## Cantor Diagonal Argument disproof

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### Re: Cantor Diagonal Argument disproof

JeffJo and Gofer are such nuisances that I find little purpose in responding to either of them.

Important question highlighted in green.

Very important question: Is there ANY circumstance where the Cantor Diagonal method DOES NOT return the result that a list is missing an element?

robert 46 wrote:String n is all 0 except for character n which is 1. Missing element is (0). Append the list onto the missing element. Repeat. New missing element is (1). Append the list onto the missing element. Repeat. New missing element is 0(1). Repeat as much as you like. Does the list become longer than wide by doing this? If not then the list remains a countable infinite, notwithstanding missing elements endlessly being added to it. Therefore it is wrong to consider a countable infinite list to ever be complete. This is contradictory.

Important question highlighted in green.

Very important question: Is there ANY circumstance where the Cantor Diagonal method DOES NOT return the result that a list is missing an element?

- robert 46
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### Re: Cantor Diagonal Argument disproof

No, it was a completely irrelevant question. And despite robert's implication, it has already been addressed. Many times.robert 46 wrote:robert 46 wrote:Does the list become longer than wide by doing this?.

Important question highlighted in green.

Asking it, yet again, ignores the fact that the concepts "wider" and "longer" have no meaning when the "width" and "length" are both countably infinite. That, with the same infinite list of infinite strings, you can make a finite comparison where:

- The widths of the intermediate strings exceed the length of the intermediate list by a monotonically increasing amount.
- The length of the intermediate list exceeds the widths of the intermediate strings by a monotonically increasing amount.
- The length of the intermediate list equals the widths of the intermediate strings.

You can make a list S(), based on the ordered set of natural numbers N, where S(n) is the binary representation (i.e., a SFSC) of n padded with infinite zeros. The "intermediate length" - a meaningless concept, but robert won't consider that - of the list after n entries is n. So just take the "intermediate width" to also be n, even if the bits aren't all significant, and you get intermediate arrays that are always square. Property #3 applies, and robert's intermediate anti-diagonals (IADs, not iads) are all defined.

But to understand this, robert will have to consider the possibility that what he wants to be true, may not be. And it seems he cannot do that. If it were, he would have addressed this point by now, since it has been thrown in his face almost every time he brings his green-colored, fallacious argument up.

Very important rebuttal; presented often to, and always ignored by robert because it is a truth he doesn't want to admit is a truth:robert 46 wrote:Very important question: Is there ANY circumstance where the Cantor Diagonal method DOES NOT return the result that a list is missing an element?

"Missing from" means "is supposed to be in, but is not in". The anti-diagonal that CDM finds is "supposed to be," and in fact is, a member of the set of all infinite binary strings T. The set that CDM uses to find it, is any subset S of T that has the property that S can be ordered by the function S(). So it is incorrect to use the word "missing" - specifically, the implication of "supposed to be in" - unless you understand which set "supposed to be in" applies to. And robert intentionally ignores that caveat.

Once this distinction is understood, the contradiction robert implies, but won't explicitly identify since it doesn't support his claim of a disproof, is seen as the lie it is intended to be. CDM always "returns" a string that is "supposed to be" in T, in fact is in T, but is "missing from" the S we would have if we were to suppose S is all of T. The underlined condition is the whole point. The result is not "missing from " the S used to construct. It is "missing from" what S would be if S=T.

What robert implies, but won't say explicitly, is that it is "supposed to be" in S. And that's a lie, which he won't admit to no matter how many times it is pointed out.

+++++

[1] One of robert's main points is that "Cantor never considered point #1 and/or #2, and I am so smart and innovative because I see it where nobody else has!." Thjis fallacy has been raised many time, by many naive people like robert. Cantor knew that both were true. And even admitted that both were true. But because both were true, only #3 is significant.

- JeffJo
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### Re: Cantor Diagonal Argument disproof

JeffJo wrote:No...robert 46 wrote:robert 46 wrote:Does the list become longer than wide by doing this?.

Important question highlighted in green.

...the concepts "wider" and "longer" have no meaning when the "width" and "length" are both countably infinite...

Well, it has to remain square under this condition. Let's say the list was infinity x infinity; then it becomes (1+infinity) x infinity. To remain square it must become (1+infinity) x (1+infinity). However, nothing is greater than infinity, so it becomes (1+endless) x endless, and, indeed, this is the same as endless x endless and remains square. However, it is impossible for endless to exist as an entity. This is what Gauss was getting at: infinite sets do not have the same kind of existence as finite sets. And, similarly, infinite lists do not have the same kind of existence as finite lists.

