Why Cantor's Diagonalization Proof is Flawed.

[Divine Insight]

Here is a Youtube video, under 10 minutes, of Cantor's diagonalization argument.

Ok, I've seen this proof countless times.

And like I say it's logically flawed because it requires the a completed list of numerals must be square, which they can' t be.

~~~~

First off you need to understand the numerals are NOT numbers. They are symbols that represent numbers. Numbers are actually ideas of quantity that represent how many individual things are in a collection.

So we aren't working with numbers here at all. We are working with numeral representations of numbers.

So look at the properties of our numeral representations of number:

Well, to begin with we have the numeral system based on ten.

This includes the ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

How many different numbers can we list using a column that is a single digit wide?

Well, we can only list ten different numbers.

0
1
2
3
4
5
6
7
8
9

Notice that this is a completed list of all possible numbers. Notice also that this list is not square. This list is extremely rectangular. It is far taller than it is wide.

Let, apply Cantor's diagonal method to our complete list of numbers that are represented by only one numeral wide.

Let cross off the first number on our list which is zero and replace it with any arbitrary number from 1-9 (i.e. any number that is not zero)


[strike]0[/strike]
1
2
3
4
5
6
7
8
9

Ok we struck out zero and we'll arbitrary choose the numeral 7 to replace it.

Was the numeral 7 already on our previous list? Sure it was. We weren't able to get to it using a diagonal line because the list is far taller than it is wide.

Now you might say, "But who cares? We're going to take this out to infinity!"

But that doesn't help at all.

Why not?

Well what happens when we make the next step? We need to make the list 2 digits wide now.

What happens?

Here is a 2-digit list of all possible numbers represented by 2 numerals.

00
01
02
03
04
05
06
07
08
09
11
12
13
14
15
.
.
.

95
96
97
98
99

What happened? Well, our completed list of possible numerals that is two digits wide has incrusted in vertical height exponentially. This list is now 100 rows tall and only 2 column wide.

Now let's cross off the first two digits of our list and replace them with arbitrary numerals.

[strike]0[/strike]0
0[strike]1[/strike]

Ok, for the first zero being stuck off the list, I'll chose to arbitrarily replace that with a 5. For the second digit being struck off the list I'll replace that arbitrarily with a 7.

My new number is 57.

Is 57 already on my completed list? Yes. It's just further down the list where I couldn't possibly reach it by drawing a diagonal line.

Now you might say, "But who cares? We're going to take this out to infinity!"

But duh? We can already see that in a finite situation we are far behind where we need to be, and with every digit we cross off we get exponentially further behind the list.

Taking this process out to infinity would be a total disaster.

You could never claim to have "completed" this process because you can't move down the list fast enough using a diagonal line that crosses off each digit diagonally.

The very nature of our system of numerical representation forbids this. You can't complete this process in a finite situation, and it gets exponentially worse with every digit you add to the width, then you could never claim to have completed this process by claiming to have taken it out to infinity.

"Completed Lists" of numerical representations of numbers are NOT SQUARE.

Yet Cantor claims to be creating a "Completed List" here. It's a bogus proof that fails. Cantor didn't stop to realize that our numerical representations of number do not loan themselves to nice neat square competed lists. And that was the flaw in his logic.

By the way you can't even do this using binary representations of numbers.

In Binary Representation

A completed list of binary numbers 2 digits wide:

00
01
10
11

It's not square. It's twice as tall as it is wide.

Add another digit it gets worse:

000
001
010
011
100
101
110
111

There is no way that a completed list of numbers can be represented numerically in square lists.

Yet Cantor's diagonal argument demands that the list must be square. And he demands that he has created a COMPLETED list.

That's impossible.

Cantor's denationalization proof is bogus.

It should be removed from all math text books and tossed out as being totally logically flawed.

It's a false proof.

Cantor was totally ignorant of how numerical representations of numbers work. He cannot assume that a completed numerical list can be square. Yet his diagonalization proof totally depends on this to be the case.

Otherwise, how can he claim to have a completed list?

If he's standing there holding a SQUARE list of numerals how can he claim that he has a completed list?

Yet at what point does his list ever deviate from being square?

It never deviates from being square. It can't because he's using a diagonal line to create it. That forces his list to always be square.

Georg Cantor was an idiot.

