Cantor Diagonal Argument disproof

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Re: Cantor Diagonal Argument disproof

Postby Gofer » Mon May 01, 2017 7:26 am

The "missing sfsc" are those intermediate anti-diagonals of finite significant characters which are produced by the sequential Diagonal method. When produced they are missing from the list so far examined.
So what is your point? We already know there are an infinite number of sfsc starting with any particular sequence; so you really haven't said anything new.

How lucky for us that Cantor's method isn't sequential then, but produces ALL elements of the anti-diagonal all at once.

It can only be defined to do this: it is not demonstrable.
Sure it's demonstrable! Just like I did with the function f and g! Here it is again, to refresh your memory:

(g n m)≝{((n-1) mod 2) for m<2, 0 for n<2, (g n/2 (m-1)) for n even, (g (n+1)/2 (m-1)) for n odd},

(f n)≝((g n n)+1) mod 2

so that (g n m) gives the m'th bit of the n'th string, and (f n) gives the n'th bit of its anti-diagonal.

All this says is that the intermediate anti-diagonal produced is missing from all the rows so far examined.
Not quite! It says that if some property holds for n, it holds for n+1, and therefore for all n since n was arbitrary.

It will be found in the remainder of the list if enough rows are examined. Yet for any missing elements ultimately found there are more to find- infinitely more because the remainder of the list is always infinite. It is impossible to logically argue that the sequential method which only produces intermediate anti-diagonals of finite significant characters can produce the single anti-diagonal, ~D, of infinite significant characters.
Your problem is that you see the method as producing finite strings; it doesn't. It calculates elements of something that is already there, just like the Fibonacci sequence is already there, but its elements haven't been calculated yet.
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Re: Cantor Diagonal Argument disproof

Postby robert 46 » Mon May 01, 2017 8:22 am

JeffJo wrote:
robert 46 wrote:
[Charles Sanders] Peirce began to hold that there were two utterly distinct classes of probable inferences, which he referred to as inductive inferences and abductive inferences.
Therefore the adjective "inductive" is a legitimate qualifier for a type of "inference".

(A) reference where he got this quote,

https://plato.stanford.edu/entries/peirce/
(B) recognize that what I said was that induction was a kind of inference,

Note that Pierce referred to inductive inference as one of the "probable" inferences. Mathematical induction cannot be treated as if it was deductive. It is not generically valid, as I have shown for the sequential Diagonal method, so I am more inclined to refer to mathematical induction as problematic.
(C) that it is different from mathematical Induction, which was my entire point

Mathematicians consider mathematical induction as if it was deductive. If it was then it would work generically. It does not work in the sequential Diagonal method.
and (D) the only reason to use "inductive" as a modifier is to compare it to other kinds of inference, as this "Charles Sanders Peirce" apparently does.

Mathematicians have given "induction" the property of exactness which "inductive" does not have. Mathematical induction (to the infinite) fails as a generically valid method.
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Re: Cantor Diagonal Argument disproof

Postby robert 46 » Mon May 01, 2017 8:41 am

Gofer wrote:
The "missing sfsc" are those intermediate anti-diagonals of finite significant characters which are produced by the sequential Diagonal method. When produced they are missing from the list so far examined.
So what is your point? We already know there are an infinite number of sfsc starting with any particular sequence; so you really haven't said anything new.

The sequential Diagonal method produces intermediate, temporarily missing, anti-diagonal sfsc each of which must be found eventually in examining the list. It does not produce ~D with infinite significant characters.
How lucky for us that Cantor's method isn't sequential then, but produces ALL elements of the anti-diagonal all at once.

It can only be defined to do this: it is not demonstrable.
Sure it's demonstrable! Just like I did with the function f and g! Here it is again, to refresh your memory:

(g n m)≝{((n-1) mod 2) for m<2, 0 for n<2, (g n/2 (m-1)) for n even, (g (n+1)/2 (m-1)) for n odd},

(f n)≝((g n n)+1) mod 2

so that (g n m) gives the m'th bit of the n'th string, and (f n) gives the n'th bit of its anti-diagonal.

There is reason to believe that "n'th" is finite, not infinite.
All this says is that the intermediate anti-diagonal produced is missing from all the rows so far examined.
Not quite! It says that if some property holds for n, it holds for n+1, and therefore for all n since n was arbitrary.

But what makes n leap to infinity? For the sequential Diagonal method to produce ~D with infinite significant characters, it must make that leap to infinity. It never does: the intermediate anti-diagonals all have finite significant characters, no matter that it holds for n+1.
It will be found in the remainder of the list if enough rows are examined. Yet for any missing elements ultimately found there are more to find- infinitely more because the remainder of the list is always infinite. It is impossible to logically argue that the sequential method which only produces intermediate anti-diagonals of finite significant characters can produce the single anti-diagonal, ~D, of infinite significant characters.
Your problem is that you see the method as producing finite strings; it doesn't. It calculates elements of something that is already there, just like the Fibonacci sequence is already there, but its elements haven't been calculated yet.

