# Cantor's Diagonalization Proof of the uncountability of the real numbers

pbuk
Gold Member
Yes my last sentence was not correct. What I should have said was:

Far from contradicting Cantor's proof, you can use Cantor's diagonal argument to construct a counter-example to show that your (andrewr's) assertion that you can list all of the real numbers using some algorithm is incorrect.

Leucippus, let me offer you a variant of Cantor's proof, that show that your consideration of "slope" is irrelevant:

{... snip for brevity ....}

we continue in this way.

now the "slope" is "even worse" the diagonal is not even as steep as before. nevertheless, no matter which n we pick, the diagonal number differs from the first n numbers in our list, so it differs from every number on our list (it differs from the k-th number in the 10k-1-th digit).
Why should I be impressed by this? You still haven't convinced me that you could ever reach the "bottom" of a list that would even contain all rational numbers.

This seems to be the point that everyone is not fully understanding.

I'm not the slightest bit worried about the list that you care "creating" as you are crossing numbers out. That list is totally meaningless.

Why?

Well it should be obvious.

Stop your diagonalization process at any point. You claim to now have a number that is not on the list above where you are working. So? Big deal.

What so you have at that point? What number have you just created thus far?

Well, clearly since you've stopped you have a truncated decimal number. I can absolutely guarantee that you are standing there holding a valid Rational number. In other words, you haven't created a "new number" that isn't already on the list Rational numbers.

All you've succeeded in doing it proving that the number you've created thus far is not on the list above where you are working. But you haven't demonstrated that it is a "new number" that can't already be on the list of Rational numbers.

In fact, it can easily be shown that this will always be the case.

Let say that you have gone out ten places and you have created the following "new number":

0.3854736394

That number cannot possibly be on your list above where you are working. But clearly as a truncated number it must necessarily already be on the list of rational number.

Moreover, no matter which numeral your process chooses for the next supposedly "New Number" I can guarantee that the number you create will be on the list of Rational Numbers.

This should be extremely ease to see.

How many digits do we have to chose from to use as our replacement digists? Well we have ten of them 0 thru 9. So let's try that and see if we can create a "New Number" that isn't already a valid Rational Number using this process:

Here's the number we're claiming is already a "New Number" that can't already be on the list of Rational Numbers:

0.3854736394

Granted this is cheating because I've stopped the process. But bear with me,....

What can we possible add to this number next? Well we can tack on anything between or including 0 thru 9. So lets do that,...

0.38547363940
0.38547363941
0.38547363942
0.38547363943
0.38547363944
0.38547363945
0.38547363946
0.38547363947
0.38547363948
0.38547363949

I've just created ten supposedly "New Numbers" that potentially could not be on the list above where I'm working. So big deal? All of these numbers (ever single last one of them) must necessarily be on a list of Rational numbers. In fact, you can take that even further and proclaim that there must necessarily be infinitely many Rational numbers that begin with any one of the numbers listed above and have a different sequence of numerals after that.

Take the original number we've created and just add any arbitrary digits to it that you care to chose on a whim and you've got yet another Rational number:

0.3854736394abcdefghijklmnopqrstuvwzyz etc. (potentially to infinity, what's going to stop you ever)

Chose any arbitrary digit from 0 to 9 for the letter variables above and you'll have a perfectly legitimate Rational number.

So at not point in this process can you claim to have created a "New Number" that can't be on the list of Rational Numbers. In fact, you can be absolutely guaranteed that you will always have a valid Rational Number at ALL TIMES in your process.

So what's Cantor's point? That if you continue this process out to "infinity" you will have somehow miraculously created a sequence of numerals that can't be Rationalized.

But wait? When would this occur? When could Cantor ever actually COMPLETE this infinite process in order to claim to have created a NEW NUMBER that isn't already on the list of Rational Numbers?

He hasn't proven anything at all. All he's done is rely on the "the definition" that we claim that real decimal expansion never quist going whereas a supposedly "Rational Number" must truncate.

But what sense is there truly to even claiming that a Rational Number "must truncate"?

What sense does that truly make? What would ever prevent you from adding yet another numeral digit to a string of numerals representing a Rational Number?

