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.

[Divine Insight]

Dear DI,

Watch carefully:

1
2
3
33
333
3333
33333
...

Please note your newly revised "counting" algorithm contradicts your previous counting algorithm, and now presents itself as a set of counting numbers that never arrives at 4.

What in the world are you talking about?

I'm not using any newly revised "counting" algorithm.

Can you count up to 33?

Do you want me to right it out for you entirely?

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, etc., until you reach 33.

The number 4 is certainly in the counting algorithm. As are all the other integers between 0 and 33.

Where did I ever suggest that I was proposing a new way to count?

And if you can count to 33, then you can probably count to 333, and then to 3333, and so on. In other words you can produce an integer of arbitrary length consisting entirely of 3's in decimal or tens notation. And there is no reason why you should ever be limited. What reason would you ever have for needing to stop at any particular string of 3's.

Therefore 3333333333333333... forever, is a valid integer.

There is no known reason why you should ever reach a maximum limit where you need to stop.

The entire argument was absurd, as it's been known for a century now that the integers cannot be made to line up 1-1 with the real numbers. Redefining the integers by inserting enough other objects to bring them up to size ... well, that's cheating. It didn't work, either, even for the rational numbers. You've left yourself no way to handle rationals that repeat after 2 digits, or 3 digits, and so on. Not to mention the untimely demise you've perpetrated upon the counting numbers beyond 4, which must now pine away forsaken.

We already know that the rational numbers can be put into a one-to-one correspondence with the integers. Therefore matching up endlessly repeating decimals that go on forever with the integers has already been done and is already accepted in modern mathematics.

Therefore the number 0.33333333333333333333... Most certainly can be put into a one-to-one correspondence with the integers. That's a given even in modern mathematics already.

And after that much is done, there remains the task of matching up all of the irrationals.

That's not a problem at all.

If you can match up a number like 0.33333333333333333... with an integer, then there's no reason why you can't match up a number like 3.14159265358979... with an integer too.

After all 314159265358979 is certainly a valid integer, and that looks like a good match right there. In fact, you can find a match for any irrational number that has been truncated at any point because the rationals have already been matched up with the integers.

So the only time you would get into trouble is if you actually allow the irrational number to run on to infinity, but even then it's questionable of precisely when you would actually run into trouble. If you stop at any point you've got a truncated decimal and therefore you have a valid rational number. So the only time you could claim to not be able to match up the irrational number is if you never stopped calculating it. But there would never be a point during that calculation when the number you have at the moment is not a rational number at that point in time.

This whole problem stems precisely from Georg Cantor's absurd idea that you can treat an endless process as though you could finish it.

As there are ways to match the integers with the rationals without creating ad hoc redefinitions, and you haven't managed even that much — note there are algebraic numbers (roots of polynomials over the rationals) in the queue as well — it's time to give up, and just perhaps acknowledge that mathematicians do know what we're doing. Cantor's proof does actually work using the definitions we use, and if it no longer works after you've changed the definitions, that doesn't make Cantor an idiot or create leave for you to speak of his work as nonsense.

Cantor's proof fails even with the standard acceptable definitions of mathematics.

Cantor's list is square. Period.

I've already shown, using totally valid mathematical truths, that a square numerical list of numbers can never be complete. It's extremely impossible using the decimal or tens system. And it's even impossible if you go all they down to binary, the simplest possible numerical system we have.

No complete list of numerals can ever be square.

That's a mathematical truth that holds no matter what.

So Cantor's proof if invalid. Just because he is producing a number that's not on his square list doesn't prove anything. A square list cannot be a complete list. So the fact that he has show that he can create a number that's not on his list only prove what we already know. Any complete list of numbers must be highly rectangular and far taller than it is wide.

That's all he has show. A trivial fact that we already know. Or at least I know. If mathematicians can't even figure out that their numerical system of notation doesn't loan itself to complete lists of numbers that are square, then they just aren't paying attention to their own tools.

