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 February 4th, 2019, 05:55 PM #11 Senior Member   Joined: Sep 2016 From: USA Posts: 685 Thanks: 462 Math Focus: Dynamical systems, analytic function theory, numerics Delete this garbage already. Kick this idiot off his pedestal.
 February 4th, 2019, 09:24 PM #12 Senior Member   Joined: May 2016 From: USA Posts: 1,310 Thanks: 552 I have said it before. Zylo's disproof of Cantor starts by ASSUMING that Cantor is wrong. If the real numbers and natural numbers can be put into 1-to-1 correspondence then, Cantor is wrong. But you cannot prove that Cantor is wrong by simply asserting that such correspondence is obvious, particularly not when Cantor has demonstrated by overwhelming argument that such a correspondence is impossible. Now I do not believe that anyone can persuade zylo that he is simply blowing smoke out of his ass, but it may be possible to keep others from being confused. If you want to think about real numbers, then you must accept that numbers can be represented by infinite strings of digits. The positive integers can also be represented by infinite strings of digits, provided that the identifying string is preceded by an infinite number of zeros. (The necessity for that number of preceding zeros being zero is lost on Zylo.) Thus, there is no basis for saying that the number of integers (as usually defined) is the same as the number of all imaginable representations of real numbers (as usually defined). Last edited by skipjack; February 5th, 2019 at 01:12 AM.
 February 5th, 2019, 02:01 AM #13 Global Moderator   Joined: Dec 2006 Posts: 21,132 Thanks: 2340 I think zylo usually refers, as in this thread, to natural numbers rather than integers, but he always ignores their usual definition and implicitly substitutes the idea that they include, even when leading zeros aren't present, non-terminating "numbers", even though such numbers aren't usable for normal counting. If somebody questions the countability of these, zylo tends to ignore the question until something else is posted, or reply to it without answering it, perhaps merely restating his earlier post with minor rewording, but no significant improvement. Without exception, zylo lists at most a few of his "numbers", as though it's obvious that all the rest are somewhere in the list. In his recent attempts, though, he ignores the possibility that the list doesn't exist at all. He has tried two ways of starting his presentation: (1) starting with 1, 2, 3, 4, etc., which lists only finite decimal representations without leading zeros, or (2) starting with a set of reals with decimal points removed, but "forgetting" to provide a way to list them. Thanks from JeffM1
February 5th, 2019, 06:25 AM   #14
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 Originally Posted by skipjack I think zylo usually refers, as in this thread, to natural numbers rather than integers, but he always ignores their usual definition and implicitly substitutes the idea that they include, even when leading zeros aren't present, non-terminating "numbers", even though such numbers aren't usable for normal counting. If somebody questions the countability of these, zylo tends to ignore the question until something else is posted, or reply to it without answering it, perhaps merely restating his earlier post with minor rewording, but no significant improvement. Without exception, zylo lists at most a few of his "numbers", as though it's obvious that all the rest are somewhere in the list. In his recent attempts, though, he ignores the possibility that the list doesn't exist at all. He has tried two ways of starting his presentation: (1) starting with 1, 2, 3, 4, etc., which lists only finite decimal representations without leading zeros, or (2) starting with a set of reals with decimal points removed, but "forgetting" to provide a way to list them.
I think you have it, except that he is not so much ignoring the conceivable impossibility of constructing a "list" of the real numbers in the interval (0, 1) as assuming its constructibility.

I get a bit mentally queasy about denying assumptions, but perhaps there is a legitimate reason to refuse countenance to the word "list." It seems to me that Zylo's argument could be re-stated without any mention of lists and without hidden assumptions.

Assume that infinity is a legitimate concept.

Assume that both the infinite set of natural numbers (as usually defined) and and the infinite set of real numbers are legitimate concepts.

Assume the legitimacy of representing natural numbers as infinite strings of decimal digits.

Assume that the infinite set of such representations exists. Call it set A.

Assume the legitimacy of representing real numbers in the interval (0, 1) as 0 followed by a decimal point followed by an infinite string of decimal digits.

Assume that the infinite set of such representations exists. Call it set B.

Perhaps I am missing something, but I do not yet see that such assumptions are illegitimate in and of themselves.