He didn't even understand how numerical representations of numbers work.

His so-called "proof" doesn't prove anything. It's totally bogus.

He can't claim to have a "completed list" by the way he is generating his list. Claiming to take this out to infinity doesn't help. With every new digit he creates he falls exponentially behind where he would need to be to create a "Completed List".

Yet that's what he claims to have: A Completed List.

It's a bogus proof, and I'm shocked that no mathematicians have yet recognize this extremely obvious truth.

They keep publishing this proof and teaching it like as is it has merit when in fact it's totally bogus.

[keithprosser3]

I'm still going over things, but I won't say DI is wrong.

But the fact it that most mathematicians have - in their finite wisdom - decided to adopt conventions regarding things like infinities in a particular way. Presumably as they have to work with these things to earn a crust they have arranged things for their convenience rather that to satisfy the whims of part-timers and layman like myself and - do I presume to much? - DI.

I do know that there are many mathematicians who share similar views about infinity to DI. But on the other hand mathematics does have lots of other impossible objects on its books - imaginary numbers, 4 dimensional cubes, Klein bottles, Penrose triangles and a whole host of other fantasies. Even a perfect circle or a line with no thickness is a nonsense in physical terms. So is infinity more impossible than a Penrose triangle or a Klein bottle?

To be quite honest, I don't really care.

[lao tzu]

would even say that it's basically impossible to define a meaningful irrational quantity that is not based on a self-referenced situation.

Dear DI,

The diagonal length of a unit square is an irrational quantity.

As ever, Jesse

[olavisjo]

.

The diagonal length of a unit square is an irrational quantity.

This diagonal irrational quantity corresponds to an irrational integer.

Consider the irrational integer...

...979,853,562,951,413 which corresponds to pi = 3.14159265358979...

[lao tzu]

Dear DI,

You might want to consider the fact that all of Cantor's numbers are real numbers, are less than one, and are represented by an infinite sequence of digits, whereas all of your numbers are integers, are greater than one, and are represented by a finite number of digits.

What?

You say the integers are greater than one, and are represented by a finite number of digits?

Dear DI,

No, I said, “all of your numbers … are represented by a finite number of digits.� As indeed they are.

Where do you come up with that? Are you saying that there exists a largest integer?

I did not come up with it, but it’s demonstrably true. Every integer can be written using a finite number of digits. This does not imply a largest integer.

If not, then how can you say that they are represented by a finite number of digits?

Name one that can’t.

You can always add another digit and you'll still have an integer.

You will then have another integer that can be represented using a finite number of digits. Adding one to a finite number yields a finite number.

So to say that they have to have a finite number of digits seems to be a claim with no proof?

Every integer is a finite number. Hence, for any integer k, there is a finite number n, such that 10^n > | k |. Hence, any integer k can be written using no more than n digits and a leading negative sign, if necessary.

In short, your rebuttal takes the form of arguing that if all real numbers were integers, then they, like the integers, would also be countable.

You are off track here entire. I never claimed the the real numbers are countable. I simply claimed that Cantor's Diagonalization Proof is flawed.

This is a minimization. In fact, you have claimed that Cantor was an idiot. You further claim that I am off track. My counter is that this is the track that you have built, and if it is off, we should look to the builder.

I'm am not arguing that all real numbers need to be countable. However, I can actually show that they necessary have to be. But that is a whole other argument unrelated to the topic of this thread.

In anything other than the informal sense, you are not arguing anything. The integers (and all finite dimensional cartesian products of the integers — which includes the rationals) are countable. This has been proven — repeatedly, with multiple diverse proofs ranging from the abstract to direct construction.

Nor can you show the real numbers are countable. Because they’re not. This also has been proven — repeatedly, with multiple diverse proofs ranging from the abstract to direct construction, such as the method of Cantor’s diagonal, considered by many, including myself, to be the most elegant.

Have you ever seen the proof by parity that the square root of two cannot be represented as a quotient of whole numbers?

That's totally irrelevant to Cantor's diagonalization proof.

We can prove there are real numbers that are not rational.

And I have no problem with that. I even understand why this is the case.

, and hence, that your rebuttal is irrelevant, and more, that your opinion of Cantor's proof does not rise beyond what Pauli would dismiss as "not even wrong."

