"Hidden Assumption in the [Cantor] Diagonal Argument" link

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"Hidden Assumption in the [Cantor] Diagonal Argument" link

Postby phobos rising » Tue Jan 26, 2016 12:48 pm

This appears to be somewhat controversial. It is four and a half pages long.
I am not taking a stand on it.

You should be able to go to "View" at the top of your computer screen
and click on a larger percent number to enlarge the text for easier reading.


Maybe you can also print it if it is desired to be in a handy form.

http://pengkuanonmaths.blogspot.com/
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Re: "Hidden Assumption in the [Cantor] Diagonal Argument" li

Postby JeffJo » Wed Jan 27, 2016 8:24 am

phobos rising wrote:This appears to be somewhat controversial. It is four and a half pages long.
I am not taking a stand on it.
I will, but just the introduction, which is all that is needed:

For proving that the set of real numbers is bigger than that of natural numbers, Georg Cantor proposed the diagonal argument.
In 1874, Georg Cantor proved that the real numbers - that is, the combination of the rational numbers and the irrational numbers - were uncountable. He used an entirely different argument to do so. But his point wasn't so much about real numbers, it was that there are uncountable sets. He just used real numbers because they represent a continuum, and it is a continuum that is uncountable. He expected controversy, which is why he made the title of the paper about real numbers.

He got the expected controversy, so he developed a second proof. In 1891, he published the Diagonal Proof which, in his own words translated from the German here, he said "there is a proof of this proposition that is much simpler, and which does not depend on considering the irrational numbers." (Emphasis added).

In other words, the Diagonal Proof is not about real numbers in any way. Not even incidentally, like the 1874 proof. It is about sets of certain strings; specifically, strings that are infinite lists of two characters. In his examples, these characters were 'm' and 'w'. An example is 'mwmwwmwwwmwwwwm...", continuing on endlessly, adding one more 'w' between the 'm's each time.

One could change the characters to '0' and '1'. These strings can then represent, in binary, the real numbers that exist between 0 and 1. But it would be a mistake to do so, since every rational number in that range whose simplest-form denominator is a power of 2 has two such representations. That is, 1/2 in binary can be represented by 0.1000(0) or 0.01111(1).

But it seems numbers are easier for adolescent students to understand than abstract strings, so the proof is often presented to them in an invalid form (because of that ambiguity) that uses numbers instead of strings[1]. Before reading the introduction of the paper you found, I thought this might be the hidden assumption that was meant. I was wrong.

If a sequence is found out of the list, then the real numbers are more numerous than the natural numbers.
This is not correct. It is not the existence of a sequence that is not in the list that proves the theorem. It is always possible to produce an infinite list that is missing some sequences. If you think you have one that doesn't, all I need to do to make one that does, is add a '0' to the start of every member of your list.

What Cantor showed, is that it is impossible to make such a list that is not missing some sequences. This may sound like the same thing, but it is most definitely not. All of the roses in my garden may be red, but that is not the same thing as saying all roses are red (or, closer to the simile, saying there are no roses that are not red).

The use of the diagonal digits imposes a condition unnoticed until now: the list must possess a diagonal.
Not an assumption at all. It is proven.

Each string is an infinite list of characters - that is, a character is associated with every counting number n. Said another way, you can count the characters. Any set of strings that you can count is a similar list. Thus, for any counting number n, there is a diagonal character d(n) that is the nth character of the nth string.

QED.

The paper you found seems (I didn't read it all) to go on and claim that width and length are different things. This may seem true under a naive interpretation, like the one robert 46 takes. What it misses is that, in the sets Cantor uses, neither has an upper bound. So it is irrelevant that they are different things, since the nth string always has an nth character.

It was tacitly supposed true because the numbers of digits and sequences are both infinite and infinite number should equal infinite number.
This was never "supposed true." In fact, the emphasized part is what Cantor is trying to prove false.

+++++

In short, this paper was written by somebody who probably has never read, and certainly does not understand, Cantor's Diagonal Proof. It belongs in the same rubbish heap with robert 46's claimed disproof, which is there for similar reasons.

+++++

[1] This so=-called controversy is a testament that this "dumbing down" of mathematics is a bad idea. Too many people seem to accept such simplifications as the actual proof. Similar examples can be found in the field of probability, where conditional probability problems are often represented as unconditional ones. As a result, naive students seem to accept that you can treat possible occurrences, that you observed did not happen in a single instance, as necessarily having been impossible from the start.
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Re: "Hidden Assumption in the [Cantor] Diagonal Argument" li

Postby robert 46 » Wed Jan 27, 2016 12:54 pm

The author comes to some of the same insights which I have found.
JeffJo wrote:Each string is an infinite list of characters - that is, a character is associated with every counting number n. Said another way, you can count the characters. Any set of strings that you can count is a similar list. Thus, for any counting number n, there is a diagonal character d(n) that is the nth character of the nth string.