They should have recognized the fallacy of Cantor's proof right off the bat. The fact that they have been worshiping it for over a century is truly amazing and only shows that they aren't even paying attention.

That last is really annoying. He's dead. Don't be taking cheap shots at him.

Hey, he was in error. Dead or not. Besides, he was a nut case. He told people that God was telling him this stuff.

And besides the fact that he's dead doesn't mean that his methods can't be challenged. Henri Poincare had this to say about Cantor's work, "His transfinite set theory is a disease from which the mathematical community will someday be cured."

Those words are going to come back and haunt the mathematical community at some point because Poincare was right.

There are reasons why we use the definitions we do, and why those definitions are fixed. For one thing, it gives scads of us, working together, the chance to one-up each other by finding the errors in each others arguments, arguments that would never arise were we redefining common terms in the middle of a discussion.

Who is asking anyone to redefine anything in common terms in the middle of a discussion?

I'm challenging the status quo of modern mathematics. I'm saying that the mathematical community made a "wrong turn" when they chose to treat irrational numbers as though they are cardinal quantities.

This may have actually happened unofficially thousands of years ago, but it wasn't formalized until the introduction of Cantor's Empty Set theory. And now it has become the basis of modern day mathematics.

On your path lies mayhem and madness! Turn back, I say, turn back! Think of the children!

I fear no evil. Besides it's in the favor of the mathematical community to learn of the truth. As long as they continue down the path they are on they will never understand the true nature of irrational relationships.

So I'm only trying to help.

This is why the set of Integers is infinite (i.e. endless).

This process clearly has a predecessor as you can always count up to where your dots begin.

And the process clearly always as a successor because you can always continue counting beyond any point where you have temporarily stopped.

This is precisely why the set of Integers is infinite to being with.

We already have an infinite set. There is no reason to pretend that we need a larger infinity.

Infinity = endless. Period.

There is no such thing as larger and smaller infinities. That's Cantor's nonsense.

In regards to whatever redefinition of infinity you might have in mind, please be advised that we already own that word, and have owned it since before you were born. It is not your word. Go find one of your own.

I thought mathematicians were interested in truth?

That's a rumor I always heard.

But more directly, we need larger infinities because there are sets we encounter naturally in our investigations with members which cannot be encompassed merely by counting up to infinity once. They require us to to do so again and again, an infinite number of times, in order to encounter all of their members. Their members can't be counted. This surprising result should be noted, and to do so with elegance, there are the uncountably infinite cardinals.

As ever, Jesse

You are absolutely correct. With their current formalism that's what they are stuck with.

But that's because they are treating zero as a number and an empty set as a valid unity of cardinal quantity. That is what is causing all of their multiple infinity problems.

If they went back and redefined zero to be the absence of a set (and therefore the absence of quantity) and redefined the all important number one correctly, then they would no longer have these silly problems of multiple infinities.

They would also understand why the irrational relationship are the way they are. And they might even have an actual insight into why complementarity must exist at some level of reality.

Right now they are totally clueless to all of these things.

How many mathematicians do you know that are aware of the fact that all meaningful irrational qualities arise from self-referenced situations?

Probably none. Because their formalism, as it is currently designed, doesn't expose this truth.

As long as they continue to treat irrational numbers as though they are cardinal properties of collections of individual things they will never see the true nature of what's actually going on. They will just think that irrationality seems to be a weird mystery.

Instead they could actually see precisely what's causing it and understand precisely what it is revealing to them about the real world.

But then again, I guess mathematicians aren't scientists so they have no clue (or even care) what the real world might be capable of revealing to them if they simply pay attention. Instead they seem to prefer getting lost in their own arbitrary whimsical definitions.

[lao tzu]

Dear DI,

Watch carefully:

1
2
3
33
333
3333
33333
...

Please note your newly revised "counting" algorithm contradicts your previous counting algorithm, and now presents itself as a set of counting numbers that never arrives at 4.

What in the world are you talking about?

