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-   -   Summary of Cantor's Diagonal Argument is Wrong (http://mymathforum.com/topology/330011-summary-cantors-diagonal-argument-wrong.html)

zylo April 4th, 2016 07:53 AM

Summary of Cantor's Diagonal Argument is Wrong
 
1) Cantor's Diagonal Argument is wrong because countably infinite binary sequences are natural numbers.

2) Cantor's Diagonal Argument fails because there is no natural number greater than all natural numbers.

3) Cantor's Diagonal Argument is not applicable for infinite binary sequences because countably infinite is a subset of infinite.

4) Without the point, every real number in [0,1) can be uniquely associated with a countably infinite binary sequence which is a natural number.
.00110....... -> 00110.......
0x2^{-1} + 0x2^{-2} + 1x2^{-3}+.... -> 0x2^{0} + 0x2^{1} + 1x2^{2} + ....
The reals are countable.

This is summarized here because it's buried in:
http://mymathforum.com/topology/3288...ot-number.html

skipjack April 4th, 2016 08:25 AM

Quote:

Originally Posted by zylo (Post 529152)
Let T be the set of all infinite binary sequences. T is obviously uncountable

Quote:

Originally Posted by zylo (Post 530721)
The reals are countable.

You seem to be contradicting yourself. Please explain which statement is incorrect.

You haven't proved your opening assertion (numbered 1), or given any examples to support it.

You haven't explained why your second and third points have any bearing on Cantor's diagonal argument. Also, you haven't explained which particular part of Cantor's argument is affected.

Do your points all occur in the thread you linked? I can't find them.

zylo April 4th, 2016 09:01 AM

Quote:

Originally Posted by skipjack (Post 530727)
Do your points all occur in the thread you linked? I can't find them.

Post #57:
s has more digits than any sequence in the list.
s is not in the list.
s is not in T.
Cantor's proof is wrong.

Post #86 and Post #90.

Also, Posts 35,42,50,53,61,71,81

v8archie April 4th, 2016 09:27 AM

So this is yet another thread for you to post the same old rubbish with exactly the same stupid mistakes. This thread is pointless and utterly without value.

zylo April 4th, 2016 12:12 PM

Let L be the list of ALL countably infinite binary sequences.

S is not in L
S is larger than any element of L because, whatever Sn is, S has "1" digits past the last "1" digit of Sn. (There is no natural number greater than all natural numbers).

If you don't accept the sentence in parentheses, you will never get it.

v8archie April 4th, 2016 12:19 PM

Quote:

Originally Posted by zylo (Post 530744)
(There is no natural number greater than all natural numbers).

If you don't accept the sentence in parentheses, you will never get it.

I realise that this is quite a groundbreaking discovery for you, but for most of us it is entirely obvious. Unfortunately, you have no idea how to use this fact correctly.

skipjack April 4th, 2016 12:31 PM

Quote:

Originally Posted by zylo (Post 530721)
. . . countably infinite binary sequences are natural numbers.

You originally gave this as a generalization of the corresponding statement for finite binary sequences, but you never justified making the generalization. Not every generalization from finite to infinite is correct.

Quote:

Originally Posted by zylo (Post 530744)
Let L be the list of ALL countably infinite binary sequences.

You haven't proved that such a list exists, so that is not a valid definition.

v8archie April 4th, 2016 12:52 PM

It's worth pointing out that even if such a list existed (which it doesn't) it would not be a unique list. This is a very trivial fact, but isn't reflected by zylo's statements.

zylo April 4th, 2016 02:12 PM

Quote:

Originally Posted by v8archie (Post 530735)
So this is yet another thread for you to post the same old rubbish with exactly the same stupid mistakes. This thread is pointless and utterly without value.

From
https://en.wikipedia.org/wiki/Cantor...gonal_argument

"The diagonal argument was not Cantor's first proof of the uncountability of the real numbers; it was actually published much later than his first proof, which appeared in 1874.
Cantor's diagonal argument, was published in 1891 by Georg Cantor."
_____________________________________

In over 100yrs, the mathematicians still haven't gotten it right.
So it took me a few posts and a couple of weeks. Cut me some slack.

============================

"You haven't proved that such a list exists, so that is not a valid definition:" skipjack
As you well know, that's Cantor's assumption from which he tries (unsuccessfully) to draw a contradiction. Perhaps you should review Cantor's proof. There you will find:
https://en.wikipedia.org/wiki/Cantor...gonal_argument
"He assumes for contradiction that T was countable. Then all its elements could be written as an enumeration s1, s2, … , sn, … ."

topsquark April 4th, 2016 02:41 PM

Quote:

Originally Posted by zylo (Post 530758)
In over 100yrs, the mathematicians still haven't gotten it right.

Wow. Just. Wow.

-Dan


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