As ever, Jesse

My observations of the flaw in Cantor's proof has nothing at all to do with whether the real numbers can be placed in a one-to-one correspondence with the integers.

Per my earlier remarks, it is your observations which are flawed, in the ways previously enumerated.

First, Cantor places all real numbers between zero and one. There’s a subset argument that validates this, but also a 1-1 correspondence possible by mapping all real numbers (-∞, ∞) into the set (0,1) via the function f(x) = 1 / (1+e^x) with inverse function g(x) = ln(1/x - 1). To complete the validation, we choose to exclude equivalent representations of real numbers that end in terminal repeating nines.

On the other hand, your argument is restricted to integers, and more, to positive integers, and more, their representations are arbitrarily chopped at the decimal point. Had you written all of your n-digit integers using 10^n - n digits beyond the decimal, your counter-example would fail. Your counter-example does not even acknowledge the issue of alternate digital representations, let alone justify your choice of representation.

Specifically, beginning with a list of 1-digit positive integers:

0.000000000...
1.000000000...
2.000000000...
3.000000000...
4.000000000...
5.000000000...
6.000000000...
7.000000000...
8.000000000...
9.000000000...

You chop to create the list:

0
1
2
3
4
5
6
7
8
9

You look at the first element: 0
Construct a contrary element: 5

Chop the list again:

0

And "observe" that 5 is not on the list! Your maimed creation is maimed. Well done, sir.

All I'm saying is that Cantors Proof is flawed. Even if the conclusion implied by his so-called proof is correct, his proof is still flawed and not a valid proof at all.

Just because you happen to come up with the right answer doesn't mean that the proof you used was correct.

But the conclusion too could be wrong. Just because irrational numbers can't be expressed neatly as a ratio of integers doesn't automatically mean that they can't be placed into a one-to-one correspondence with the integers. Where is there any logical reasoning to support such a conclusion?

I have no need to place the "Real Numbers" in a direct correspondence with the integers. But that's beside the point.

I"m addressing the validity of Cantor's proof here. I'm not arguing about whether or not the reals can be placed in a one-to-one correspondence with the integers.

That is a totally different argument.

Whatever.

As ever, Jesse

[lao tzu]

.

The diagonal length of a unit square is an irrational quantity.

This diagonal irrational quantity corresponds to an irrational integer.

Consider the irrational integer...

...979,853,562,951,413 which corresponds to pi = 3.14159265358979...

Dear o,

Cute. But these are "integers," not integers, and moreover, necessarily unsigned. Hence, not numbers, as they cannot admit an additive inverse. And still, irrational quantities. But cute, none the less. Thank you.

As ever, Jesse

[Divine Insight]

Dear DI,

No, I said, “all of your numbers … are represented by a finite number of digits.� As indeed they are.

They don't need to be. I only did that for the sake of brevity in my presentation.

I could have used integers that are infinitely long just as easily.

Where do you come up with that? Are you saying that there exists a largest integer?

I did not come up with it, but it’s demonstrably true. Every integer can be written using a finite number of digits. This does not imply a largest integer.

You'll need to show a proof of this please.

If not, then how can you say that they are represented by a finite number of digits?

Name one that can’t.

I don't need to.

All I need to do is start it, then you need to tell me why you need to truncate it.

Here's my infinitely long integer,...

5672356378262517823635...

Continue it on for as long as you like, and when you get to a point where you NEED to stop please give me a call and explain to me why you had to stop.

You can always add another digit and you'll still have an integer.

You will then have another integer that can be represented using a finite number of digits. Adding one to a finite number yields a finite number.

That's exactly right. So why would you ever need to stop adding digits?

How can you show that there is a reason why you would need to stop adding digits?

So to say that they have to have a finite number of digits seems to be a claim with no proof?

Every integer is a finite number. Hence, for any integer k, there is a finite number n, such that 10^n > | k |. Hence, any integer k can be written using no more than n digits and a leading negative sign, if necessary.

Where is your proof that n needs to be a finite number?

In short, your rebuttal takes the form of arguing that if all real numbers were integers, then they, like the integers, would also be countable.

Or you could put this the other way around. My argument basically states that since there can be no such thing as a largest integer then the set of integers must necessarily be just as endless as any set of any other numbers that are also endless.

Thus an endless set is "cardinally equivalent" to any other endless set, (if you want to think of endless as being a "Cardinal Quality")

And that is indeed my position.