Dear DI,

I'm doing my best to extract meaning from what is otherwise an incoherent argument. When I noted that your newly minted string-quasi-whatever-integer 33333...on forever ... had no successor or predecessor, you demurred and produced a sequence, which I interpreted as an attempt to display predecessors and successors.

Of course, even that doesn't scan, but at least it was something to work with. As you're now adamantly claiming they were not predecessors and successors, you once again have none for your sqwi3s...forever.

And you are once again missing 1/3.

Can you count up to 33?

Do you want me to right it out for you entirely?

Please do, and go all the way up to sqwi3s...forever so we can see its predecessor and successor, if you don't mind.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, etc., until you reach 33.

Still no sqwi3s...forever.

The number 4 is certainly in the counting algorithm. As are all the other integers between 0 and 33.

But no sqwi3s...forever.

Where did I ever suggest that I was proposing a new way to count?

When you suggested sqwi3s...forever had a predecessor and successor.

And if you can count to 33, then you can probably count to 333, and then to 3333, and so on. In other words you can produce an integer of arbitrary length consisting entirely of 3's in decimal or tens notation. And there is no reason why you should ever be limited. What reason would you ever have for needing to stop at any particular string of 3's.

The fact that you do not need to stop does not imply you will reach Alpha Centauri B. Or sqwi3s...forever. In fact, your sqwi3s...forever will never be reached, because it has no predecessor.

Therefore 3333333333333333... forever, is a valid integer.

It's also cheating to employ a new meaning for therefore.

There is no known reason why you should ever reach a maximum limit where you need to stop.

Go on forever and you won't reach -1, either.

The entire argument was absurd, as it's been known for a century now that the integers cannot be made to line up 1-1 with the real numbers. Redefining the integers by inserting enough other objects to bring them up to size ... well, that's cheating. It didn't work, either, even for the rational numbers. You've left yourself no way to handle rationals that repeat after 2 digits, or 3 digits, and so on. Not to mention the untimely demise you've perpetrated upon the counting numbers beyond 4, which must now pine away forsaken.

We already know that the rational numbers can be put into a one-to-one correspondence with the integers. Therefore matching up endlessly repeating decimals that go on forever with the integers has already been done and is already accepted in modern mathematics.

Therefore the number 0.33333333333333333333... Most certainly can be put into a one-to-one correspondence with the integers. That's a given even in modern mathematics already.

Indeed it can, as can the algebraic extensions. Which doesn't mean you can do it. You certainly haven't, so far.

And after that much is done, there remains the task of matching up all of the irrationals.

That's not a problem at all.

You're going to redefine your way out of this issue, too?

If you can match up a number like 0.33333333333333333... with an integer, then there's no reason why you can't match up a number like 3.14159265358979... with an integer too.

After all 314159265358979 is certainly a valid integer, and that looks like a good match right there.

Bingo. So your solution on how to deal with the rationals is to redefine them as integers, and your solution on how to deal with the irrationals is to redefine them as rationals. DI, no. Just no. You are never going to overthrow a century's work on mathematics by redefining your way out of every crisis.

In fact, you can find a match for any irrational number that has been truncated at any point because the rationals have already been matched up with the integers.

So the only time you would get into trouble is if you actually allow the irrational number to run on to infinity, but even then it's questionable of precisely when you would actually run into trouble. If you stop at any point you've got a truncated decimal and therefore you have a valid rational number. So the only time you could claim to not be able to match up the irrational number is if you never stopped calculating it. But there would never be a point during that calculation when the number you have at the moment is not a rational number at that point in time.

This whole problem stems precisely from Georg Cantor's absurd idea that you can treat an endless process as though you could finish it.

You are in no position to speak of anyone else's argument as absurd. Arrogance is not argument. The whole problem is that you don't understand Cantor. There is no suggestion of "finishing" this process. In fact, we know we can't. Hence we use tools to make sure we don't miss any counting numbers. One of those tools, and one that's obviously needed, is to ensure a pathway to any counting number. If there is no pathway to sqwi3s...forever, you will not reach it. If there is such a pathway, for any counting number, then you need never finish.