- Hermann Weyl… classical logic was abstracted from the mathematics of finite sets and their subsets …. Forgetful of this limited origin, one afterwards mistook that logic for something above and prior to all mathematics, and finally applied it, without justification, to the mathematics of infinite sets. This is the Fall and original sin of [Cantor's] set theory ….

https://en.wikipedia.org/wiki/Controver ... 27s_theory

JeffJo wrote:You can make a list S(), based on the ordered set of natural numbers N, where S(n) is the binary representation (i.e., a SFSC) of n padded with infinite zeros. The "intermediate length" - a meaningless concept, but robert won't consider that - of the list after n entries is n. So just take the "intermediate width" to also be n, even if the bits aren't all significant, and you get intermediate arrays that are always square. Property #3 applies, and robert's intermediate anti-diagonals (IADs...) are all defined.

Not only are the IADs defined, they are constructible (notwithstanding implied trailing 0s).

Very important rebuttal...robert 46 wrote:Very important question: Is there ANY circumstance where the Cantor Diagonal method DOES NOT return the result that a list is missing an element?

"Missing from" means "is supposed to be in, but is not in". The anti-diagonal that CDM finds is "supposed to be," and in fact is, a member of the set of all infinite binary strings T. The set that CDM uses to find it, is any subset S of T that has the property that S can be ordered by the function S(). So it is incorrect to use the word "missing" - specifically, the implication of "supposed to be in" - unless you understand which set "supposed to be in" applies to.

Given all strings of 4 characters put in a list, CDM can only examine 4 elements to determine a missing anti-diagonal. However, this missing anti-diagonal is in the remainder of the list, 12 elements, which CDM cannot examine. This applies for all lists of n characters. The inference is necessarily that for lists of strings of infinite characters, CDM cannot examine the entire list. In the finite case the width and length are both finite but of different size: width<length. In the case of infinite width this relationship cannot change. Thus the implication is that the width is a countable infinite, and the length is also a countable infinite, but of different magnitude. Necessarily, and identical to the finite case, CDM cannot process the entire list for producing ~D.

It always fails in the finite case, so the implication is that it fails in the infinite case for the same reason.

What robert implies, but won't say explicitly, is that it is "supposed to be" in S.

Yet in the finite case, the missing anti-diagonal not only is supposed to be in S but actually is in S at a location which CDM cannot examine.

- robert 46
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### Re: Cantor Diagonal Argument disproof

Robert's whole argument seems to boil down to: if the SDM has examined n items, there are infinitely many left, and can therefore not complete the examination, which, in turn, is just another way of saying: one cannot calculate infinitely many items. The rant about 2^n, intermediate and retired string is really just a smokescreen.

Robert of course fails to realize that that is not what Cantor's theorem is about. It is about whether an arbitrary item of ~D can be constructed, of effectively enumerated, which all of them can, and NOT about whether their totality could ever be computed.

Robert of course fails to realize that that is not what Cantor's theorem is about. It is about whether an arbitrary item of ~D can be constructed, of effectively enumerated, which all of them can, and NOT about whether their totality could ever be computed.

- Gofer
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### Re: Cantor Diagonal Argument disproof

Gofer wrote:Robert's whole argument seems to boil down to: if the SDM has examined n items, there are infinitely many left, and can therefore not complete the examination, which, in turn, is just another way of saying: one cannot calculate infinitely many items.

That is not it. What I am saying is that consequent to the production of a monotonically increasing number of missing IAD, which must be in a list of SFSC, it is necessarily the case that the examination of the list of SFSC cannot end: i.e. all the missing IAD must be present in the list of SFSC, and the monotonically increasing count of the missing thereby ensures that the examination cannot be completed to find all of them. So, the argument is not that an endless examination cannot end, but that it would be contradictory if it did end for reason other than its endlessness.

The rant about 2^n, intermediate and retired string is really just a smokescreen.

This is disingenuous. All it shows is a lack of understanding of the seriousness of the problem.

Robert of course fails to realize that that is not what Cantor's theorem is about. It is about whether an arbitrary item of ~D can be constructed, of [or] effectively enumerated, which all of them can, and NOT about whether their totality could ever be computed.

Wouldn't it be nice if Gofer could express thoughts clearly? A good idea to reread and consider how to rephrase for comprehensibility.