That is beside the point. The sequential Diagonal method does not produce an anti-diagonal of infinite significant characters. It produces intermediate anti-diagonals of finite significant characters. There is no leap to the infinite: inductive inference to the infinite is invalid.
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Re: Cantor Diagonal Argument disproof

Postby JeffJo » Mon May 01, 2017 8:44 am

robert 46 wrote:Note that Pierce referred to inductive inference as one of the "probable" inferences.
Note that Mathematical Induction is recognized as a form of direct proof, and so is "deductive."
Wikipedia wrote:Although its name may suggest otherwise, mathematical induction should not be misconstrued as a form of inductive reasoning (also see Problem of induction). Mathematical induction is an inference rule used in proofs. In mathematics, proofs including those using mathematical induction are examples of deductive reasoning, and inductive reasoning is excluded from proofs.
It is generically valid, as I have shown repeatedly and you ignore.

Mathematical Induction: If n is a natural number, so is n+1.

What you keep trying to change it into isn't deductive. But that's irrelevant. It's just a similarly-rooted word that takes on a completely different meaning.

Your meaning: The last 100 firetrucks I've seen were red. So all firetrucks are red.

My friends at the Long Green Fire Department would strongly disagree.

Mathematicians consider mathematical induction as if it was deductive. If it was then it would work generically. It does not work in the sequential Diagonal method.
Your "sequential Diagonal method" doesn't work, because it compares two different lists that are not comparable. Like my fire truck analogy.

Mathematicians have given "induction" the property of exactness which "inductive" does not have. Mathematical induction (to the infinite) fails as a generically valid method.
Mathematicians recognize that the conclusion based on Mathematical Induction are logical facts that are necessarily true. Your usage is just an observation that you haven't proven must follow in every case; just those that conform to the biased way you chose to make the observations. I can call a skunk a polecat, but that doesn't make it a cat.
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Re: Cantor Diagonal Argument disproof

Postby JeffJo » Mon May 01, 2017 9:13 am

Robert's "inductive reasoning":

Let S(n) be the set of all natural numbers from 1 to n. Let L(X) be the largest member of the set X (note that bold-italics represents a set).

L(S(n)) is finite for all n; in fact, it is equal to n. By "inductive reasoning," L(S(n)) is finite even as n->infinity. So there is a largest member of the set of all natural numbers.

Please, robert, what is that value?

Mathematical Induciton:

1 is a natural number.

If n is a natural number, then n+1 is a natural number.

Please, robert, tell me what natural number is not defined, in a truly "deductive" sense, by this?

+++++
robert refuses to try to understand:
But what makes n leap to infinity?
Nothing. "n" is a finite number, and is never "infinity." Or "infinite," for that matter.

But every possible finite number is defined, by mathematical Induction. All of them are finite, but the set itself is infinite.

For the sequential Diagonal method to produce ~D with infinite significant characters, it must make that leap to infinity.
No, it doesn't need to make any such leap. As you say, such a leap is impossible.

Still, just like the natural numbers, every character in it is defined by the function described by Cantor's Diagonal Method.

It never does: ...
Correct, but irrelevant.

... the intermediate anti-diagonals all have finite significant characters,...
Your intermediate anti-diagonals are irrelevant. The full, infinite, explicitly defined anti-diagonal has infinite significant characters, is not an sfsc, and is not in the list that was well-defined.

... no matter that it holds for n+1. ...
Which is another example of not being able to infer the infinite from the finite. Cantor's S is infinite and assumed to be listable, your sfsc's are a well-defined set that is a valid S, its anti-diagonal is well-defined, and produces a string that is in T but not in S. Get that? The only time Cantor uses "Missing" is "missing from T." Never "missing from S."

So every time you use the term "missing element," where you refuse to define "element of what" and imply it is S, you are misrepresenting Cantor's proof.
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Re: Cantor Diagonal Argument disproof

Postby robert 46 » Mon May 01, 2017 2:21 pm

JeffJo wrote:
robert 46 wrote:Note that Pierce referred to inductive inference as one of the "probable" inferences.
Note that Mathematical Induction is recognized as a form of direct proof, and so is "deductive."
Wikipedia wrote:Although its name may suggest otherwise, mathematical induction should not be misconstrued as a form of inductive reasoning (also see Problem of induction). Mathematical induction is an inference rule used in proofs. In mathematics, proofs including those using mathematical induction are examples of deductive reasoning, and inductive reasoning is excluded from proofs.