You are off track here entire. I never claimed the the real numbers are countable. I simply claimed that Cantor's Diagonalization Proof is flawed.

This is a minimization. In fact, you have claimed that Cantor was an idiot. You further claim that I am off track. My counter is that this is the track that you have built, and if it is off, we should look to the builder.

Ok, Cantor may not have been an idiot in general, but he was an idiot in terms of his failure to understanding that completed lists of numerals cannot be square. He totally blew it on this proof.

Moreover are you aware that Cantor was a fruitcake who claimed that God Himself had revealed all these things to Cantor?

I think a God would have known that a completed list of numerals cannot be square.

A God would have known that this "proof" is no proof at all.

I'm am not arguing that all real numbers need to be countable. However, I can actually show that they necessary have to be. But that is a whole other argument unrelated to the topic of this thread.

In anything other than the informal sense, you are not arguing anything. The integers (and all finite dimensional cartesian products of the integers — which includes the rationals) are countable. This has been proven — repeatedly, with multiple diverse proofs ranging from the abstract to direct construction.

That's not a problem. It also doesn't help Cantor's proof that the real numbers are NOT countable.

Nor can you show the real numbers are countable. Because they’re not. This also has been proven — repeatedly, with multiple diverse proofs ranging from the abstract to direct construction, such as the method of Cantor’s diagonal, considered by many, including myself, to be the most elegant.

But they are countable. Especially the list of Reals that Cantor was working with from 0 to 1.

Here you go.

0: 0.0
1: 0.1
2: 0.2
3: 0.3
4: 0.4
5: 0.5
.
.
.
4678: 0.4678
4679: 0.4679
4680: 0.4680
.
.
.
9823736252782899340467: 0.9823736252782899340467
9823736252782899340468: 0.9823736252782899340468
9823736252782899340469: 0.9823736252782899340469
9823736252782899340470: 0.9823736252782899340470
.
.
.
99999999.... : 0.99999999....


There I just put the integers in a one-to-one correspondence with the reals from 0 to 1.

Please tell me which real number I've missed.

My observations of the flaw in Cantor's proof has nothing at all to do with whether the real numbers can be placed in a one-to-one correspondence with the integers.

Per my earlier remarks, it is your observations which are flawed, in the ways previously enumerated.

Cantor's proof is flawed. It necessarily produces a square list. Therefore he can never claim that his list is complete. Therefore he can't claim that this phantom number he has created is NOT on a genuinely completed list.

First, Cantor places all real numbers between zero and one. There’s a subset argument that validates this, but also a 1-1 correspondence possible by mapping all real numbers (-∞, ∞) into the set (0,1) via the function f(x) = 1 / (1+e^x) with inverse function g(x) = ln(1/x - 1). To complete the validation, we choose to exclude equivalent representations of real numbers that end in terminal repeating nines.

On the other hand, your argument is restricted to integers, and more, to positive integers, and more, their representations are arbitrarily chopped at the decimal point. Had you written all of your n-digit integers using 10^n - n digits beyond the decimal, your counter-example would fail. Your counter-example does not even acknowledge the issue of alternate digital representations, let alone justify your choice of representation.

It doesn't matter that they are chopped at the decimal point. That's totally irrelevant. The decimal point is meaningless in this numerical representation. The decimal point only serves to hide the error, it doesn't justify the error.

It's a trick of numeral representation is has nothing to do with the actual idea of quantity that these numbers actually represent.

Specifically, beginning with a list of 1-digit positive integers:

0.000000000...
1.000000000...
2.000000000...
3.000000000...
4.000000000...
5.000000000...
6.000000000...
7.000000000...
8.000000000...
9.000000000...

You chop to create the list:

0
1
2
3
4
5
6
7
8
9

You look at the first element: 0
Construct a contrary element: 5

Chop the list again:

0

And "observe" that 5 is not on the list! Your maimed creation is maimed. Well done, sir.

This doesn't apply to me because I'm not claiming that this is a valid way to prove anything.

On the contrary, I'm arguing that it's an invalid method of trying to map quantities. It's a numerical illusion that has more to do with numeral and numeral representation of quantity than it has to do with the actual idea of quantity.

In fact, mathematicians often loose sight of the idea of quantity and get bogged down in numeral representations instead which can often lead to problems.