At the very BEST, all Cantor has done is claim that Real numbers must be expressed using infinite decimal expansions and all Rational Numbers must truncate. But that's not a proof of anything. It's just an assumption of an arbitrary man-made definition.

Unless Cantor can actually show that he can successfully "complete" this infinite process and actually have obtained a "new sequence of numerals" that can't already be on a list of Rational numbers, then he hasn't proven anything.

This is why I agree with Henri Poincare on this point. Pretending that an infinite process could be "completed" is an absurd idea to begin with and will always result in utterly meaningless or absurd results.

Cantor hasn't proven anything with this proof other than he's still clinging to the idea of a "completed infinity". He's trying to treat infinity as though it can be thought of as being finite.

It makes not sense for Cantor to even proclaim to have created a "new number" unless he can "complete" his process and show us this "new number" that he has created.

But he can NEVER do that because his process requires that it must never STOP, if it ever is said to STOP what does he have? He's standing there holding perfectly valid Rational Number.

So his so-called "proof" is nothing short of utter absurdity.

@ micromass: i disagree with the contention that if a proof can only be demonstrated graphically, it isn't valid. for what is a formal system, if not a few squiggles on paper? is that not graphical? i don't know about you, but i read most of my mathematics.

more to the point: many proofs in algebraic topology or category theory amount to: diagram-chasing. the diagram IS the proof, in some cases. there's no logical reason why "pictures" can't be made every bit as unambiguous and consistent as well-formed formulas. constructing a length with a compass and straight-edge IS a proof of constructibility (a concept which can be expressed purely algebraically).

there is many people who think that a mathematical result "isn't true" until it is proven formalizable in (first- or second-) order logic in a formal language that has encoded the axioms of Zermelo-Frankel set theory. this is unfortunate.
Well, I am in agreement with you on the, quite passionately actually.

In fact, I take precisely the opposite view of Micromass. If you can show me graphically, or physically something the totally defines mathematical formalism, then as far as I'm concerned it's the mathematical formalism that must then necessarily be wrong. Certainly not physical reality.

And currently we have no physical proof (or even reason to believe) that anything can be infinite much less that multiple cardinal sizes of infinities should exist.

Sigh...

Why should I be impressed by this? You still haven't convinced me that you could ever reach the "bottom" of a list that would even contain all rational numbers.
There is no bottom, that's the point. There is no end point.

This seems to be the point that everyone is not fully understanding.

I'm not the slightest bit worried about the list that you care "creating" as you are crossing numbers out. That list is totally meaningless.

Why?

Well it should be obvious.

Stop your diagonalization process at any point. You claim to now have a number that is not on the list above where you are working. So? Big deal.
You don't stop your diagonalization process. It continues for an infinite number of times.

What so you have at that point? What number have you just created thus far?

Well, clearly since you've stopped you have a truncated decimal number. I can absolutely guarantee that you are standing there holding a valid Rational number. In other words, you haven't created a "new number" that isn't already on the list Rational numbers.

All you've succeeded in doing it proving that the number you've created thus far is not on the list above where you are working. But you haven't demonstrated that it is a "new number" that can't already be on the list of Rational numbers.
You haven't demonstrated that because you can't. Clearly, if you stop your process then you will get a valid rational number. If you don't stop your process then you get a real number not on your list.

In fact, it can easily be shown that this will always be the case.

Let say that you have gone out ten places and you have created the following "new number":

0.3854736394

That number cannot possibly be on your list above where you are working. But clearly as a truncated number it must necessarily already be on the list of rational number.

Moreover, no matter which numeral your process chooses for the next supposedly "New Number" I can guarantee that the number you create will be on the list of Rational Numbers.

This should be extremely ease to see.

How many digits do we have to chose from to use as our replacement digists? Well we have ten of them 0 thru 9. So let's try that and see if we can create a "New Number" that isn't already a valid Rational Number using this process:

Here's the number we're claiming is already a "New Number" that can't already be on the list of Rational Numbers:

0.3854736394

Granted this is cheating because I've stopped the process. But bear with me,....
Yes. It is cheating. So don't stop the process. You have to continue it.

I'm not even reading the rest since it is so comically wrong. Instead of giving all these arguments, why not find an error in the formal proof??

