
Originally Posted by zylo View Post
There is no such thing as n=infinity.
Learn some mathematics!

You do not have competence to say whether any disproves your ridiculous claims because you don't understand basic logic or mathematics.

Last edited by v8archie; July 2nd, 2016 at 07:25 AM.
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