SDM: Sequential Diagonal Method

IAD: Intermediate Anti-Diagonal

SFSC: String/s of Finite Significant Characters

~D: ultimate anti-diagonal

- robert 46
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### Re: Cantor Diagonal Argument disproof

No, "square" has no meaning, because you can't compare the lengths of two endless lists. In order to compare them you have to find an end.robert 46 wrote:Well, it has to remain square under this condition.

So once again, robert demonstrates complete ignorance of the topics he tries to lecture others about.

No. "infinity" is not a number, and you cannot subtract it. If you try using "potential" infinity[1], you can get any and all of 0, any positive natural number, any negative natural number, positive "potential" infinity, or negative "potential" infinity. So that you think you one found examples in some of these categories is still irrelevant, since Cantor, I, probably Gofer, and most modern mathematicians admit that all are possible.Let's say the list was infinity x infinity;

And once again, robert demonstrates complete ignorance of the topics he tries to lecture others about.

It doesn't need to "remain square." And once again, robert demonstrates complete ignorance of the topics he tries to lecture others about.To remain square it must ...

Then apparently Hermann Weyl doesn't fall into the category of those who understand it. Which you can tell by the quote, where he says the field of infinite sets is an extension of finite sets, just like robert does; and his objection is founded in rejecting the Axiom of Infinity. Since such rejection requires finding an end to the natural numbers, it is wrong to reject it.… classical logic ...

And finding such a quote is not significant, since there are many others who do understand it, and why Hermann Weyl is wrong.

What robert ignored in his next reply is colored in red:

The gist of what he ignored, is the description of how to do what robert says can't be done. So he is kinda forced to ignore it, while also ignoring that the problem he describes - even if his conclusion was right, which it still isn't - is a known property of infinite sets, and still irrelevant. And yet again robert demonstrates his complete ignorance of the topics he tries to lecture others about.Not only are the IADs defined, they are constructible (notwithstanding implied trailing 0s).JeffJo wrote:You can make a list S(), based on the ordered set of natural numbers N, where S(n) is the binary representation (i.e., a SFSC) of n padded with infinite zeros. The "intermediate length" - a meaningless concept, but robert won't consider that - of the list after n entries is n. So just take the "intermediate width" to also be n, even if the bits aren't all significant, and you get intermediate arrays that are always square. Property #3 applies, and robert's intermediate anti-diagonals (IADs, not iads) are all defined.

Again, robert resorts to lying when he can't support his assertions. The fact is that CDM looks at the entire string, and has no problem here.robert 46 wrote:Very important rebuttal...[elided]robert 46 wrote:Very important question: Is there ANY circumstance where the Cantor Diagonal method DOES NOT return the result that a list is missing an element?

Given all strings of 4 characters put in a list, CDM can only examine 4 elements ...

Given all strings of 4 characters put in a list, you have a list with 16 strings. Since you are padding them with infinite zeros, just take the first 16 characters of each for SDM. But this is still irrelevant (another point robert ignores completely) because it is just a form of pointing out that, while the infinite list of natural numbers appears to be smaller that the infinite list of perfect squares, they can in fact be put into a 1:1 correspondence.

Not if you use the right algorithm.However, this missing anti-diagonal is in the remainder of the list, 12 elements,

Since the algorithm is endless, this is false, and the entire paragraph that follows is rendered moot.which CDM cannot examine.

And this is the fallacy that robert refuses to address.It always fails in the finite case, so the implication is that it fails in the infinite case for the same reason.

No, it isn't. It is never supposed to be in S. This isn't even based on marginally subtle abstract concepts. THE STRING IS NOT SUPPOSED TO BE IN S.the missing anti-diagonal not only is supposed to be in S but actually is in S at a location which CDM cannot examine.

And if you use every string in S, it isn't.

+++++

[1] Which is not what Cantor talks about, but closer to what you do here.

- JeffJo
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### Re: Cantor Diagonal Argument disproof

That is not it. What I am saying is that consequent to the production of a monotonically increasing number of missing IAD, which must be in a list of SFSC, it is necessarily the case that the examination of the list of SFSC cannot end: because all the missing IAD must be present in the list of SFSC, and the monotonically increasing count of the missing thereby ensures that the examination cannot be completed to find all of them. So, the argument is not that an endless examination cannot end, but that it would be contradictory if it did end for reason other than its endlessness.

As I was saying, all that you're really saying is that there's an infinite 'yet to be examined' number of string left in the list, which really is not that surprising. And furthermore, what you are really implying is that the set of SFSC is not really countable, meaning that there doesn't exist an injection from it to the naturals. However, I have already demonstrated such an injection, namely sum(s1*2^0+s2*2+s3*2^2 ...), where sn is bit n of the sequence. It is an exercise to prove that it is indeed an injection.