And currently we have no physical proof (or even reason to believe) that anything can be infinite much less that multiple cardinal sizes of infinities should exist.
This is important. Infinite and cardinal sizes indeed do not exist in reality (I think). But this is COMPLETELY irrelevant. You have to see mathematics as a series of symbols which don't carry meaning. Mathematics is about manipulating those symbols. Whatever meaning the symbols have, is completely irrelevant. We interpret it as some kind of infinity. But this is simply an interpretation and not the theory itself.

Cantor asks us to consider a list of ALL the real numbers. He then constructs a real number not on that list, showing that ANY such list is necessarily incomplete. Therefore there can be no list of all real numbers.
Surely you meant to say, Cantor asks us to consider a list of ALL the "rational" numbers. He then constructs a real number not on that list.

But how can he claim to have ever created a number that is not already on the list of ALL rational numbers?

The way he can possibly claim to have done this if he does indeed take his process out finitely (without ever terminating it).

But if he never terminates it, then how can he claim to have "created" a new number?

It seems to me that all he's doing is trying to depend on an idea that "real" numbers must be infinite whilst rational numbers are thought of as somehow needing to terminate.

But in truth, why should a Rational number need to ever terminate?

What would cause a Rational number need to terminate? Aren't the Rational numbers themselves supposed to be infinite in size? (i.e. unbounded, unending)

All a Rational number is, is a fraction of Natural Numbers.

So we can write:

123/321

That's a rational number.

That's a rational number too.

Again a rational number.

In fact, is there any reason why we would need to stop this process?

Clearly there is no end to the Rational numbers we can create. That's why they are considered to be an infinite set of numbers.

So what sense does it even make to claim that a Rational Number should necessarily "terminate"?

We have no reason to even place this restriction on the numerators and denominators of the fractions that define them.

In short, we actual have no logical reason to claim that a Rational number must terminate.

That's just an arbitrary idea made up by mankind to try to make a distinction between a "Rational Number" and a "Real Number".

But do we truly even need a concept of "Real Numbers" at all?

I say no, we don't.

The very idea of "Real Numbers" is an erroneous idea that is not even required to describe physical reality. There is nothing in physical reality that even requires this concept.

pbuk
Gold Member
Leucippus you are still insisting that there are more rows than there are columns: do you think that there are flaws in each of these arguments that I have presented?

pbuk
Gold Member
Surely you meant to say, Cantor asks us to consider a list of ALL the "rational" numbers. He then constructs a real number not on that list.
NOOOOOO - Cantor asks us to construct a list of all the real numbers. He then constructs a real number not on that list, proving that such a list cannot exist.

Surely you meant to say, Cantor asks us to consider a list of ALL the "rational" numbers. He then constructs a real number not on that list.
No, Cantor asks us to consider a list of all REAL numbers.

But how can he claim to have ever created a number that is not already on the list of ALL rational numbers?
Take:

0.1
0.11
0.111
0.1111
0.11111
0.111111
...

then Cantor diagonalization yields 0.1111111111... which is not on this list. This is a simple example. Cantor diagonalization ALWAYS yields a number not on the list. Cantor wants you to look at a list of ALL real numbers and it will construct a number not on the list.

The way he can possibly claim to have done this if he does indeed take his process out finitely (without ever terminating it).

But if he never terminates it, then how can he claim to have "created" a new number?
The formal proof clarifies this. There is no question of "terminating". Cantor diagonalization is not a dynamical process.

It seems to me that all he's doing is trying to depend on an idea that "real" numbers must be infinite whilst rational numbers are thought of as somehow needing to terminate.

But in truth, why should a Rational number need to ever terminate?
It doesn't. 1/3=0.33333... never terminates.

What would cause a Rational number need to terminate? Aren't the Rational numbers themselves supposed to be infinite in size? (i.e. unbounded, unending)

All a Rational number is, is a fraction of Natural Numbers.

So we can write:

123/321

That's a rational number.

That's a rational number too.

Again a rational number.

In fact, is there any reason why we would need to stop this process?
What process?? Do you mean taking the limit?? The limit of rational numbers doesn't need to be rational.