Keeping an eye on the difference between numeral representations and the actual concepts of quantity is quite important.

All I'm saying is that Cantors Proof is flawed. Even if the conclusion implied by his so-called proof is correct, his proof is still flawed and not a valid proof at all.

Just because you happen to come up with the right answer doesn't mean that the proof you used was correct.

But the conclusion too could be wrong. Just because irrational numbers can't be expressed neatly as a ratio of integers doesn't automatically mean that they can't be placed into a one-to-one correspondence with the integers. Where is there any logical reasoning to support such a conclusion?

I have no need to place the "Real Numbers" in a direct correspondence with the integers. But that's beside the point.

I"m addressing the validity of Cantor's proof here. I'm not arguing about whether or not the reals can be placed in a one-to-one correspondence with the integers.

That is a totally different argument.

Whatever.

As ever, Jesse

Well, I'm only arguing about the specifics of Cantor's proof here. To get side-tracked into other arguments would to be off topic. Those arguments would not justify Cantor's proof. Even if the conclusion of Cantor's proof could be show to be accidentally and coincidentally true, that still wouldn't justify his proof.

[Divine Insight]

I would even say that it's basically impossible to define a meaningful irrational quantity that is not based on a self-referenced situation.

Dear DI,

The diagonal length of a unit square is an irrational quantity.

As ever, Jesse

And that's based on a self-referenced geometric situation just as I had said.

So there you go.

I can also show that it's always based on a self-referenced situation no matter how it is defined.

If you simply define it as the "Square Root of Two", then it's nothing more than that number that when added to itself the same quantity of times that it represents, it equals 2.

That's a self-referenced situation as well.

It does not represent the cardinality of individual things in a collection.

Now you may claim that it does. You may claim that it represents a collection of individual units of distance. But a unit of distance is arbitrary. It's a totally abstract notion that is not physical. You cannot produce an absolute "unit of distance" physically that is not permitted to change.

Moreover, you can PUSH the irrationality of a this geometrical figure around thus proving that it is a property of the self-reference of the geometrical construction.

In other words, I can easily make the diagonal of a square rational. Just let it be 1.

What happened? I just rationalized the diagonal of a square!!!!

Well, what happened is that in order for that to be true, the sides of the square must now have an irrational length.

Thus I have just proven that the irrationality between the sides and the diagonal of a square is a "Self-referenced" situation that can even be pushed around at will by my free choice to arbitrarily define what I choose to use as a UNIT of distance.

The irrationally associated with a square is a self-referenced geometric property of the square. It's not an absolute quantitative property of the diagonal.

It's not a "Cardinal Number". It's a relative relationship that is dependent upon a self-referenced situation.

Just as I had stated it must be.

I had said, "I would even say that it's basically impossible to define a meaningful irrational quantity that is not based on a self-referenced situation."

And you just gave me an example of an irrational number that is based on a self-referenced situation.

So I'm still waiting,....

[lao tzu]

I would even say that it's basically impossible to define a meaningful irrational quantity that is not based on a self-referenced situation.

Dear DI,

The diagonal length of a unit square is an irrational quantity.

As ever, Jesse

And that's based on a self-referenced geometric situation just as I had said.

No, it's not.

[keithprosser3]

0.23234792340340584359834502408234098734953734535930087..... and so on.

The digits are entirely random, produced by rattling my numeric key pad.

I haven't thought it through myself, but I don't see any self-referencing going on at first glance. If DI demolishes that irrational number, there are plenty more. There are after all more irrational numbers than there are rational numbers, so we could be at this for some time!

[lao tzu]

Dear DI,

No, I said, “all of your numbers … are represented by a finite number of digits.� As indeed they are.

They don't need to be. I only did that for the sake of brevity in my presentation.

Dear DI,

Yes, they do.

I could have used integers that are infinitely long just as easily.

No, you couldn't. Whatever such an object might be, it's not an integer.

You'll need to show a proof of this please.

I just did.

I don't need to.

No, you don't. But without one, you've little left but assertion.

All I need to do is start it, then you need to tell me why you need to truncate it.

Here's my infinitely long integer,...

5672356378262517823635...

It's not an integer. In fact, it's not even a number.

Continue it on for as long as you like, and when you get to a point where you NEED to stop please give me a call and explain to me why you had to stop.