All this shows is that mathematicians have attempted to define the method as being deductive without actually proving that it works, and then using a definition as a basis for belief.
It is generically valid, as I have shown repeatedly and you ignore.

Posturing. Just try to explain how the sequential Diagonal method actually produces ~D. Whereas every intermediate missing anti-diagonal can subsequently be found later in the list, where is ~D found???
Mathematicians consider mathematical induction as if it was deductive. If it was then it would work generically. It does not work in the sequential Diagonal method.
Your "sequential Diagonal method" doesn't work, because it compares two different lists that are not comparable.

What two different lists? Recall that the list S can have any infinite strings. The strings of finite significant characters are a proper subset of the infinite strings. The infinite strings can be partitioned into the set of sfsc and the set of strings of infinite significant characters. As long as the sfsc are in S, any other strings of infinite significant characters can be in it as well. All this does is stretch out retiring the intermediate anti-diagonals, but each of them will be retired after enough elements are examined. There is a method for putting the sfsc into S, so that is a good place to start. Thus the sequential Diagonal method does not apply only to the list of sfsc, but to the sfsc and any other infinite strings in S.

So now hypothesize that all the strings of infinite characters are in S (i.e. all of T), and that the sfsc are put in in order, but interspersed with any number of strings of infinite significant characters between them. Clearly, this does not change the situation that any missing intermediate anti-diagonal will eventually be found. Nor does it change the fact that the examination continues endlessly. Thus even if all of T is in S, the sequential Diagonal method cannot produce ~D. This, of course, is necessary if all of T is in S to prevent a contradiction: as Gofer phrases it per Cantor, An element would be shown to both be in T and not be in T. We do not get into this contradiction if ~D cannot be produced.

So there are two arguments:
1. List S cannot contain all of T if there exists a ~D not in S under all circumstances.
2. List S can contain all of T if it is impossible to produce ~D.

Whereas the "existence proof" for ~D is based on the faulty belief in mathematical induction as a deductive method, 1. is rejected. 2. prevents contradictions. Problem solved: the Diagonal method does not prove that T is uncountable.
Mathematicians have given "induction" the property of exactness which "inductive" does not have. Mathematical induction (to the infinite) fails as a generically valid method.
Mathematicians recognize that the conclusion based on Mathematical Induction are logical facts that are necessarily true.

Perhaps if they pay attention to what I am saying about the sequential Diagonal method they will recognize that they have been deluding themselves for all these years (back to Newton and Leibnitz, undoubtedly).
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Re: Cantor Diagonal Argument disproof

Postby JeffJo » Mon May 01, 2017 3:07 pm

robert 46 wrote:All this shows is that mathematicians have attempted to define the method as being deductive without actually proving that it works, and then using a definition as a basis for belief.
What it shows is that mathematicians have proven that it works. If you don't like that, find another field to look into, since you don't understand this one.

It is generically valid, as I have shown repeatedly and you ignore.

Posturing.
Demonstratable. You use the word "posturing" to ignore what you can't find an invalid argument against.

Just try to explain how the sequential Diagonal method actually produces ~D.
Just try to define what "actually produces" means, and justify why it is needed when the entire string is defined. See viewtopic.php?f=5&t=2150&p=24215#p24215.

Whereas every intermediate missing anti-diagonal can subsequently be found later in the list, where is ~D found???
Whereas every intermediate missing anti-diagonal is has finite significant bits, and ~D has infinite significant bits, it is found somewhere other than your list of intermediate strings.

What two different lists?
You compare rows, and columns, when the rows are defined by a different indexing method than the columns. There are other ways where they are the same apparent "size," and still other ways where the discrepancy is switched. All prove nothing. See all the posts you've ignored.

Recall that the list S can have any infinite strings.
Recall that your list of sfsc's has has fundamentally finite strings only.

The strings of finite significant characters are a proper subset of the infinite strings.
Yes they are. So?

As long as the sfsc are in S, any other strings of infinite significant characters can be in it as well.
And your point is... ?

Cantor's is that if there is a list, it defines a string not in that list. It can be put in a new list, but that defines yet another string not in it.

There is a method for putting the sfsc into S, so that is a good place to start. Thus the sequential Diagonal method does not apply only to the list of sfsc, but to the sfsc and any other infinite strings in S ...
... AS LONG AS THAT SET CAN BE PUT INTO A LIST.

And Cantor's method finds another string not in any list you can provide.

So now hypothesize that all the strings of infinite characters are in S (i.e. all of T)
Okay. It is it untrue, and doesn't accomplish anything, so there isn't much point.

But you still need to remember that you are also assuming it can be listed. Anything you demonstrate after this just shows the two assumptions don't play well together. Not that there is something wrong with the method when you don't add this second assumption.