- Gofer
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### Re: Cantor Diagonal Argument disproof

JeffJo wrote:No, "square" has no meaning, because you can't compare the lengths of two endless lists. In order to compare them you have to find an end.robert 46 wrote:Well, it has to remain square under this condition.

An end can't be found because endless is impossible to realize. Therefore ~D cannot be produced because it is endless and limit theory does not apply.

Then apparently Hermann Weyl doesn't fall into the category of those who understand it. Which you can tell by the quote, where he says the field of infinite sets is an extension of finite sets, just like robert does; and his objection is founded in rejecting the Axiom of Infinity. Since such rejection requires finding an end to the natural numbers, it is wrong to reject it.… classical logic ...

Find a reputable source online who agrees with you.

And finding such a quote is not significant, since there are many others who do understand it, and why Hermann Weyl is wrong.

Then produce these "many others" who can explain why the quote has not been elided from the Wikipedia article.

The gist of what he ignored, is the description of how to do what robert says can't be done.Not only are the IADs defined, they are constructible (notwithstanding implied trailing 0s).JeffJo wrote:You can make a list S(), based on the ordered set of natural numbers N, where S(n) is the binary representation (i.e., a SFSC) of n padded with infinite zeros. The "intermediate length" - a meaningless concept, but robert won't consider that - of the list after n entries is n. So just take the "intermediate width" to also be n, even if the bits aren't all significant, and you get intermediate arrays that are always square. Property #3 applies, and robert's intermediate anti-diagonals (IADs, not iads) are all defined.

I am saying that the IADs can be produced. What is the point of contention?

So he is kinda forced to ignore it, while also ignoring that the problem he describes - even if his conclusion was right, which it still isn't - is a known property of infinite sets, and still irrelevant.

What is this rambling, vague sentence supposed to mean?

Given all strings of 4 characters put in a list, you have a list with 16 strings. Since you are padding them with infinite zeros,

I am not padding them with infinite zeros, I am addressing diagonalization applied to the finite list of all strings of a finite n characters in width.

just take the first 16 characters of each for SDM.

That would be considering only the first 16 elements of a list of all strings of 16 characters, which is 2^16 elements long. So it still cannot examine the entire list.

Not if you use the right algorithm.However, this missing anti-diagonal is in the remainder of the list, 12 elements,

Your "algorithm" blows up with endless expansion: it goes from strings of 4 characters to strings of 16 characters to strings of 2^16=65536 characters to strings of 2^65536 characters to....

Since the algorithm is endless, this is false, and the entire paragraph that follows is rendered moot.which CDM cannot examine.robert 46 wrote:Remainder of paragraph:

This applies for all lists of n characters. The inference is necessarily that for lists of strings of infinite characters, CDM cannot examine the entire list. In the finite case the width and length are both finite but of different size: width<length. In the case of infinite width this relationship cannot change. Thus the implication is that the width is a countable infinite, and the length is also a countable infinite, but of different magnitude. Necessarily, and identical to the finite case, CDM cannot process the entire list for producing ~D.

JeffJo attempted to redefine what I said, and did not address the problem of diagonalization only processing a square portion of a rectangular array.

And this is the fallacy that robert refuses to address.It always fails in the finite case, so the implication is that it fails in the infinite case for the same reason.

Diagonalization can only work with what it is given. JeffJo's idea of padding the 16x4 array with 0s to make it 16x16 ignores the fact that now he is only considering the initial square portion of what is a 65536x16 array of all strings of 16 characters.

No, it isn't. It is never supposed to be in S. This isn't even based on marginally subtle abstract concepts. THE STRING IS NOT SUPPOSED TO BE IN S.[Yet in the finite case,] the missing anti-diagonal not only is supposed to be in S but actually is in S at a location which CDM cannot examine.

Note JeffJo's blatant quoting out of context. I think he is equivocating over S being a generic list, or S only being a list of strings of infinite characters.

Given all strings of 4 characters, the 4 character wide anti-diagonal is in this list, S, having 16 elements.

- robert 46
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### Re: Cantor Diagonal Argument disproof

And this is the exact reason why you can't make any arguments based on whether an endless-in-two-dimensions array is "square" or not. Thanks for playing our game, now go away.robert 46 wrote:An end can't be found because endless is impossible to realize.

