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February 4th, 2016, 07:48 PM   #21
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Quote:
Originally Posted by zylo View Post
Induction applies for all n from 1 to infinity.
That's simply incorrect, as the phrase "all natural numbers" doesn't mean "from 1 to infinity".

The article you linked to doesn't use the word "infinity". It refers to "all values of $n$". There are infinitely many possible values of $n$, but none of them is "infinity".
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February 4th, 2016, 07:48 PM   #22
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Originally Posted by Azzajazz View Post
A number with $n$ decimal places, for $n\in\mathbb{N}$. Also known as a rational number.

What's that got to do with your proof that the irrationals are countable?
Some respondents seem not to have heard of or understand the fundamental principle of mathematical induction.
https://en.wikipedia.org/wiki/Mathematical_induction

For those who don't like or understand my proofs that the reals are countable, there is a "proof" that the reals are uncountable:
https://en.wikipedia.org/wiki/Cantor...tability_proof
Take your choice.
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February 4th, 2016, 07:53 PM   #23
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There is no such choice, as your assertions rely on a phrase that doesn't appear in the article you linked to. The linked article doesn't use the word "infinity".
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February 4th, 2016, 07:53 PM   #24
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Quote:
Originally Posted by zylo View Post
Nobody seems to have heard of or understand the fundamental principle of mathematical induction.
No, it's just you that doesn't have a clue.

Quote:
Originally Posted by zylo View Post
For those who don't like or understand my proofs that the reals are countable, there is a "proof" that the reals are uncountable.
Proofs aren't something about which you can "take your choice". They are either correct or they are not. Moreover, if there were to exist valid proofs of both a statement and its negation, the entire logical structure of mathematics would collapse.

Everyone else here both knows and understands Cantor's proof. It's, again, just you that doesn't.

Last edited by skipjack; February 4th, 2016 at 07:58 PM.
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February 4th, 2016, 09:35 PM   #25
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Originally Posted by zylo View Post
Some respondents seem not to have heard of or understand the fundamental principle of mathematical induction.
https://en.wikipedia.org/wiki/Mathematical_induction
Tell me whether this is true or false:
$$\aleph_0\in\mathbb{N},$$
Or maybe just
$$\infty\in\mathbb{N}.$$
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February 5th, 2016, 06:55 AM   #26
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Originally Posted by Azzajazz View Post
Tell me whether this is true or false:
$$\aleph_0\in\mathbb{N},$$
Or maybe just
$$\infty\in\mathbb{N}.$$
Do you believe in the principle of mathematical induction?

Last edited by skipjack; February 5th, 2016 at 07:25 AM.
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February 5th, 2016, 07:19 AM   #27
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The principal of mathematical induction is not like Father Christmas. It's a method that exists by logical deduction, not a story that you can choose to believe or not.

If it were a story, you could make up whatever conclusion you wanted, but since it's not, you can't - which is something you don't seem to be able to come to terms with.

Last edited by skipjack; March 5th, 2016 at 11:31 AM.
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February 5th, 2016, 07:26 AM   #28
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Quote:
Originally Posted by zylo View Post
Do you believe in the principle of mathematical induction?
You've already provided a link for mathematical induction, and the linked article doesn't use the word "infinity", so your proofs using induction and claiming to show that the reals are countable only show that terminating decimals are countable, which excludes all irrational reals.
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February 5th, 2016, 08:15 AM   #29
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Quote:
Originally Posted by skipjack View Post
You've already provided a link for mathematical induction, and the linked article doesn't use the word "infinity", so your proofs using induction and claiming to show that the reals are countable only show that terminating decimals are countable, which excludes all irrational reals.
The decimal places for any number can be counted:

.1234.........

That's the 1:1 association. Whether or not you want to use the terminology "infinite" or "for all n" is academic.
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February 5th, 2016, 08:23 AM   #30
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Originally Posted by zylo View Post
Whether or not you want to use the terminology "infinite" or "for all n" is academic.
That's just wrong. Because all $n$ are finite and "infinite" is not. "It's turtles all the way down."

Perhaps you should investigate the difference between a set being countable ("listable" is a better term) and being able to say how many elements it has.

Last edited by v8archie; February 5th, 2016 at 08:25 AM.
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