That's exactly right. So why would you ever need to stop adding digits?

How can you show that there is a reason why you would need to stop adding digits?

You don't. But unless you do, it's not an integer.

More, there's no way to define arithmetic operations on the object, no way to order it, and hence, it's not a number at all, merely a string of characters.

So to say that they have to have a finite number of digits seems to be a claim with no proof?

Every integer is a finite number. Hence, for any integer k, there is a finite number n, such that 10^n > | k |. Hence, any integer k can be written using no more than n digits and a leading negative sign, if necessary.

Where is your proof that n needs to be a finite number?

It's a direct implication from the fact that all integers are finite, as they are defined as the set of counting numbers, their additive inverses, and 0.

In short, your rebuttal takes the form of arguing that if all real numbers were integers, then they, like the integers, would also be countable.

Or you could put this the other way around. My argument basically states that since there can be no such thing as a largest integer then the set of integers must necessarily be just as endless as any set of any other numbers that are also endless.

Thus an endless set is "cardinally equivalent" to any other endless set, (if you want to think of endless as being a "Cardinal Quality")

And that is indeed my position.

On the contrary, not only are there distinct infinite cardinals, but they form an endless chain. As a simple exercise, consider the size of the set of all integers, that is, the smallest infinite cardinal.

Now consider the set of all subsets of the counting numbers. Representing the real numbers in [0,1) in binary, it's trivial to construct a 1-1 correspondence between these subsets and the real numbers by placing a "1" at the digit representing the 10^-k place if k is in the subset and a 0 there otherwise.

While it's trivial to injectively map the singleton subsets from this set of subsets back to the counting numbers, it is not possible to construct a surjective map from the counting numbers back onto this set of its subsets. In fact, a similar diagonal argument can be used to display a subset which was not included.

This argument can be extended recursively using any transfinite cardinal, and hence shows not merely that there are distinct infinite cardinals but that there is no largest infinite cardinal.

DI, this is settled math. You're tilting at windmills here.

Ok, Cantor may not have been an idiot in general, but he was an idiot in terms of his failure to understanding that completed lists of numerals cannot be square. He totally blew it on this proof.

Moreover are you aware that Cantor was a fruitcake who claimed that God Himself had revealed all these things to Cantor?

I think a God would have known that a completed list of numerals cannot be square.

A God would have known that this "proof" is no proof at all.

Please refrain from aspersions cast on the dead you would refrain from making on the living.

On the contrary, if there is a god, and that god is omniscient, and that god created Cantor, then that god did indeed reveal these things to Cantor. And yes, that god also made him a "fruitcake." It seems that god is a bit of a joker, handing the best fruits to his nuts. You should read some of Newton's excursions into mysticism sometime.

Of course, I don't believe in any gods, so I agree with you in spirit, though perhaps only ironically. No god, and certainly not one that can't distinguish one from three, inspired Cantor.

Also, Cantor's list is not square. Nor is the list "completed." Repeating these statements will only continue to mislead you. The list is ordered and has a diagonal. That's all, and all that's needed.

That's not a problem. It also doesn't help Cantor's proof that the real numbers are NOT countable.

I've personally found great benefit to a thing being true when trying to construct a proof, but I do understand your point. Still, as the issue here is with your understanding of Cantor's proof, and not his proof itself, all of these objections are irrelevant.

But they are countable. Especially the list of Reals that Cantor was working with from 0 to 1.

Here you go.

0: 0.0
1: 0.1
2: 0.2
3: 0.3
4: 0.4
5: 0.5
.
.
.
4678: 0.4678
4679: 0.4679
4680: 0.4680
.
.
.
9823736252782899340467: 0.9823736252782899340467
9823736252782899340468: 0.9823736252782899340468
9823736252782899340469: 0.9823736252782899340469
9823736252782899340470: 0.9823736252782899340470
.
.
.
99999999.... : 0.99999999....


There I just put the integers in a one-to-one correspondence with the reals from 0 to 1.

Please tell me which real number I've missed.

There are an uncountably infinite number of real numbers missing from your list, but we don't need to go beyond the rationals to find one.

You missed 1/3.

Cantor's proof is flawed. It necessarily produces a square list. Therefore he can never claim that his list is complete. Therefore he can't claim that this phantom number he has created is NOT on a genuinely completed list.

No, it's not. No, it doesn't. And again, whatever.

Regards, J