Clearly there is no end to the Rational numbers we can create. That's why they are considered to be an infinite set of numbers.

So what sense does it even make to claim that a Rational Number should necessarily "terminate"?
It doesn't.

We have no reason to even place this restriction on the numerators and denominators of the fractions that define them.

In short, we actual have no logical reason to claim that a Rational number must terminate.
Agreed.

That's just an arbitrary idea made up by mankind to try to make a distinction between a "Rational Number" and a "Real Number".

But do we truly even need a concept of "Real Numbers" at all?

I say no, we don't.

The very idea of "Real Numbers" is an erroneous idea that is not even required to describe physical reality. There is nothing in physical reality that even requires this concept.
Take a a square with side 1m. What is the size of its diagonal? Is this rational?

Sigh...
You haven't demonstrated that because you can't. Clearly, if you stop your process then you will get a valid rational number. If you don't stop your process then you get a real number not on your list.
Talk about a comical statement. If you don't stop your process you'll never be able to claim have ever 'gotten' anything.

I'm telling you that Henri Poincare is right. The very idea that you can pretend to have completed an infinite process without stopping it is an utterly absurd and meaningless idea.

It's simply nonsense.

It makes no sense.

This is important. Infinite and cardinal sizes indeed do not exist in reality (I think). But this is COMPLETELY irrelevant.
As a physicist it's not irrelevant to me. As a physicist why should I even be remotely interested in a mathematical formalism that is built on band-aides and crutches that have absolutely nothing at all to do with the physical reality that I might be interested in studying?

Why should I apply a mathematical formalism to reality if that formalism has nothing to do with reality?

You have to see mathematics as a series of symbols which don't carry meaning. Mathematics is about manipulating those symbols. Whatever meaning the symbols have, is completely irrelevant. We interpret it as some kind of infinity. But this is simply an interpretation and not the theory itself.
Well, that's fine for someone who might be interested in totally abstract meaningless logical systems.

But as a physicist why should I be interested in playing such games?

I may as well play dominoes if I'm going to do that.

I'm more interesting in finding (or creating) a "mathematical formalism" that can actually be used to properly describe the true nature of reality.

pbuk
Gold Member
The very idea of "Real Numbers" is an erroneous idea that is not even required to describe physical reality. There is nothing in physical reality that even requires this concept.
If I want to put a diagonal brace in a frame with sides of 1 unit I require √2 which is not rational.

If I want to put a tyre on a wheel I will need ∏ which is not rational.

If I want to calculate how much 235U I have left in a rector after 1 month I need e which is not rational.

This thread is going off-topic. Therefore:

Moderator note: This thread will only deal with Cantor's diagonalization proof. The invalidity of infinity and real numbers will be considered off-topic and the post will be deleted. If you want to discuss this topic, create a new thread.

Furthermore, since informal arguments are clearly not convincing.

Moderator note: This thread will only deal with the rigorous version of Cantor's proof which is found in post 24 or in any serious set theory book.

NOOOOOO - Cantor asks us to construct a list of all the real numbers. He then constructs a real number not on that list, proving that such a list cannot exist.
Ok MrAnchovy. I get it now!

You've answered my question and solved the whole problem. I was indeed viewing the proof incorrectly.

Unfortunately it doesn't help.

Why does it not help?

Because it would be a totally futile and meaningless "proof". In fact, he would be supposedly 'proving' something that necessarily falls out of how he is defining a "Real Number".

He's simply demanding that a "Real Number" is necessarily an "infinite" decimal expansion by definition and assuming that a "Rational Number" cannot be described by an "infinite" decimal expansion.

Another problem is that his "proof" would obviously work for any numbers. Not just reals.

Think of it this way. Just take the decimal point in his proof in his proof away.

What do you have left? A list of natural numbers.

You can go down that list in precisely the same way and proof that even the Natural Numbers can't be put into a correspondence with themselves.

How so?

Well, let's try it.

Start our list with the arbitrary natural number of 1

1

We cross that out and replace it with any arbitrary digit (say 9) So we have a "new number" 9.

Lets continue, we'll need a two digit number to work on next so lets choose 23 our list so far looks as follows:

1
23

Replace the 3 with 5. Our newly constructed number is 95.