Clearly, this does not change the situation that any missing intermediate anti-diagonal will eventually be found.
What is clear, is that intermediate strings being found, or not found, is irrelevant. If S is listed, that listing defines a ~D that is not in S. Period.

Nor does it change the fact that the examination continues endlessly.
Nor does it change the fact that "examination" is not a part of the proof. Definition is.

Thus even if all of T is in S, the sequential Diagonal method cannot produce ~D.
All of ~D is defined by the fact that you can list S.

Whereas the "existence proof" for ~D is based on the faulty belief in mathematical induction
Whereas the definition of the string ~D does not use mathematical induction, this is a non-sequitur. And whereas you keep calling it mathematical induction and/or inductive inference interchangeably, it demonstrates that you do not understand what mathematical induction is. What you think it is, is indeed faulty.

Perhaps if they pay attention to what I am saying about the sequential Diagonal method they will ....
... recognize that you are combining words that you do not understand, and merely asserting that they produce the conclusions you want to reach but don't understand.
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Re: Cantor Diagonal Argument disproof

Postby robert 46 » Tue May 02, 2017 8:45 am

JeffJo wrote:
robert 46 wrote:All this shows is that mathematicians have attempted to define the method as being deductive without actually proving that it works, and then using a definition as a basis for belief.
What it shows is that mathematicians have proven that it works.

Mathematical induction to the infinite does not work in the sequential Diagonal method. Try to prove otherwise and I will continue to argue against it.
Just try to explain how the sequential Diagonal method actually produces ~D.
Just try to define what "actually produces" means, and justify why it is needed when the entire string is defined.

If something is produced then it can be agreed that it exists. The sequential Diagonal method does not produce ~D because this method is an endless process producing and retiring intermediate sfsc anti-diagonals. Furthermore, this method does not even imply the existence of an ultimate anti-diagonal ~D, of infinite significant characters, which is categorically different from what this method does produce- strings of finite significant characters.
Whereas every intermediate missing anti-diagonal can subsequently be found later in the list, where is ~D found???
Whereas every intermediate missing anti-diagonal is has finite significant bits, and ~D has infinite significant bits, it is found somewhere other than your list of intermediate strings.

Do you agree that ~D is not even implied by the sequential Diagonal method?
What two different lists?
You compare rows, and columns, when the rows are defined by a different indexing method than the columns. There are other ways where they are the same apparent "size," and still other ways where the discrepancy is switched.

Nowhere in my argument about the sequential Diagonal method have I introduced the issue of the length being longer than the width. I have confined it to the production and retirement of intermediate anti-diagonal elements.
Recall that the list S can have any infinite strings.
Recall that your list of sfsc's has has fundamentally finite strings only.

What does interspersing strings of infinite significant characters do to change the nature of the sequential Diagonal method? Does it not continue to produce intermediate anti-diagonal elements of an increasing number of significant characters, and later finding them?
The strings of finite significant characters are a proper subset of the infinite strings.
Yes they are....
As long as the sfsc are in S, any other strings of infinite significant characters can be in it as well.
And your point is... ?

See paragraph immediately above.
Cantor's is that if there is a list, it defines a string not in that list. It can be put in a new list, but that defines yet another string not in it.

The problem with this is that a belief in induction to the infinite must be invoked to support the existence of ~D. If induction to the infinite was valid generically it would apply to the sequential Diagonal method. Yet even you agree that the sequential method never ends, so induction to the infinite is not realized.
There is a method for putting the sfsc into S, so that is a good place to start. Thus the sequential Diagonal method does not apply only to the list of sfsc, but to the sfsc and any other infinite strings in S ...
... AS LONG AS THAT SET CAN BE PUT INTO A LIST.

Can any strings of infinite significant characters be put into a list? If so, then they can be interspersed with the sfsc which provably can be put into a list.
And Cantor's method finds another string not in any list you can provide.

It doesn't find anything: rather it assumes existence from a problematical principle which is defined to be correct.
So now hypothesize that all the strings of infinite characters are in S (i.e. all of T)
Okay. It is it untrue, and doesn't accomplish anything, so there isn't much point.

But you still need to remember that you are also assuming it can be listed. Anything you demonstrate after this just shows the two assumptions don't play well together. Not that there is something wrong with the method when you don't add this second assumption.

Can elements of infinite significant characters be interspersed with elements of finite significant characters?
Clearly, this does not change the situation that any missing intermediate anti-diagonal will eventually be found.
What is clear, is that intermediate strings being found, or not found, is irrelevant. If S is listed, that listing defines a ~D that is not in S. Period.

Not consequent to the sequential Diagonal method, which does not use the problematical principle of induction to infinity.
Nor does it change the fact that the examination continues endlessly.
Nor does it change the fact that "examination" is not a part of the proof. Definition is.