As there are ways to match the integers with the rationals without creating ad hoc redefinitions, and you haven't managed even that much — note there are algebraic numbers (roots of polynomials over the rationals) in the queue as well — it's time to give up, and just perhaps acknowledge that mathematicians do know what we're doing. Cantor's proof does actually work using the definitions we use, and if it no longer works after you've changed the definitions, that doesn't make Cantor an idiot or create leave for you to speak of his work as nonsense.

Cantor's proof fails even with the standard acceptable definitions of mathematics.

Cantor's list is square. Period.

So once again, in even greater detail: A square is a geometric figure composed of four line segments. A line segment, DI, is a portion of a line between two fixed points. The points of any line are in 1-1 correspondence with the real numbers via the ruler axiom. Infinity is not a real number, and hence can not be a vertex of a line segment. And hence, a line segment cannot go on forever. Therefore a square can not go on forever.

Therefore, Cantor's construction is not a square. Period.

I've already shown, using totally valid mathematical truths, that a square numerical list of numbers can never be complete. It's extremely impossible using the decimal or tens system. And it's even impossible if you go all they down to binary, the simplest possible numerical system we have.

The simplest possible numerical system is unary, not binary. You can create it with notches on a stick.

No complete list of numerals can ever be square.

See above, re: what is a square.

That's a mathematical truth that holds no matter what.

Ice cream has no bones is a physical truth, and as relevant.

So Cantor's proof if invalid. Just because he is producing a number that's not on his square list doesn't prove anything. A square list cannot be a complete list. So the fact that he has show that he can create a number that's not on his list only prove what we already know. Any complete list of numbers must be highly rectangular and far taller than it is wide.

Instead of a Holiday Inn Express, for your next trip I recommend the Hilbert Hotel. Infinite rectangles can be scaled out as wide or tall as you like.

de = e, for e an infinite cardinal and any cardinal d < e.

That's all he has show. A trivial fact that we already know. Or at least I know. If mathematicians can't even figure out that their numerical system of notation doesn't loan itself to complete lists of numbers that are square, then they just aren't paying attention to their own tools.

They should have recognized the fallacy of Cantor's proof right off the bat. The fact that they have been worshiping it for over a century is truly amazing and only shows that they aren't even paying attention.

We don't worship it. But even if we did, how can worshiping something you don't understand for a century be a point of ridicule for a member of a group engaged in precisely the same activity for nearly 2000 years?

That last is really annoying. He's dead. Don't be taking cheap shots at him.

Hey, he was in error. Dead or not. Besides, he was a nut case.

DI, it is highly inappropriate to use epithets to make fun of the mentally ill.

He told people that God was telling him this stuff.

So did Jesus.

And besides the fact that he's dead doesn't mean that his methods can't be challenged. Henri Poincare had this to say about Cantor's work, "His transfinite set theory is a disease from which the mathematical community will someday be cured."

Those words are going to come back and haunt the mathematical community at some point because Poincare was right.

Yes, challenge his work and his methods. No, do not be obnoxious about it.

There are reasons why we use the definitions we do, and why those definitions are fixed. For one thing, it gives scads of us, working together, the chance to one-up each other by finding the errors in each others arguments, arguments that would never arise were we redefining common terms in the middle of a discussion.

Who is asking anyone to redefine anything in common terms in the middle of a discussion?

You, with your sqwi3s...forever "integer." With your infinite-length-edged squares. With your counting numbers that can't be used for counting.

And admittedly so, only a few posts ago in your response to micatala.

I'm challenging the status quo of modern mathematics. I'm saying that the mathematical community made a "wrong turn" when they chose to treat irrational numbers as though they are cardinal quantities.

This may have actually happened unofficially thousands of years ago, but it wasn't formalized until the introduction of Cantor's Empty Set theory. And now it has become the basis of modern day mathematics.