Move down another row pick another arbitrary natural number, we'll need one that is at least 3 digits wide or more, lets pick 3476 just for fun.

Here's our new list

1
23
3476

Now we need to replace the 7 with something other than 7, let's choose 8.

Our new number is now 958, and so on.

So we currently have a list of natural numbers

1
23
3476

And thus far we've created a "new number" 958 that isn't on this list.

And we can continue down the list like this just like Cantor and end up with precisely the same situation he has. (i.e. we can create a supposedly "new" natural number that isn't already on the list of numbers that we have created in this way)

Does that mean that we can create a new Natural Number that isn't already on the list of Natural Numbers? And that the Natural Numbers must then be "uncountably infinite" just like the Reals?

This so-called "proof" doesn't prove anything. In fact, if it proves anything at all it proves that the Natural Numbers must have the same weird cardinality as the reals because we can use the very same process as Cantor used to prove it for the Natural Numbers too.

So how are we any further ahead toward proving anything unique?

Ok MrAnchovy. I get it now!

You've answered my question and solved the whole problem. I was indeed viewing the proof incorrectly.

Unfortunately it doesn't help.

Why does it not help?

Because it would be a totally futile and meaningless "proof". In fact, he would be supposedly 'proving' something that necessarily falls out of how he is defining a "Real Number".

He's simply demanding that a "Real Number" is necessarily an "infinite" decimal expansion by definition and assuming that a "Rational Number" cannot be described by an "infinite" decimal expansion.

Another problem is that his "proof" would obviously work for any numbers. Not just reals.

Think of it this way. Just take the decimal point in his proof in his proof away.

What do you have left? A list of natural numbers.

You can go down that list in precisely the same way and proof that even the Natural Numbers can't be put into a correspondence with themselves.

How so?

Well, let's try it.

Start our list with the arbitrary natural number of 1

1

We cross that out and replace it with any arbitrary digit (say 9) So we have a "new number" 9.

Lets continue, we'll need a two digit number to work on next so lets choose 23 our list so far looks as follows:

1
23

Replace the 3 with 5. Our newly constructed number is 95.

Move down another row pick another arbitrary natural number, we'll need one that is at least 3 digits wide or more, lets pick 3476 just for fun.

Here's our new list

1
23
3476

Now we need to replace the 7 with something other than 7, let's choose 8.

Our new number is now 958, and so on.

So we currently have a list of natural numbers

1
23
3476

And thus far we've created a "new number" 958 that isn't on this list.

And we can continue down the list like this just like Cantor and end up with precisely the same situation he has. (i.e. we can create a supposedly "new" natural number that isn't already on the list of numbers that we have created in this way)

Does that mean that we can create a new Natural Number that isn't already on the list of Natural Numbers? And that the Natural Numbers must then be "uncountably infinite" just like the Reals?

This so-called "proof" doesn't prove anything. In fact, if it proves anything at all it proves that the Natural Numbers must have the same weird cardinality as the reals because we can use the very same process as Cantor used to prove it for the Natural Numbers too.

So how are we any further ahead toward proving anything unique?
No. If you do that same prace on the natural numbers then you get a number which extends indefenitely to the left. For example ....999999999. This is NOT a natural number, so this doesn't apply. (instead, it's a so called p-adic number, Cantors diagonalization shows that the p-adics are not countable).

The difference with Cantor diagonalization on the reals is that you end up with a number which extends indefinitely after the decimal point. This IS a real number.

pbuk
Gold Member
Moderator note: This thread will only deal with the rigorous version of Cantor's proof which is found in post 24 or in any serious set theory book.
Awww, I was getting so close to this:

If there are as many rows (real numbers) as there are columns then you can construct the diagonal number which is not in any row, hence the contradiction that shows that this cannot be the case.

If there are more rows (real numbers) than there are columns (natural numbers) then... we don't need to construct anything - this is exactly what we are trying to prove!

Because it would be a totally futile and meaningless "proof". In fact, he would be supposedly 'proving' something that necessarily falls out of how he is defining a "Real Number".
Yes, that is how proofs generally work.