As I have said, Cantor only attempts to define his argument to be correct.
Thus even if all of T is in S, the sequential Diagonal method cannot produce ~D.
All of ~D is defined by the fact that you can list S.

One cannot list an infinite set in practice, but only in principle.
Whereas the "existence proof" for ~D is based on the faulty belief in mathematical induction
Whereas the definition of the string ~D does not use mathematical induction, this is a non-sequitur.

What about: if one can flip the character at row n, column n, one can flip the character at row n+1, column n+1?
And whereas you keep calling it mathematical induction and/or inductive inference interchangeably, it demonstrates that you do not understand what mathematical induction is.

I am merely pointing out that "induction" does not derive from "inductive inference" in that "inductive inference" is not "deductive". "Induction" is claimed to be deductive merely as a matter of belief.
Perhaps if they pay attention to what I am saying about the sequential Diagonal method they will ....
... recognize that you are combining words that you do not understand, and merely asserting that they produce the conclusions you want to reach but don't understand.

This topic has been receiving over 190 views/day for the past month [1]. If this unusual interest includes mathematicians, I invite them to comment.


[1] 48888 views as of 2017-03-31; 54612 views as of 2017-04-30.
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Re: Cantor Diagonal Argument disproof

Postby JeffJo » Tue May 02, 2017 9:23 am

robert 46 wrote:Mathematical induction to the infinite does not work in the sequential Diagonal method.
Mathematical Induction doe not apply as you tried to use it, so this is a non sequitur.

Just try to define what "actually produces" means, and justify why it is needed when the entire string is defined.
If something is produced then it can be agreed that it exists.
So, you can't define what it means? Just the result you want it to have?

But here you asserted "If A, then B", where A="something is produced" and B="it can be agreed that it exists." But you use "actually produced" in the context "If not A, then not B" which is equivalent to "If B, then A." This is the logical fallacy known as affirming the consequent.

Diagonalization is an endless process; so if "produce" requires an end, then it doesn't "produce" anything. Not even the intermediate results, since they are, well, intermediate.

Mathematical Induction defines every natural number in N. If you accept that, then Mathematical Induction is valid. If you don't, then Cantor Diagonalization is not "wrong" in your fantasy world, it is inapplicable to anything.

Diagonalization defines an infinite string. It does not do it by Mathematical Induction, or even "inductive inference" as you call it. It takes one definition (of N) and defines another infinite collection from it without any form of induction. Because of that, your attempt to make a comparison that uses "inductive inference" of your sequence of intermediate results is meaningless gibberish.

And I suspect you know this, or else you would respond to this flaw.

Do you agree that ~D is not even implied by the sequential Diagonal method?
I agree that anything implied by the "sequential Diagonal method" is unrelated to anything having to to with Cantor's Theorem. ~D is defined, period.

What does interspersing strings of infinite significant characters do to change the nature of the sequential Diagonal method?
Nothing. But the fact that you start with a list of strings does change your invalid conclusions.

Does it not continue to produce intermediate anti-diagonal elements of an increasing number of significant characters, and later finding them?
If you list the squares of natural numbers, do you not get a similar intermediate result that you skipped an increasing number of non-square elements? Both facts are irrelevant.

Cantor's [point] is that if there is a list, it defines a string not in that list. It can be put in a new list, but that defines yet another string not in it.
The problem with this is that a belief in induction to the infinite must be invoked to support the existence of ~D.
The problem with your objection is that induction is not a part of the definition of ~D.[/quote]

Can any strings of infinite significant characters be put into a list? If so, then they can be interspersed with the sfsc which provably can be put into a list.
Yes, but irrelevant to your "sequential Diagonal method" which only looks at a finite portion.

Can elements of infinite significant characters be interspersed with elements of finite significant characters?
Yes, but irrelevant to your "sequential Diagonal method" which only looks at a finite portion.

Not consequent to the sequential Diagonal method, which does not use the problematical principle of induction to infinity.
Consequent, because you keep insisting that defining ~D needs to, and using the "sequential Diagonal method" to claim it can;t.

As I have said, Cantor only attempts to define his argument to be correct.
As I have said, Cantor doesn't "define his argument to be correct," whatever that means. His argument defines the string you say he can't use, and he can.

What about: if one can flip the character at row n, column n, one can flip the character at row n+1, column n+1?
And this is an example of induction, how? Nobody says that one follows form another; just that it you can define a character, you can flip it.

I am merely pointing out that "induction" does not derive from "inductive inference" in that "inductive inference" is not "deductive".
And you are ignoring that "Mathematical Induction" is an example of this sentiment, so comparing it to "inductive inference" is irrelevant.

"Induction" is claimed to be deductive merely as a matter of belief.
No, it is demonstrated to be deductive because the properties follow logically.