Non sequitur. The reason that we both cited ("An end can't be found") tells us only that we can't define the aspect ratio of the array (my point), and says nothing about whether the entire anti-diagonal is "demonstrated" (your inapplicable attempt to define "produced.")Therefore ~D cannot be produced because it is endless and limit theory does not apply.

The fact is that every element of N is assumed to be "produced," so every character in ~D is "demonstrated" from it.

Too many. Search for university explanations of "infinity."Find a reputable source online who agrees with you.

Weyl is part of the controversy over Cantor, which is what your article is about. Not the theorem itself.Then produce these "many others" who can explain why the quote has not been elided from the Wikipedia article.

I didn't contend the statement, as you clearly implied. You used the confirmation of what I said to (again) evade the rest of the paragraph, shown in red.I am saying that the IADs can be produced. What is the point of contention?

And that's why your algoritm "blows up," but mine doesn't. The nth sequential diagonalization requires n strings that at least n characters long. To compare your fallacious SDM to CDM, you pad it with infinite zeros, and you can do the same here. But you use this point to evade that fact that a finite comparison to an infinite algorithm is invalid.I am not padding them with infinite zeros, I am addressing diagonalization applied to the finite list of all strings of a finite n characters in width.

And it still doesn't need to. All this is, is a way to point out that an infinite set can be put into a 1:1 correspondence with a strict subset OF ITSELF. This is a point you refuse to address, and it invalidates your entire argument.That would be considering only the first 16 elements of a list of all strings of 16 characters, which is 2^16 elements long. So it still cannot examine the entire list.

No, only yours does. Mine takes n strings of length n. But it is still not comparable to an infinite list of infinite strings, a point ylu refuse to address because it invalidates your entire argument.Your "algorithm" blows up with endless expansion:

Still moot.Remainder of paragraph:Since the algorithm is endless, this is false, and the entire paragraph that follows is rendered moot.which CDM cannot examine.

Robert continues to ignore why "square" is not a word that can be applied to an infinite array.JeffJo attempted to redefine what I said, and did not address the problem of diagonalization only processing a square portion of a rectangular array.

Correct. CDM doesn't "work with" more than it is given, and it also doesn't "work with" less.Diagonalization can only work with what it is given.

It is given an infinite list of infinite strings.

And robert is ignoring that fact that in the proof, S represents any list that is countable (look up the definition of "generic") and the list contains on;ly infinite strings.I think he is equivocating over S being a generic list, or S only being a list of strings of infinite characters.

Given all infinite strings of 4 significant characters at the beginning of an infinite list, the infinite anti-diagonal is never found in the list.Given all strings of 4 characters, the 4 character wide anti-diagonal is in this list, S, having 16 elements.

+++++

All robert's SDM demonstrates, is that only an injective mapping can exist between a list of n strings taken from the set of binary strings with n significant bits, and the set of 2^n such strings. His error, which he steadfastly refuses to even acknowledge has been pointed out, is thinking that when he applies is flawed notion of "taking it to infinity," that the same property must be preserved.

All robert's flawed analysis proves, is that only a injective mapping can exist between the

the finite sets {1,2,3,...,n} and {2,4,8,...,2^n}.

But it is a known, and accepted by those who understand modern set theory, FACT that any two countably-infinite sets can be demonstrated to have injective, bijective, and surjective mappings. For example, the sets N={1,2,3,...} and P2={2,4,8,...} are bijective by the relation p2=2^n. This demonstrates that robert's conclusions, about square arrays and "missing elements found later," do not extend, from the finite sets he uses, to endless sets.

And in fact, robert refuses to even look at the bijective mapping I constructed for him, because he found one that is injective. He refuses that I admit BOTH mappings exist, not that his can't. Yet his point is that only his can, and he is wrong.

- JeffJo
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### Re: Cantor Diagonal Argument disproof

What JeffJo seems to be saying is that an array 2^n x n can be padded out to 2^n x 2^n. This of course makes it square. However, it is countable in both dimensions, so if this is extended to n=infinity, 2^infinity x 2^infinity, it should remain countable in both dimensions. This would mean that ~D (with countable infinite characters) is missing from a countable infinite list, because it necessarily has countable infinite characters/columns and remains square, and that is contradictory. So the resolution is that ~D cannot be produced by CDM: i.e. the endless list cannot be processed. This backs up the conclusion reached from SDM.

If JeffJo objects that the whole point is to show that ~D is missing from a countable infinite list, then what of the fact that the list has 2^infinity elements: which is all of T?