This thread is going off-topic. Therefore:

Moderator note: This thread will only deal with Cantor's diagonalization proof. The invalidity of infinity and real numbers will be considered off-topic and the post will be deleted. If you want to discuss this topic, create a new thread.
With all due respect doesn't this seem to be a bit restrictive?

You just asked me yourself, "What process?? Do you mean taking the limit?? The limit of rational numbers doesn't need to be rational."

A question like that demands addressing the very concept of LIMIT.

You, as a mathematician, should be fully aware that the LIMIT process of calculus does not prove the existence of what the limit equals. On the contrary all that is required to prove a limit is that certain trends and conditions have been proven. The actual result (i.e. the number that the limit is equal to) cannot be said to necessarily "exist".

Surely you're aware of this. You can prove that a "limit exists" for functions where the value of the limit is undefined and therefore in terms of the "function" itself that point does not exist.

So a taking any process to a limit does not imply that the result has any actual "existence".

That may seem to have nothing to do with Cantor's diagonalization proof, but it's very much a part of it. Cantor is claiming that because he can take something to a limit that necessarily proves that the thing the limit is pointing too exists.

That's actually a false use of Limits anyway.

The epsilon-delta definition of limits won't even support any such conclusions.

So is that "off-topic" or is it required information concerning the proof in question?

The proof basically demands that something has been taken to a limit.

So how could that topic not be part of the proof?

Furthermore, since informal arguments are clearly not convincing.

Moderator note: This thread will only deal with the rigorous version of Cantor's proof which is found in post 24 or in any serious set theory book.
Well, I've already voiced my views on this.

If an informal intuitive or graphical argument can be shown to trump a mathematical axiom, then which should be accepted as being more reasonable?

Actually if you demand that we stick solely to the axioms of set theory then how could we ever show that they are flawed? They would necessarily contain those flaws.

Finally, and very sincerely,...

If you don't feel that this forum is the proper venue for fleshing out these kinds of ideas, then may I ask if you can point to an appropriate forum or web site where an intuitive approach to reason is deemed acceptable to discuss.

It not my intent to step on anyone's toes.

But I seriously would like to flesh out these ideas with people who are at least intelligent and professional enough to be capable of comprehending the points I'm attempting to address.

Clearly since these are concerns associated with mathematical formalism, it only makes sense to discuss them with people who at least have some understanding of mathematics.

But it seems to be rather fruitless if they are just going to demand that the status quo cannot be challenged without first accepting the very axioms that are in question.

Where can I propose ideas like these without having them immediately dismissed simply because they are indeed challenging sacred axioms?

Or has mathematics become a religion where any suggestion that its axioms might contain flaws is considered totally unacceptable blaspheme?

You can't very well use a formal axiomatic system to disprove itself.

If there are "flaws" in the formal system they must necessarily be address from an external intuitive approach.

That's my position on that.

So can you suggest a web site math forum where intuitive reasoning has not yet been cast asunder as being totally worthless?

With all due respect doesn't this seem to be a bit restrictive?

You just asked me yourself, "What process?? Do you mean taking the limit?? The limit of rational numbers doesn't need to be rational."

A question like that demands addressing the very concept of LIMIT.

You, as a mathematician, should be fully aware that the LIMIT process of calculus does not prove the existence of what the limit equals. On the contrary all that is required to prove a limit is that certain trends and conditions have been proven. The actual result (i.e. the number that the limit is equal to) cannot be said to necessarily "exist".

Surely you're aware of this. You can prove that a "limit exists" for functions where the value of the limit is undefined and therefore in terms of the "function" itself that point does not exist.

So a taking any process to a limit does not imply that the result has any actual "existence".

That may seem to have nothing to do with Cantor's diagonalization proof, but it's very much a part of it. Cantor is claiming that because he can take something to a limit that necessarily proves that the thing the limit is pointing too exists.

That's actually a false use of Limits anyway.

The epsilon-delta definition of limits won't even support any such conclusions.

So is that "off-topic" or is it required information concerning the proof in question?

The proof basically demands that something has been taken to a limit.