You've certainly got your work cut out for you then. I know of half a dozen independent proofs of the fact the real numbers are uncountable, just off the top of my head, even if you were able to eliminate Cantor's.

On your path lies mayhem and madness! Turn back, I say, turn back! Think of the children!

I fear no evil. Besides it's in the favor of the mathematical community to learn of the truth. As long as they continue down the path they are on they will never understand the true nature of irrational relationships.

So I'm only trying to help.

So am I, DI, so am I.

I thought mathematicians were interested in truth?

That's a rumor I always heard.

Yes, and we found it, and discovered it wasn't enough. But that's an entirely different discussion.

But more directly, we need larger infinities because there are sets we encounter naturally in our investigations with members which cannot be encompassed merely by counting up to infinity once. They require us to to do so again and again, an infinite number of times, in order to encounter all of their members. Their members can't be counted. This surprising result should be noted, and to do so with elegance, there are the uncountably infinite cardinals.

As ever, Jesse

You are absolutely correct. With their current formalism that's what they are stuck with.

But that's because they are treating zero as a number and an empty set as a valid unity of cardinal quantity. That is what is causing all of their multiple infinity problems.

If they went back and redefined zero to be the absence of a set (and therefore the absence of quantity) and redefined the all important number one correctly, then they would no longer have these silly problems of multiple infinities.

Having done so, however, addition would lose its identity and subtraction would no longer exist. Multiplication would be decoupled at the distributive law. Arithmetic would end, and with it all technological progress. This would not be an improvement over the status quo.

Alternative formulations crop up from time to time, and even displace their predecessors on the odd occasion. But if they cannot accomplish at least as much as the current formulation, they are rejected, for cause.

They would also understand why the irrational relationship are the way they are. And they might even have an actual insight into why complementarity must exist at some level of reality.

Right now they are totally clueless to all of these things.

There are endless examples of mathematicians making use of complementarity.

How many mathematicians do you know that are aware of the fact that all meaningful irrational qualities arise from self-referenced situations?

Probably none. Because their formalism, as it is currently designed, doesn't expose this truth.

And here again, to make sense of what you're saying, it's necessary to redefine "self-referential" away from its common usage, and "meaningful" as well.. There's a geometric example above. You've been given an algebraic example earlier.

As long as they continue to treat irrational numbers as though they are cardinal properties of collections of individual things they will never see the true nature of what's actually going on. They will just think that irrationality seems to be a weird mystery.

Instead they could actually see precisely what's causing it and understand precisely what it is revealing to them about the real world.

But then again, I guess mathematicians aren't scientists so they have no clue (or even care) what the real world might be capable of revealing to them if they simply pay attention. Instead they seem to prefer getting lost in their own arbitrary whimsical definitions.

No, mathematics is not science, but science would not exist without it. And no, mathematical abstractions are not dependent on any real world phenomena, but history shows that whenever one of these mathematical abstractions can be formalized, scientists will eventually find a way to use that formalized abstraction to model something in the real world.

Like George Boole's algebra, for example. Without which, the computer upon which you're presently typing would not exist, as its logic circuits are the product of George's algebra.

As ever, Jesse

[olavisjo]

.

Please do, and go all the way up to sqwi3s...forever so we can see its predecessor and successor, if you don't mind.

Let X = sqwi3s...forever
Then X - 1 is it's predecessor,
And X + 1 is the successor.

If you understand Hilbert’s Hotel, you will understand where Georg Cantor went wrong.

“If Hilbert’s Hotel could exist, it would have to have a sign posted outside saying, ‘No Vacancy. Guests Welcome’.�

Read more: http://www.reasonablefaith.org/defender ... z2iozLbewX

Cantors diagonal argument is simply accommodating a new guest in a hotel with no vacancies.

[Divine Insight]

Please do, and go all the way up to sqwi3s...forever so we can see its predecessor and successor, if you don't mind.

You are falling into the very same trap that Georg Cantor fell into. You are thinking in terms of "completed infinity".