This topic has been receiving over 190 views/day for the past month. If this unusual interest includes mathematicians, I invite them to comment.
Do you really what to follow in Don Blazys' argumentative footsteps?

Most are probably us paging back and forth.
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Re: Cantor Diagonal Argument disproof

Postby Gofer » Tue May 02, 2017 11:45 am

Robert, does the function (*2) exist regardless the number of calculations that have been performed?

If I use it on ℕ, and calculate the first three numbers {2,4,6}, does that mean there are numbers missing?
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Re: Cantor Diagonal Argument disproof

Postby robert 46 » Tue May 02, 2017 2:46 pm

JeffJo wrote:
robert 46 wrote:Mathematical induction to the infinite does not work in the sequential Diagonal method.
Mathematical Induction doe not apply as you tried to use it, so this is a non sequitur.

If the sfsc are listed in order:
1. Is it true that one can examine the first element and produce an intermediate missing anti-diagonal of 1 significant character?
2. Is it true that one can examine the second element and extend the intermediate missing anti-diagonal to 2 significant characters?
3. Is it true that one can by this process extend the intermediate missing anti-diagonal to n significant characters?
4. Is it true that one can by this process extend the missing anti-diagonal to infinite significant characters?

If you find 1 to 3 true, does mathematical induction imply 4 is also true?
[robert 46 wrote: Just try to explain how the sequential Diagonal method actually produces ~D.]

Just try to define what "actually produces" means, and justify why it is needed when the entire string is defined.
If something is produced then it can be agreed that it exists.

...here you asserted "If A, then B", where A="something is produced" and B="it can be agreed that it exists." But you use "actually produced"

You did not quote the part where I used "actually produces". You have reverted to quoting out of context.

The sequential Diagonal method neither defines nor implies ~D. I challenged you to explain how it can produce ~D in order to assert existence for ~D. You have evaded this challenge using diversion.
Diagonalization is an endless process; so if "produce" requires an end, then it doesn't "produce" anything. Not even the intermediate results, since they are, well, intermediate.

Of course it produces the intermediate anti-diagonals because it produces all the significant characters of each of them.
Mathematical Induction defines every natural number in N. If you accept that, then Mathematical Induction is valid. If you don't, then Cantor Diagonalization is not "wrong" in your fantasy world, it is inapplicable to anything.

Does mathematical induction go to the infinite to define infinity as a natural number? Would the sequential Diagonal method have to go to the infinite to produce an anti-diagonal of infinite significant characters? Is this possible by induction?
Diagonalization defines an infinite string. It does not do it by Mathematical Induction, or even "inductive inference" as you call it. It takes one definition (of N) and defines another infinite collection from it without any form of induction. Because of that, your attempt to make a comparison that uses "inductive inference" of your sequence of intermediate results is meaningless gibberish.

This fast-talk of yours is classic. N is not defined to include infinity as a number. But having infinite significant characters is fundamentally different from having finite significant characters because in this usage infinite is treated as if it is a counting number.
Do you agree that ~D is not even implied by the sequential Diagonal method?
I....

Evasion.
What does interspersing strings of infinite significant characters do to change the nature of the sequential Diagonal method?
Nothing. But the fact that you start with a list of strings does change your invalid conclusions.

Does Cantor start with a list of strings?
Does it not continue to produce intermediate anti-diagonal elements of an increasing number of significant characters, and later finding them?
If you list the squares of natural numbers, do you not get a similar intermediate result that you skipped an increasing number of non-square elements? Both facts are irrelevant.

No- your diversion introduces a false analogy, and is irrelevant.
Cantor's [point] is that if there is a list, it defines a string not in that list. It can be put in a new list, but that defines yet another string not in it.

The problem with this is that a belief in induction to the infinite must be invoked to support the existence of ~D.
The problem with your objection is that induction is not a part of the definition of ~D.

Does induction take no part in Cantor's Diagonal argument?
Can any strings of infinite significant characters be put into a list? If so, then they can be interspersed with the sfsc which provably can be put into a list.
Yes, but irrelevant to your "sequential Diagonal method" which only looks at a finite portion.

It looks at an endlessly increasing portion constructively. Cantor doesn't look at anything constructively- all of it is a consequence of definition.
[robert 46 wrote: Clearly, this does not change the situation that any missing intermediate anti-diagonal will eventually be found.

JeffJo wrote: What is clear, is that intermediate strings being found, or not found, is irrelevant. If S is listed, that listing defines a ~D that is not in S. Period.]

Not consequent to the sequential Diagonal method, which does not use the problematical principle of induction to infinity.
Consequent, because you keep insisting that defining ~D needs to, and using the "sequential Diagonal method" to claim it can't.

I am saying that defining ~D to exist is nothing more than a ploy.
As I have said, Cantor only attempts to define his argument to be correct.
As I have said, Cantor doesn't "define his argument to be correct," whatever that means. His argument defines the string you say he can't use, and he can.