If JeffJo objects that the whole point is to show that ~D is missing from a countable infinite list, then what of the fact that the list has 2^infinity elements: which is all of T?

- robert 46
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### Re: Cantor Diagonal Argument disproof

Blue indicates nonsensical statements. Red are flat-out wrong. Green are where robert misinyterprets, as commented.

[1] If it is finite only.The word "square" cannot be applied to an infinite one.

[2] Then it is countable infinite, which is not what robert means. There is no difference between a countably infinite n, and countably infinite 2^n, and the array cannot be called "square."

[3] Correct, ~D is not in the list;. It is an infinite string, but it is not based on an SFSC that is padded to infinity.

[4] "The whole point" is to show that if you have an infinite list, EVEN WITHOUT KNOWING HOW IT IS DEFINED, that there is a string that isn't in it. I don't know what is so hard to robert to understand about that.

robert 46 wrote:What JeffJo seems to be saying is that an array 2^n x n can be padded out to 2^n x 2^n. This of course makes it square[1]. However, it is countable in both dimensions, so if this is extended to n=infinity, 2^infinity x 2^infinity[1], it should remain countable[2] in both dimensions. This would mean that ~D (with countable infinite characters) is missing[3] from a countable infinite list, because it necessarily has countable infinite characters/columns and remains square, and that is contradictory. So the resolution is that ~D cannot be produced by CDM: i.e. the endless list cannot be processed. This backs up the conclusion reached from SDM.

If JeffJo objects that the whole point is to show that ~D is missing from a countable infinite list[4], then what of the fact that the list has 2^infinity elements: which is all of T?

[1] If it is finite only.The word "square" cannot be applied to an infinite one.

[2] Then it is countable infinite, which is not what robert means. There is no difference between a countably infinite n, and countably infinite 2^n, and the array cannot be called "square."

[3] Correct, ~D is not in the list;. It is an infinite string, but it is not based on an SFSC that is padded to infinity.

[4] "The whole point" is to show that if you have an infinite list, EVEN WITHOUT KNOWING HOW IT IS DEFINED, that there is a string that isn't in it. I don't know what is so hard to robert to understand about that.

- JeffJo
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**Posts:**2607**Joined:**Tue Mar 10, 2009 11:01 am

### Re: Cantor Diagonal Argument disproof

JeffJo wrote:Blue indicates nonsensical statements. Red are flat-out wrong. Green are where robert misinyterprets, as commented.robert 46 wrote:What JeffJo seems to be saying is that an array 2^n x n can be padded out to 2^n x 2^n. This of course makes it square[1]. However, it is countable in both dimensions, so if this is extended to n=infinity, 2^infinity x 2^infinity[1], it should remain countable[2] in both dimensions. This would mean that ~D (with countable infinite characters) is missing[3] from a countable infinite list, because it necessarily has countable infinite characters/columns and remains square, and that is contradictory. So the resolution is that ~D cannot be produced by CDM: i.e. the endless list cannot be processed. This backs up the conclusion reached from SDM.

If JeffJo objects that the whole point is to show that ~D is missing from a countable infinite list[4], then what of the fact that the list has 2^infinity elements: which is all of T?

[1] If it is finite only.The word "square" cannot be applied to an infinite one.

Whereas anti-diagonals are consequent to negating characters at row n and column n, the processed array is square no matter whether finite or infinite.

[2] Then it is countable infinite, which is not what robert means. There is no difference between a countably infinite n, and countably infinite 2^n, and the array cannot be called "square."

See above comment.

[3] Correct, ~D is not in the list;. It is an infinite string, but it is not based on an SFSC that is padded to infinity.

Which shows that JeffJo's idea that the rectangular arrays 2^n x n are padded out to 2^n x 2^n is invalid, but that the arrays remain rectangular where the processed portion is only the first square section n x n, and not the whole list; which also holds in the infinite case. If the arrays were padded to 2^n then in the infinite case the length of a list of all of T would be 2^(2^infinity), which is nonsensical.

Yet JeffJo seems to think that a countable finite n extended to infinity becomes a countable infinite, and a countable finite 2^n extended to infinity is also a countable infinite. Yet T has 2^infinity elements and the claim is made that it is an uncountable infinite. What is the difference between 2^n extended to infinity and T having 2^infinity elements? It can't be both a countable and uncountable infinite.

[4] "The whole point" is to show that if you have an infinite list, EVEN WITHOUT KNOWING HOW IT IS DEFINED, that there is a string that isn't in it. I don't know what is so hard to robert to understand about that.