That's the problem right there. You'll never see my pov until you can recognize that infinity is an endless process that can never be completed nor treated as a finite cardinal property of a set.

Now if you try to claim that Cantor (and all of modern mathematics) isn't treating infinity as a finite cardinal property of a set, then you're wrong, because that's precisely what they need to do if they are going to proclaim that there are larger and smaller infinite "cardinality".

That's the very idea right there that I am saying is a flawed and illogical idea. Yet this is the very same flawed idea that you are demanding that I produce for you. You want me to produce a "Completed" infinite list of 3's.

It can't be done. The three dots at the end of the string of digits is there precisely to indicate that this process is endless.

So for you do even demand that I produce this as a finite finished number is absurd. You're not understanding the problem with Cantor's completed infinities.

No, mathematics is not science, but science would not exist without it. And no, mathematical abstractions are not dependent on any real world phenomena, but history shows that whenever one of these mathematical abstractions can be formalized, scientists will eventually find a way to use that formalized abstraction to model something in the real world.

Like George Boole's algebra, for example. Without which, the computer upon which you're presently typing would not exist, as its logic circuits are the product of George's algebra.

As ever, Jesse

The mere fact that Boolean algebra is considered to be "mathematics" is yet another wrong turn that the mathematical community took. Instead of lumping Boolean logic in with the concept of number (which is a quantitative idea based on the cardinality of sets) then should have instantly recognized that Boolean Algebra is a different kind of logic altogether based on a totally different concept. Boolean Algebra has nothing at all to do with ideas of quantity or cardinal properties of collections of individual things. It's a totally different kind of logic using gates with operation such as AND, OR, XOR, etc.

This was placed under the "Umbrella of Mathematics" for Academic reasons only.

Mathematicians tended to work with logic. The logicians themselves where already lost in philosophy without worrying about real world application. The mathematicians, although not scientists themselves, worked closely with the scientists and engineers.

So when Boolean Algebra came along the "Mathematics Departments" in academia were the people who naturally gravitated toward taking on this new logical system and thus it became recognized as "Just more Mathematics".

This is the problem with mathematics today. It will take on any kind of logical system without truly recognizing and differentiating it from the basic idea of quantity, arithmetic, and cardinality that was the foundation of mathematical pursuits in the first place.

Mathematics has lost it's focus, and has just become a hodgepodge of anything that falls under the umbrella of "Logic". In fact, many people today tend to think that there is no difference between "Logic" and "Mathematics" precisely for this reason.

Yes, Boolean Algebra is great. It's very useful, and we need it. But we don't need to label it "mathematics" that was our mistake. And like I say, the only reason it historically worked out that way is because the mathematics departments in educational institutions were best suited to embrace this type of logical analysis. So rather than starting up a whole new discipline and department named "Boolean Logic", they just shoved it under the umbrella of mathematics.

And this is yet another problem with modern day "mathematics". It takes on too many unrelated logical concepts and that only serve to obscure things rather than making things clearer.

Boolean Algebra has nothing to do with ideas of quantity or the cardinality of sets. Sure, it can be used to create logic gates and systems that can then do arithmetical computations, but still, it's an entirely different logical formalism in it's own right.

So it's not that these fields of study should not be pursued. It's just that they shouldn't all just arbitrarily be continually shoved under the umbrella of mathematics.

The very term "Mathematics" has come to be nothing more than another word for "Logic". They take on anything that has to do with logic whether it has anything to do with the idea of number, or quantity, or cardinality at all.

Mathematics is a "lost" discipline for this very reason. They have lost their focus.

Boolean Algebra, and the Mathematics of Numbers as concepts of quantity, are two entirely different things. But now we call them both "Mathematics".

[Divine Insight]

Cantors diagonal argument is simply accommodating a new guest in a hotel with no vacancies.

But it's not. Cantor is claiming to be turning away at least ONE guest! The number he creates that "cannot be on his list" (i.e. cannot be in his Hilbert Hotel)

But Cantor's hotel is square!