He can only use it by defining it to be usable!
What about: if one can flip the character at row n, column n, one can flip the character at row n+1, column n+1?
And this is an example of induction, how? Nobody says that one follows from another; just that it you can define a character, you can flip it.

So if you define an infinite string, you can flip any character? Where does the infinite string get its existence from? Let met guess- definition.
I am merely pointing out that "induction" does not derive from "inductive inference" in that "inductive inference" is not "deductive".
And you are ignoring that "Mathematical Induction" is an example of this sentiment, so comparing it to "inductive inference" is irrelevant.

Then why do they call it "induction" if it is unrelated to "inductive"?
"Induction" is claimed to be deductive merely as a matter of belief.
No, it is demonstrated to be deductive because the properties follow logically.

It doesn't work in the sequential method.
This topic has been receiving over 190 views/day for the past month. If this unusual interest includes mathematicians, I invite them to comment.

Most are probably us paging back and forth.

During the month there were 130=688-558 new posts. This is 42 views per post. I doubt the three participants account for this. It appears many others are looking with nothing to say.
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Re: Cantor Diagonal Argument disproof

Postby JeffJo » Wed May 03, 2017 8:23 am

robert 46 wrote:If the sfsc are listed in order:
1. Is it true that one can examine the first element and produce an intermediate missing anti-diagonal of 1 significant character?
2. Is it true that one can examine the second element and extend the intermediate missing anti-diagonal to 2 significant characters?
3. Is it true that one can by this process extend the intermediate missing anti-diagonal to n significant characters?
4. Is it true that one can by this process extend the missing anti-diagonal to infinite significant characters?
Sure, but irrelevant.

If S(n) is a function that returns a unique, infinite binary string for each natural number n in N, and D(s,n) is a function that returns the flipped character of the nth character of the infinite string s:
1) Is it true that D(S(n),n), is a function that defines an infinite string?
2) Is it true that there is no n where D(S(n),n) is equal to S(n) ?

You did not quote the part where I used "actually produces".
You use claims that I avoided something, to avoid what you can't address. Define "actually produced."

The sequential Diagonal method neither defines nor implies ~D.
If you think so, use Cantor's Diagonal Method (CDM), which is not based on Mathematical Induction.

I challenged you to explain how it can produce ~D in order to assert existence for ~D.
And I have explained it many times. I just did, again. I just didn't use your term "actually produce", which you refuse to define yet insist I use.

Of course it produces the intermediate anti-diagonals...
Then "(actually) produces", as you use it here, means something that isn't significant to CDM.

An infinite binary string is a function whose domain is the set of natural numbers N, and that returns one of two characters for each number in that domain.

CDM is not a sequential process. It is a function whose domain is N, and that returns one of two characters for each number in that domain. That is, an infinite binary string.

Any "method" that limits the domain to a finite subset of N is not comparable to CDM. So it is impossible to "explain" the result of CDM by such a sequential process. Not even by Mathematical Induction.

Does mathematical induction go to the infinite to define infinity as a natural number?
Infinity is not a natural number. As you use the word, it means the size[1] of N, and is not defined by Mathematical Induction. It is defined by the definition of cardinality, and the existence of the set N.

Would the sequential Diagonal method have to go to the infinite to produce an anti-diagonal of infinite significant characters? Is this possible by induction?
"Go to the infinite" is a meaningless expression.[2]

This fast-talk of yours is classic.
Your refusal to accept concepts that you don't like, and subsequent evasion of simple explanations ("fast-talk," "just definitions," "posturing") is your classic ploy.

N is not defined to include infinity as a number.
You finally said something right. But since I never said that "infinity" was a member of N, and in fact have sadi the opposite many times, I have no idea what your point is. Unless it is to, once again, change what I actually said, to what you want me to have said, so you can point out an error.

But having infinite significant characters is fundamentally different from having finite significant characters because in this usage infinite is treated as if it is a counting number.
NO. You keep trying to treat it like one, and I keep telling you that you can't.

Evasion.
Now you are the one who didn't quote what you replied to. I had said that your Sequential Diagonal Method (SDM) was not akin to CDM, for the same reason that infinity is not a natural number. This is not evasion (on my part), it is repetition of the fact that you keep evading.

Does Cantor start with a list of strings?
CDM starts with in infinite list of infinite strings. SDM only uses finite lists, and only a finite part of each string. Even "going to infinity" (whatever that means) does not make the list infinite, or allow SDM to use the part of any string that makes it infinite. So just like "N is not defined to include infinity as a number," SDM is not defined to include the infinite lists and infinite strings that CDM uses.