Now construct set C by removing the initial 0 and decimal point from each element of set B.

Sets B and C are in 1-to-1 correspondence by construction.

To the extent that talking about infinite sets is legitimate, I do not yet see what is illegitimate about that construction.

So, until someone can show me the flaw in this admittedly revised version of Zylo's argument, I am on board up to this point. Of course, I am open to correction.

Now Zylo says A and C are in 1-to-1 correspondence.

I say nonsense. Every element in A is indeed matched by a corresponding element of C, but not every element of C is matched by a corresponding element of A. There is no 1-to-1 correspondence.

Last edited by skipjack; February 5th, 2019 at 10:10 AM.

 February 5th, 2019, 09:29 AM #15 Banned Camp   Joined: Jun 2014 From: USA Posts: 650 Thanks: 55 Zylo is going to assume the set A contains all infinite strings of decimal digits. It can’t if it’s 1-to-1 with the natural numbers though. That’s an important distinction in this case as it’s how we conclude that A cannot be surjected onto C. I would ask for a mapping of the naturals onto A that is 1-to-1. Last edited by skipjack; February 5th, 2019 at 10:09 AM.
 February 5th, 2019, 10:28 AM #16 Global Moderator   Joined: Dec 2006 Posts: 21,132 Thanks: 2340 I wouldn't mind zylo implicitly assuming that any subset of the natural numbers (as usually defined) is listable, but zylo seems to be deliberately trying to cause confusion by doing this when using a non-standard meaning of "natural number" that allows 545454545... , for example, to be included, which is tantamount to assuming what he wishes to prove, or by stating that 1, 2, 3, 4, . . . includes all natural numbers, whilst ignoring that this doesn't hold if his meaning of "natural numbers" applies.
February 5th, 2019, 10:51 AM   #17
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 Originally Posted by skipjack I wouldn't mind zylo implicitly assuming that any subset of the natural numbers (as usually defined) is listable…
Any subset of $\mathbb{N}$ will have a cardinality $\leq$ $|\mathbb{N}|$, so we can assume all subsets are countable (jic, no zylo, this doesn't mean the set of all subsets of $\mathbb{N}$ is countable).

Despite that, there are plenty of subsets $p$ of $\mathbb{N}$ for which there is no formula for computing which natural numbers are in $p$ and which are not. Where there is no formula for determining whether a given natural number is in a given $p$, it seems logical to ask how we might "list" the elements of $p$. What do we mean by list zylo?

February 5th, 2019, 01:01 PM   #18
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 Originally Posted by AplanisTophet Any subset of $\mathbb{N}$ will have a cardinality $\leq$ $|\mathbb{N}|$, so we can assume all subsets are countable (jic, no zylo, this doesn't mean the set of all subsets of $\mathbb{N}$ is countable). Despite that, there are plenty of subsets $p$ of $\mathbb{N}$ for which there is no formula for computing which natural numbers are in $p$ and which are not. Where there is no formula for determining whether a given natural number is in a given $p$, it seems logical to ask how we might "list" the elements of $p$. What do we mean by list zylo?
You'd list them by putting them into bijection with the natural numbers.

You have some other definition?

February 5th, 2019, 06:01 PM   #19
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 Originally Posted by Maschke You'd list them by putting them into bijection with the natural numbers. You have some other definition?
No, but zylo usually does.

February 5th, 2019, 06:52 PM   #20
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 Originally Posted by AplanisTophet Zylo is going to assume the set A contains all infinite strings of decimal digits. It can’t if it’s 1-to-1 with the natural numbers though. That’s an important distinction in this case as it’s how we conclude that A cannot be surjected onto C. I would ask for a mapping of the naturals onto A that is 1-to-1.
Well that of course is the flaw in his argument. Skipjack's (I think it was skipjack) choice of 1/9 was brilliant. An infinite string of 1's is certainly in set C, but it does not represent a natural number as they are usually defined. In other words, it can be demonstrated that there is no bijection between A and C.

Zylo would be better off to declare that he is finitist and that, from that viewpoint, Cantor's diagonal proof is meaningless rather than incorrect. But we all know that Zylo is unable to do anything that sensible.

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