And a square hotel cannot hold the infinity of the natural numbers, much less any new guests we might wish to take in.

I've already proven that any complete list of numbers represented by our system of numeral representation must be highly rectangular.

A Hilbert Hotel constructed as a representation of numerical lists must be a skyscraper that is far taller than it is wide.

It cannot be a square building.

Cantor's diagonal hotel is perfectly square. It cannot hold a complete list of numerals.

Cantors needs to REJECT new guests, at least the ONE that he has constructed diagonally cannot be admitted into his hotel.

And the reason is because his hotel is square, and completed lists of numbers represented in numerical notation cannot be square.

All Cantor proved here is what we already knew (or at least should have known). Completed lists of numbers written out in numerical notation cannot be square.

Clearly Cantor was unaware of this fact, for if he were aware of it he would see the folly of his own proof.

Clearly, also, even mathematicians don't appear to be aware of the limitations of their own numerical notational systems. Because they have accepted and embraced Cantor's diagonal "proof" for well over a century now, and no one has caught this trivial all-important error yet. Except for me, and I'm not even a mathematician.

The system of numerical notation that Cantor is using in his diagonal argument does not loan itself to his conclusion.

"Complete lists" of numbers cannot be represented in numerical notation in a square format.

Therefore, the fact that Cantor can prove that some number must be missing from his list hasn't proven anything other than this trivial fact about the limitations of numerical representations of number.

He can't do it using numerals in base ten by a long shot. And his method can't even be done in binary, because even in binary notation completed lists of numbers are necessarily at least twice as tall as they are wide for even just 2 digits. Add more digits and the situation become increasingly worse. And you can't do any better than binary. There is nothing lower to work with. So it can't be done in any system of numerical notation.

Cantor's entire method is totally useless for proving anything other than square lists of numerals are necessarily incomplete. But we already knew this, (or at least we should have known this if we are paying attention to the limitations of our systems of notation)

[olavisjo]

.

Cantors diagonal argument is simply accommodating a new guest in a hotel with no vacancies.

But it's not. Cantor is claiming to be turning away at least ONE guest! The number he creates that "cannot be on his list" (i.e. cannot be in his Hilbert Hotel)

I was just saying that both hotels are full, yet it is possible to find and accommodate a new guest in both hotels. It is possible to accommodate all the real numbers in the natural hotel, because you can continue checking in any amount of real number guests.

But you are right about the shape of the hotel.

The width to height will be n/2ⁿ, and when n approaches infinity the ratio will approach zero. So it would be impossible to find a diagonal to form a new guest that is not already on the list. Georg Cantor simply slips in the fallacious assumption that an infinite width is equal to an infinite height.

[keithprosser3]

Sorry to be away so long - I've been writing down an infinite square of numbers for calculating is diagonal and I only just finished.

Therefore 3333333333333333... forever, is a valid integer.

I'll just say 33... if I may.

Its not clear if 33... divided by 33... is 1, 10, 100, 0.1.

33... isn't a valid integer - it is several valid integers; an infinite number of valid integers in fact.

btw, I have discovered today many mathematicians that reject completed infinities so DI is not without followers in the mathematical community. There are even mathematicians who reject numbers that are simply ridiculously large. I suppose in any other discipline they'd be considered cranks, but to a first approximation all mathematicians are cranks. It seems practically every famous mathematician was a fruitcake.

[olavisjo]

Sorry to be away so long - I've been writing down an infinite square of numbers for calculating is diagonal and I only just finished.

Therefore 3333333333333333... forever, is a valid integer.

I'll just say 33... if I may.

Its not clear if 33... divided by 33... is 1, 10, 100, 0.1.

33... isn't a valid integer - it is several valid integers; an infinite number of valid integers in fact.

btw, I have discovered today many mathematicians that reject completed infinities so DI is not without followers in the mathematical community. There are even mathematicians who reject numbers that are simply ridiculously large. I suppose in any other discipline they'd be considered cranks, but to a first approximation all mathematicians are cranks. It seems practically every famous mathematician was a fruitcake.