Merely that it is invalid to believe that such example is either produced or defined by an algorithm. JeffJo and Gofer seem to think that an access function is equivalent to an algorithm. This is not correct because it cannot be shown that an access function accesses more than finite positions: i.e. the concept of endless access is invalid. This is unlike an algorithm which converges to some value without an actual endless iteration. It is well-known that a recursive function without an exit test will never return a result. Similarly, an access function without an exit test will never return a result. Nowhere does CDM imply an exit test, so there is no reason to believe that diagonalization would return a result. To get around this, infinite/endless parallel access is hypothesized to define a result. This mystical invocation has no rational explanation.

JeffJo's chromaticity is posturing silliness.

- robert 46
- Intellectual
**Posts:**2838**Joined:**Mon Jun 18, 2007 9:21 am

### Re: Cantor Diagonal Argument disproof

And "whereas" the CDM algorithm only requires a character from position n, in string n? All it requires is that character to be defined. Which it always is, with infinite lists of infinite strings. So robert is, again, lying when he makes this claim.robert 46 wrote:Whereas anti-diagonals are consequent to negating characters at row n and column n, the processed array is square no matter whether finite or infinite.

See above reply to said comment.See above comment.

... which shows that robert is still clinging the the invalid assertion that infinite arrays have an aspect ratio; and...Which shows that JeffJo's idea that the rectangular arrays 2^n x n are padded out to 2^n x 2^n is invalid,

... that "extended to infinity" has any meaning whatsoever.Yet JeffJo seems to think that a countable finite n extended to infinity

)(Emphasis added.) To defend this statement, robert needs to define what he means by "produced." Since he has refused to do so, or even acknowledge that he was asked to do so, it is clear that he cannot.... it is invalid to believe that such example is either produced or defined by an algorithm.

So again, this is a lie.

- JeffJo
- Intellectual
**Posts:**2607**Joined:**Tue Mar 10, 2009 11:01 am

### Re: Cantor Diagonal Argument disproof

Below in blue is what I said which JeffJo evaded:

Evasion through summary dismissal is JeffJo's consistent ploy.

robert 46 wrote:Whereas anti-diagonals are consequent to negating characters at row n and column n, the processed array is square no matter whether finite or infinite.

...

See above comment.

...

Which shows that JeffJo's idea that the rectangular arrays 2^n x n are padded out to 2^n x 2^n is invalid, but that the arrays remain rectangular where the processed portion is only the first square section n x n, and not the whole list; which also holds in the infinite case. If the arrays were padded to 2^n then in the infinite case the length of a list of all of T would be 2^(2^infinity), which is nonsensical.

...

Yet JeffJo seems to think that a countable finite n extended to infinity becomes a countable infinite, and a countable finite 2^n extended to infinity is also a countable infinite. Yet T has 2^infinity elements and the claim is made that it is an uncountable infinite. What is the difference between 2^n extended to infinity and T having 2^infinity elements? It can't be both a countable and uncountable infinite.

...

Merely that it is invalid to believe that such example is either produced or defined by an algorithm. JeffJo and Gofer seem to think that an access function is equivalent to an algorithm. This is not correct because it cannot be shown that an access function accesses more than finite positions: i.e. the concept of endless access is invalid. This is unlike an algorithm which converges to some value without an actual endless iteration. It is well-known that a recursive function without an exit test will never return a result. Similarly, an access function without an exit test will never return a result. Nowhere does CDM imply an exit test, so there is no reason to believe that diagonalization would return a result. To get around this, infinite/endless parallel access is hypothesized to define a result. This mystical invocation has no rational explanation.

JeffJo's chromaticity is posturing silliness.

Evasion through summary dismissal is JeffJo's consistent ploy.

- robert 46
- Intellectual
**Posts:**2838**Joined:**Mon Jun 18, 2007 9:21 am

### Re: Cantor Diagonal Argument disproof

Robert still thinks every argument he writes must be responded to directly, especially when the foundation for the arguments are already discredited and so the parts "in blue" are irrelevant. He uses these claims to evade how they are already discredited. For example, the comment robert didn't address at all:

The fact is, Mathematical "construction" does not mean the equivalent of the English "actual production." For example, it is impossible to (in English) "actually produce" a circle, but in geometry it is considered possible to construct one by using axioms and definitions alone. Get that, robert? All we need to do, to construct an infinite set or string, is to demonstrate that every element of the set follows from our axioms and definitions.