Except that there is no d(infinity); it is delusional to think there is; and it is invalid to infer d as existing.
The paper you found seems (I didn't read it all) to go on and claim that width and length are different things. This may seem true under a naive interpretation, like the one robert 46 takes. What it misses is that, in the sets Cantor uses, neither has an upper bound. So it is irrelevant that they are different things, since the nth string always has an nth character.

JeffJo continues to evade that for strings of n bits there 2^n possible sequences; giving rise to an array n wide by 2^n long. JeffJo also evades that the diagonal method can only examine a square section of any n x 2^n array.
In short, this paper was written by somebody who probably has never read, and certainly does not understand, Cantor's Diagonal Proof. It belongs in the same rubbish heap with robert 46's claimed disproof, which is there for similar reasons.

More of JeffJo's fallacious argumentation style involving posturing.
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Re: "Hidden Assumption in the [Cantor] Diagonal Argument" li

Postby JeffJo » Wed Jan 27, 2016 3:46 pm

robert 46 wrote:The author comes to some of the same insights which I have found.
Yep, that's one way to tell how wrong it is.

Except that there is no d(infinity);
Nobody ever said there was. There is, however, a character associated with every counting number n. And that association defines d, just like e=2n defines the infinite set of all even numbers.

JeffJo continues to evade that for strings of n bits there 2^n possible sequences; ...
Robert continues to evade two facts: that we are talking about infinite strings, and that finding we haven't covered all possible sequences isn't the point.

For example, the above comparison of natural numbers to even natural numbers also produces, in a finite analysis, an array that is twice as long in one dimension as the other. With endless dimensions, tho, this is irrelevant.

More of JeffJo's fallacious argumentation style involving posturing.
Here's an example of what robert calls a "fallacious argumentation style."

In his diagonal argument (a translation of which I linked to above), Cantor explicitly says the point is to prove that there can exist sets that can't be put into 1:1 correspondence with the natural numbers 1,2,3,... . That is, that some infinite sets are bigger than others. He also says that the point of the diagonal argument is to not use the set of real numbers as the example. The only sets involved in the proof contain infinite-length strings of two characters, either "m" or "w."

The paper claiming an unknown assumption, on the other hand, starts with "For proving that the set of real numbers is bigger than that of natural numbers, Georg Cantor proposed the diagonal argument." This is blatantly untrue. It goes on to claim the elements in the subject set are strings of "digits of real numbers." This is close to what is true, if several conditions are added. But the paper doesn't state them, or adhere to them. In fact, the supposed "hidden assumption" boils down to the paper using finite strings of 1's and 0's, where the diagonal argument never uses finite strings. It goes on to claim that a step in the diagonal argument is based on the assertion "infinite number should equal infinite number," when that is the exact opposite of what Cantor is trying to prove.

Yes, it is only an opinion that the author of the paper never read Cantor's diagonal argument, and clearly doesn't understand it. That is a magnanimous assessment of the blatant discrepancies. The alternative is that the author has read it, does understand it, and is deliberately lying about its contents.

However, robert 46 is posturing. As he pointed out, his claims against Cantor are much the same. He has been informed of these discrepancies many times, and refuses to acknowledge any of them because he needs to pretend they don't exist in order to feel superior to others.
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Re: "Hidden Assumption in the [Cantor] Diagonal Argument" li

Postby robert 46 » Fri Jan 29, 2016 10:22 am

Peng Kuan makes an interesting observation:
111…*5=555… -> 111…<555…
But if we do another operation, we obtain the contrary:
555…*2+1=111… -> 111…>555…

I would express it:
(1).(0) * (0)5.(0) = (5).(0) -> (1).(0) < (5).(0)
(5).(0) * (0)2.(0) + (0)1.(0) = (1).(0) -> (5).(0) < (1).(0)

This says that a*5=b -> a<b, but b*2+1=a -> b<a; so a*10+1=a. The only way this can be valid is if a=infinity, but infinity is not a counting number so the expressions are invalid. The difficulty is that neither (1).(0) nor (5).(0) are counting numbers because they do not have leading zeros.

(a<b) & (b<a)=false, of logical necessity. [1] Therefore a*10+1<>a, so a=(1).(0) is invalid, necessarily.

Previously:
While the last sequence [...11111111111 in binary] does not exist, its form is known: an infinite
string of 1
(emphasis added)
This shows the absurdity of dealing with things which do not exsit as if they do exist.


[1] proof:
given (a<b) & (b<a)

Recognizing that a<b and b<a are mutually exclusive:

assume a<b=true, then (true) & (~true)=true & false=false
assume b<a=true, then (~true) & (true)=false & true=false
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