It is an infinite integer, so these operations are not valid.

0 × ∞
0 × -∞
∞ + -∞
∞ - ∞
∞ / ∞
∞⁰
1^∞

[Divine Insight]

btw, I have discovered today many mathematicians that reject completed infinities so DI is not without followers in the mathematical community.

Thank you.

Although, I'm sure they aren't following me.

I would love to give a presentation to a large group of serious mathematicians though. More so on the topic of irrational relationships than on Cantor's diagonalization folly.

Although, to fully understand "irrational numbers" I would need to address the cardinal definition of number as a quantitative property of a collection of individual objects. In the quantitative individuality of these objects is actually quite important. And this idea become lost in a set theory where zero is defined as the empty set, and the quantitative value of individuality becomes the abstract definition of One as the set containing the empty set.

The reason this is problematic is because the empty set itself has no quantitative property of individuality. It actually has a qualitative property of individuality that is quite different from a quantitative property and this causes problems down the road.

In any case, cardinal numbers, like the integers and rationals are well within this cardinal definition of number. Negative numbers and imaginary numbers can also be justified via cardinal definitions as long as their signs (negative and imaginary) are recognized to be relative properties between sets, rather than absolute cardinal properties of a stand alone set.

However, there is no valid way to describe an irrational relationship as a cardinal quantity. We already know this, this is why these "numbers" have been classified as being irrational in the first place. They can not be described even by ratios of cardinal numbers.

One may argue in geometry that these irrational "quantities" define a set of unit lengths, and that qualifies as being a cardinal property of a set. However, this runs into problems when we try to define precisely what we mean by the quantitative property of a unit of distance.

It's far better to understand these relationship as having arises from a more complicated situation where self-reference is involved. Then we can better understand why these relative relationships are not cardinal numbers and why they cannot be expressed as such.

Then the mystery goes away and we have a far better understanding of the cause of irrational relationships.

Moreover, every time we come across an irrational number we can instantly realize that there must be a self-referenced situation going on and we can pinpoint it and understand the cause of it.

Insofar as I know there are no mathematicians who are aware that irrational relationships always arise from self-referenced situations. They aren't even aware of this trivial fact.

And the reason they aren't aware of it is because they don't yet understand irrational relationships. They just think they are totally weird and unexplainable cardinal properties of some sort of collections that can't be justified.

This is the folly of even thinking of them as being cardinal numbers in the first place. That's not what they are.

~~~~

So I would love to sit down with bunch of potentially interested mathematicians, and give a talk on this topic. Mathematicians who reject completed infinities would certainly be more promising that mathematicians who have accepted Cantor's baloney.

I have no desire to argue with mathematicians who are hell bent on defending Cantor's multiple sized infinities. If they are that determined to believe in that, then I'll just stay home and save myself the time and effort and they can believe in whatever they like.

I have no desire to argue with a bunch of Cantor Fans.

[olavisjo]

.

Take away your decimal point and you can do the same thing with integers.

1) 1
2) 12
3) 167
4) 5785


Ok, there's the beginning of my list of integers.

I need to make the integers on the right an extra digit long in every step just to facilitate the diagonalization method of elimination.

So now I run my diagonal line down that list


1) [strike]1[/strike]
2) 1[strike]2[/strike]
3) 16[strike]7[/strike]
4) 578[strike]5[/strike]

Replace the following

1 = 2
2 = 3
7 = 8
5 = 6

My new number is 2386

Therefore I've created an integer that's not on my list. And if I keep this up to infinity I will ALWAYS have a NEW integer that's not on my list.

It's numerical trickery.

It's not numerical trickery.

You have demonstrated that the set of natural numbers is uncountable, and for the same reason that the set of real numbers is uncountable.

Therefore it is wrong to conclude that the cardinality of the set of real numbers is greater than the cardinality of the natural numbers.

You have successfully demonstrated that Cantor's Diagonalization Proof is Flawed.