In the above reply, robert complains that one cannot use definitions alone to "construct" an infinite set, despite the citations we have provided that say it is. (Aside: isn't it amazing, how robert ignores citations that contradict his thesis; yet when he finds just one that he thinks supports it, no matter how isolated, he thinks we need to address it?)

+++++

I have never evaded any point robert has made, and he has evaded or misconstrued all of mine, which is why these "discussions" run on endlessly. But too many of his points are compounded erratically and redundantly on false premises, so I don't always address each line item individually. That's not evasion.

What is evasion, is how robert claims that I evaded his entire pyramid of illogic, when I pointed out how it had already collapsed at its base. He evades how I removed the base, and so invalidated the entire pyramid, by point out that I didn't directly address everything above the base. This is a form of lying.

... makes all of ...JeffJo wrote:robert is still clinging the the invalid assertion that infinite arrays have an aspect ratio;

... irrelevant. Another comment robert didn't address at all:robert 46 wrote:but that the arrays remain rectangular where the processed portion is only the first square section n x n, and not the whole list; which also holds in the infinite case. If the arrays were padded to 2^n then in the infinite case the length of a list of all of T would be 2^(2^infinity), which is nonsensical.

... makes all of ...JeffJo wrote:"extended to infinity" has [no] meaning whatsoever.

... irrelevant. And whatever robert thinks "actually produced" means, but won't share with those he wants to meet his definition, is necessary to address ...robert 46 wrote:a countable finite n extended to infinity becomes a countable infinite, and a countable finite 2^n extended to infinity is also a countable infinite. Yet T has 2^infinity elements and the claim is made that it is an uncountable infinite. What is the difference between 2^n extended to infinity and T having 2^infinity elements? It can't be both a countable and uncountable infinite.

The only "mystical invocation" in this discussion, is what robert thinks is necessary to satisfy being "actually produced," and WHY it is so necessary. I've asked for this definition only about 50 times (hyperbole), and his only response has been to provide the definition of something else (verifiable fact). Maybe he should also define what he thinks "evasion" means, because this sure looks like it.JeffJo and Gofer seem to think that an access function is equivalent to an algorithm. This is not correct because it cannot be shown that an access function accesses more than finite positions: i.e. the concept of endless access is invalid. This is unlike an algorithm which converges to some value without an actual endless iteration. It is well-known that a recursive function without an exit test will never return a result. Similarly, an access function without an exit test will never return a result. Nowhere does CDM imply an exit test, so there is no reason to believe that diagonalization would return a result. To get around this, infinite/endless parallel access is hypothesized to define a result. This mystical invocation has no rational explanation.

The fact is, Mathematical "construction" does not mean the equivalent of the English "actual production." For example, it is impossible to (in English) "actually produce" a circle, but in geometry it is considered possible to construct one by using axioms and definitions alone. Get that, robert? All we need to do, to construct an infinite set or string, is to demonstrate that every element of the set follows from our axioms and definitions.

In the above reply, robert complains that one cannot use definitions alone to "construct" an infinite set, despite the citations we have provided that say it is. (Aside: isn't it amazing, how robert ignores citations that contradict his thesis; yet when he finds just one that he thinks supports it, no matter how isolated, he thinks we need to address it?)

- No infinite set can be constructed by what appears to be robert's definition, because it seems to require an end to an endless algorithm. But I can;t say for sure,since robert won't provide the definition.
- The Axiom of Infinity says that we can choose to accept a definition of a set it is defines every element of the set, without needing to have an algorithm that ends.
- CDM provides a definition that applies to every character in the anti-diagonal, without needing to have an algorithm that ends, and so constructs an infinite string.
- Robert has no valid response to this, so he will find some other point to argue about in order to evade it. I know this will happen, because it has happened every time it has been pointed out to robert.
- I'm off on vacation for about a month, so robert's future claims that I am avoiding his inane replies are just as much evasion, as his recent posts have been.

+++++

I have never evaded any point robert has made, and he has evaded or misconstrued all of mine, which is why these "discussions" run on endlessly. But too many of his points are compounded erratically and redundantly on false premises, so I don't always address each line item individually. That's not evasion.

What is evasion, is how robert claims that I evaded his entire pyramid of illogic, when I pointed out how it had already collapsed at its base. He evades how I removed the base, and so invalidated the entire pyramid, by point out that I didn't directly address everything above the base. This is a form of lying.

- JeffJo
- Intellectual
**Posts:**2607**Joined:**Tue Mar 10, 2009 11:01 am

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