So how could that topic not be part of the proof?
There is indeed one place that you need limits, and this is in the decimal expansion of a number. Given a sequence of numbers (between 0 and 9) $(x_1,x_2,x_3,...)$, then it always induces a real number $x_1\frac{1}{10}+x_2\frac{1}{10^2}+x_3\frac{1}{10^3}+...$. If you want to discuss this, then it is possible in this thread, but the discussion will need to be formal.

Well, I've already voiced my views on this.

If an informal intuitive or graphical argument can be shown to trump a mathematical axiom, then which should be accepted as being more reasonable?

Actually if you demand that we stick solely to the axioms of set theory then how could we ever show that they are flawed? They would necessarily contain those flaws.

Finally, and very sincerely,...

If you don't feel that this forum is the proper venue for fleshing out these kinds of ideas, then may I ask if you can point to an appropriate forum or web site where an intuitive approach to reason is deemed acceptable to discuss.

It not my intent to step on anyone's toes.

But I seriously would like to flesh out these ideas with people who are at least intelligent and professional enough to be capable of comprehending the points I'm attempting to address.

Clearly since these are concerns associated with mathematical formalism, it only makes sense to discuss them with people who at least have some understanding of mathematics.

But it seems to be rather fruitless if they are just going to demand that the status quo cannot be challenged without first accepting the very axioms that are in question.

Where can I propose ideas like these without having them immediately dismissed simply because they are indeed challenging sacred axioms?

Or has mathematics become a religion where any suggestion that its axioms might contain flaws is considered totally unacceptable blaspheme?

You can't very well use a formal axiomatic system to disprove itself.

If there are "flaws" in the formal system they must necessarily be address from an external intuitive approach.

That's my position on that.

So can you suggest a web site math forum where intuitive reasoning has not yet been cast asunder as being totally worthless?
The Cantor diagonalization theorem states precisely that: under the given axioms of set theory, it is not true that the reals are countable. So in order to accept Cantor's theorem, it is necessary to accept the axioms. If you don't accept the axioms, then of course the theorem may be false!!

This thread will deal with the theorem that states: under the given axioms of set theory, it is not true that the reals are countable. So in this thread we will accept the currect axiom system and deduce Cantor's theorem. This thread will not be used to question the axioms.

If you want to challenge the axioms, you are free to do so in another thread.

But it seems to be rather fruitless if they are just going to demand that the status quo cannot be challenged without first accepting the very axioms that are in question.

Where can I propose ideas like these without having them immediately dismissed simply because they are indeed challenging sacred axioms?

Or has mathematics become a religion where any suggestion that its axioms might contain flaws is considered totally unacceptable blaspheme?
Dude, get off your pedestal. It's not that you can not question math here, you're just wrong in this case.

Dude, get off your pedestal. It's not that you can not question math here, you're just wrong in this case.
Well no need to claim that I'm on a 'pedestal'.

I agree that I was wrong.

How's that?

I AGREEnow that Cantor's proof is restricted by the the assumption of these axioms (although in truth, those axioms weren't in place in Cantor's day). They actually evolved out of the original intuitive work of Cantor. And this contributed to them becoming the formalized axioms that they have become today.

Let's not forgot that there didn't even exist any such things as a formal set theory until the turn of the 20th century and Cantor's ideas played a very large role in that development.

I have serious concerns with the whole development of set theory from that time period forward.

And of course my ideas are necessarily going to need to be based on intuitive ideas in order to address these concerns. How could they be anything other than this? That can't very well be based on formally accepted axioms that haven't yet been written.

Well no need to claim that I'm on a 'pedestal'.

I agree that I was wrong.

How's that?

I AGREEnow that Cantor's proof is restricted by the the assumption of these axioms (although in truth, those axioms weren't in place in Cantor's day). They actually evolved out of the original intuitive work of Cantor. And this contributed to them becoming the formalized axioms that they have become today.

Let's not forgot that there didn't even exist any such things as a formal set theory until the turn of the 20th century and Cantor's ideas played a very large role in that development.

I have serious concerns with the whole development of set theory from that time period forward.

And of course my ideas are necessarily going to need to be based on intuitive ideas in order to address these concerns. How could they be anything other than this? That can't very well be based on formally accepted axioms that haven't yet been written.
OK, since you indeed agree that Cantor's theorem is true under the current established axioms, then this thread is done.