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 April 27th, 2017, 09:59 AM #11 Senior Member   Joined: Dec 2015 From: Earth Posts: 337 Thanks: 44 $\displaystyle A\rightarrow$ not absolutely $\displaystyle A$
April 27th, 2017, 12:13 PM   #12
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 Originally Posted by zylo Ref previous post. (1) is CDA, the subject of this thread. You can't use CDA to prove CDA. (2) Each a_{n} is a rational number. Any rational combination of rational numbers is a rational number. The rational numbers are countable.
(1) CDA is a proof. It's not something that requires proof. The uncountability of (non-terminating, non-repeating) sequences of decimal digits doesn't require CDA.
(2) So what, false, false. The irrational numbers are by definition the limit of sequences of rationals.

 April 27th, 2017, 01:30 PM #13 Senior Member   Joined: Mar 2015 From: New Jersey Posts: 1,648 Thanks: 119 The LIMIT of an infinite binary digit is a real number. Cantor does not consider the set of limits of infinite binary digits, he considers the set of infinite binary digits, which are rational numbers, which are countable.
April 27th, 2017, 01:35 PM   #14
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Quote:
 Originally Posted by zylo The LIMIT of an infinite binary digit is a real number. Cantor does not consider the set of limits of infinite binary digits, he considers the set of infinite binary digits, which are rational numbers, which are countable.
That's absurd. Each item on the list is a real number, an infinite decimal or binary sequence. What on earth are you talking about? You sound bored. You haven't even got an argument.

April 27th, 2017, 01:49 PM   #15
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 Originally Posted by zylo The LIMIT of an infinite binary digit is a real number
Nonsense. A binary digit isn't even a sequence. A sequence isn't a number. The CDA doesn't make any reference at all to numbers. It's a proof about sequences, not limits of sequences.

April 28th, 2017, 06:33 AM   #16
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 Originally Posted by Maschke I don't know what that means. 3 + 1/10 + 4/100 + 1/1000 + ... is a combination of rational numbers that sums to $\pi$. Please make your posts self-contained if you want people to respond.
It's limit is pi, but it never equals pi. Just like Lim 1/n is zero but 1/n never equals 0. CDA is not about limits, it is about sequences.

Quote:
 Originally Posted by v8archie Nonsense. A binary digit isn't even a sequence. A sequence isn't a number. The CDA doesn't make any reference at all to numbers. It's a proof about sequences, not limits of sequences.
CDA allegedly proves the real numbers are countable (put a period before each one).

A binary sequence is a natural number. An infinite binary sequence is a natural number. The natural numbers are countable. CDA is wrong.

Last edited by zylo; April 28th, 2017 at 06:39 AM.

April 28th, 2017, 06:43 AM   #17
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Quote:
 Originally Posted by zylo An infinite binary digit is a natural number.
No, it's not a natural number.

April 28th, 2017, 06:47 AM   #18
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Quote:
 Originally Posted by zylo An infinite binary sequence is a natural number.
What natural number is .10101010101010101010101010..., with or without the binary point?

April 28th, 2017, 07:20 AM   #19
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 Originally Posted by Maschke What natural number is .10101010101010101010101010..., with or without the binary point?
$\displaystyle p=a{_n}2{^n}+......+a_{1}2{^1}+a{_0}2{^0}$

101010......... makes no sense if there is a largest natural number. Do you happen to know what it is?

Presumably all binary digits of an infinite binary sequence are known, otherwise it would make no sense to speak of AN infinite binary sequence.

April 28th, 2017, 07:56 AM   #20
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 Originally Posted by zylo $\displaystyle p=a{_n}2{^n}+......+a_{1}2{^1}+a{_0}2{^0}$ 101010......... makes no sense if there is a largest natural number. Do you happen to know what it is? Presumably all binary digits of an infinite binary sequence are known, otherwise it would make no sense to speak of AN infinite binary sequence.
You've lapsed into utter incoherence.

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