No- your diversion introduces a false analogy, and is irrelevant.
My "diversion" pointed out the exact difference between "going to infinity" with finite sets, and having a set with infinite cardinality. The finite sets cannot be in 1:1 correspondence with a strict subset of themselves, but an infinite set can. The fact that you perceived to be a contradiction because your finite sets had one HAS NO BEARING WHATSOEVER ON INFINITE SETS.

Does induction take no part in Cantor's Diagonal argument?
I've said it before, can't you read? Or do you not even look at the parts you evade, so like an ostrich with its head in the sand, you think it isn't there?

Mathematical Induction has no direct role in Cantor's Diagonalization Proof. Its only use, is to define the entirety of the natural numbers. It does not say anything about infinity, only the entirety of the natural numbers.

It is the Axiom of Infinity, and the definition of cardinality, that say something about infinity. The Axiom says that an endless collection exists as a set. Then, any class of sets that can be put into 1:1 correspondence with each other are defined to have the same cardinality. Note that there are non-injective, and non-surjective, mappings between such sets if they are infinite. So SDM, which points out non-injective and/or non-surjective mappings, IS COMPLETELY IRRELEVANT TO INFINITE SETS.

[SDM] looks at an endlessly increasing portion constructively.
Which is why it can't look at an infinite set.

Cantor doesn't look at anything constructively- all of it is a consequence of definition.
Construction only "finds" objects that are based on such definitions. N is defined constructively. The strings are defined on N. The list of S is defined on N. CDM is defined on the list of S.

I am saying that defining ~D to exist is nothing more than a ploy.
And I'm saying that the definition of ~D is just that - a definition. It takes one construct, and defines another from it. That is constructive. So evading this issued by dismissing "definitions" is a ploy.

He can only use it by defining it to be usable!
IT EXISTS BECAUSE EVERY PART OF IT IS DEFINED FROM THINGS THAT EXIST!

So if you define an infinite string, you can flip any character?
IF YOU DEFINE AN INFINITE STRING, YOU CAN ALSO DEFINE THE FLIPPED CHARACTER AT EVERY POSITION FROM THAT DEFINITION!

Where does the infinite string get its existence from? Let met guess- definition.
WHERE DOES ANY MATHEMATICAL OBJECT (ULTIMATELY) GET ITS EXISTENCE FROM? AXIOMS AND DEFINITIONS!

It doesn't work in the sequential method.
Irrelevant. It does work in CDM.

+++++

[1] "Size" is the wrong word, but it is what you mean. It means a natural number. "Cardinality" is the right word. It means the equivalent of size, but for infinite sets.

Both describe how a set can be put into a 1:1 correspondence with another set. But unlike finite sets, a infinite set can be put into a 1:1 correspondence with a strict subset of itself. So the concept has to be different for infinite sets; hence, "infinity" is not a natural number.

You refuse to even address this point, which is why you can't grasp the simple concepts behind the theories of infinite sets. Every one of your arguments, in one way or another, is trying to find a contradiction in the fact that the sets you use can be put into a 1:1 correspondence with a strict subset of itself. It is only a contradiction for finite sets.

[2] In the context of the theory of infinite sets, where "infinity" mean "actual infinity." It can be used in arithmetic, where it means the very different concept of "potential infinity." Your problem, is that you keep trying to equate the two.
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Re: Cantor Diagonal Argument disproof

Postby robert 46 » Wed May 03, 2017 1:12 pm

JeffJo wrote:[2] In the context of the theory of infinite sets, where "infinity" mean "actual infinity." It can be used in arithmetic, where it means the very different concept of "potential infinity." Your problem, is that you keep trying to equate the two.

Plato believed in the actual infinite, whereas Aristotle, his follower, only believed in the potential infinite. I see Aristotle as being more modern and realistic in his thinking.

The great problem with religions is that they define what they want, and then use their definitions as the support for argumentation. Clearly, mathematics following Plato is an idealist religion. There is no getting anywhere arguing with religionists and their dogmatic beliefs.
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Re: Cantor Diagonal Argument disproof

Postby Gofer » Wed May 03, 2017 4:27 pm

How come Robert still hasn't answered how a function such as (*2) can exist without "intermediate strings", "missing strings" and so on?
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Re: Cantor Diagonal Argument disproof

Postby robert 46 » Wed May 03, 2017 6:16 pm

Gofer wrote:How come Robert still hasn't answered how a function such as (*2) can exist without "intermediate strings", "missing strings" and so on?

2* is irrelevant to the issues at hand. Clearly, a string of infinite significant characters manifests an actual infinite. Yet this actual infinite cannot be reached using the constructive sequential Diagonal method. Any claim that Cantor's Diagonal method is constructive is nonsensical: it is entirely definitional. If the sequential method cannot construct ~D, then ~D cannot be constructed, period!
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