Refuting the Anti-Cantor Cranks

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In summary, the conversation revolves around the validity of Cantor's diagonalization proof of the uncountability of the real numbers. The person being argued with makes the same arguments against the proof, claiming that the proposed real number is not well-defined and that the definition is contradictory. The responder tries to explain that the contradiction is not in the definition, but in the assumption of a complete list of real numbers. The conversation continues in circles without reaching a resolution.
  • #1
lugita15
1,554
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I occasionally have the opportunity to argue with anti-Cantor cranks, people who for some reason or the other attack the validity of Cantor's diagonalization proof of the uncountability of the real numbers, arguably one of the most beautiful ideas in mathematics. They usually make the same sorts of arguments, so years ago I wrote up this FAQ to deal with them. Unfortunately, it's still hard to get anywhere with these people; the discussion frequently turns into something of this form:

ME: Suppose there is an ordered list containing all the real numbers. Then we can read off the diagonal entries and construct a real number that differs in the Nth decimal place from the Nth real number on the list. This real number obviously cannot be in the list. So the list doesn't contain all the real numbers.
THEM: Of course your proposed number is not on the list; it's not a well-defined real number.
ME: What do you mean? I gave you the exact procedure for constructing it. You take the Nth real number on the list, and you make it differ from that number in the Nth decimal place.
THEM: But if we really have a list of all the real numbers, then your proposed number has to be somewhere in the list, right?
ME: Yes, of course, so let's say it's in the 57th place. Then it would have to differ from itself in the 57th place, which is impossible!
THEM: Exactly, it's impossible! Your definition requires that it differs in some place from itself, which is impossible, so your definition is bad.
ME: But you're only saying that it's impossible on the basis of the assumption that there's a complete list of real numbers, and the whole point is to disprove that assumption.
THEM: But we're doing this proof under that assumption, so how can we make a definition that runs contrary to that definition?
ME: But that definition is a good one regardless of whether there are countably or uncountably many reals. It is a complete, algorithmic, unambigupus specification of the real number. What else could you want?
THEM: I want the definition to be both unambiguous and non-contradictory, and your definition is contradictory!
ME: Forget about the purported complete lists of real numbers for a moment. Don't you agree that for any list of real numbers, complete or incomplete, it's possible to construct a real number that differs in the Nth place from the Nth number on the list?
THEM: No, it's only possible to construct such a real number if that real number isn't on the list, otherwise it's a contradictory definition.
ME: Don't you see that the contradiction is not the fault of my perfectly good definition, but rather the fault of your assumption that there are countably many real numbers?
THEM: No, I don't.
ME: But what if we took our putative complete list of real numbers, and fed it in line by line into a computer with an algorithm that spits out, digit by digit, a real number that differs in the Nth digit from the Nth number on the list? Would such a computer program work?
THEM: No it wouldn't, the computer program would hit the place on the list where the number being constructed would reside, and then it would get crash, because it can't choose a digit for the number that differs in the nth place from itself.
ME: Argh!

So how do I stop going in circles and convince them that they're wrong?

Any help would be greatly appreciated.

Thank You in Advance.
 
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  • #2
lugita15 said:
I occasionally have the opportunity to argue with anti-Cantor cranks, people who for some reason or the other attack the validity of Cantor's diagonalization proof of the uncountability of the real numbers, arguably one of the most beautiful ideas in mathematics. They usually make the same sorts of arguments, so years ago I wrote up this FAQ to deal with them. Unfortunately, it's still hard to get anywhere with these people; the discussion frequently turns into something of this form:

ME: Suppose there is an ordered list containing all the real numbers. Then we can read off the diagonal entries and construct a real number that differs in the Nth decimal place from the Nth real number on the list. This real number obviously cannot be in the list. So the list doesn't contain all the real numbers.
THEM: Of course your proposed number is not on the list; it's not a well-defined real number.
ME: What do you mean? I gave you the exact procedure for constructing it. You take the Nth real number on the list, and you make it differ from that number in the Nth decimal place.
THEM: But if we really have a list of all the real numbers, then your proposed number has to be somewhere in the list, right?
ME: Yes, of course, so let's say it's in the 57th place. Then it would have to differ from itself in the 57th place, which is impossible!
THEM: Exactly, it's impossible! Your definition requires that it differs in some place from itself, which is impossible, so your definition is bad.
ME: But you're only saying that it's impossible on the basis of the assumption that there's a complete list of real numbers, and the whole point is to disprove that assumption.
THEM: But we're doing this proof under that assumption, so how can we make a definition that runs contrary to that definition?
ME: But that definition is a good one regardless of whether there are countably or uncountably many reals. It is a complete, algorithmic, unambigupus specification of the real number. What else could you want?
THEM: I want the definition to be both unambiguous and non-contradictory, and your definition is contradictory!
ME: Forget about the purported complete lists of real numbers for a moment. Don't you agree that for any list of real numbers, complete or incomplete, it's possible to construct a real number that differs in the Nth place from the Nth number on the list?
THEM: No, it's only possible to construct such a real number if that real number isn't on the list, otherwise it's a contradictory definition.
ME: Don't you see that the contradiction is not the fault of my perfectly good definition, but rather the fault of your assumption that there are countably many real numbers?
THEM: No, I don't.
ME: But what if we took our putative complete list of real numbers, and fed it in line by line into a computer with an algorithm that spits out, digit by digit, a real number that differs in the Nth digit from the Nth number on the list? Would such a computer program work?
THEM: No it wouldn't, the computer program would hit the place on the list where the number being constructed would reside, and then it would get crash, because it can't choose a digit for the number that differs in the nth place from itself.
ME: Argh!

So how do I stop going in circles and convince them that they're wrong?



*** The first question has an easy answer: just stop!

The second one is more involved yet easier to grasp after a while: you can't convince a true anticantorian crank. Period.

You see, I still have to meet an A.C.C. who is a mathematician. I've met literally hundreds in the last 11-12 years. So what

happens? Well, my theory is that these are guys (or guyettes) so bored and empty of actual content in their lives that they decide to pick

on something they can understand. Now, Cantor Diagonal Theorem, Goldbach's Conjecture, Fermat's Last Theorem, etc. are

statements that are easy to understand even to greenhorns, kids, students...and cranks.

You don't see many cranks writing nonsenses against Riemann Hypothesis, ABC Conjecture, Elliptic Functions or Dirichlet's Theorem

on Primes in Arithmetic Sequences. Why? Simple: even to understand these concepts one needs at least a thin layer of

mathematical higher education.

Now, there've always been uneducated, stupid or just bored people around, but now they all have internet to spread up their

nonsenses, and this causes them deep and overwhelming happiness.

So stop trying to convince ACC's: whenever they'll get a girl(boy)friend, begin to actually study something or, in general,

get some kind of intellectual life they'll stop by themselves...hopefully.

DonAntonio
.


Any help would be greatly appreciated.

Thank You in Advance.

...
 
  • #3
DonAntonio, a lot of them may be beyond saving, but there are some people who can otherwise reason quite well who just don't quite grasp this argument. So I'd like to know what a convincing response would be in the typical dialogue above, at least if you're dealing with a somewhat open-minded crank (they do exist!).
 
  • #4
I've been observing and sometimes arguing with the anti-Cantor cranks for years. There's no hope. Logic and reason are futile. They just don't want to get it. If you argue with an anti-Cantor crank, realize you're doing it for your own amusement. It makes no difference to them.
 
  • #5
SteveL27 said:
I've been observing and sometimes arguing with the anti-Cantor cranks for years. There's no hope. Logic and reason are futile. They just don't want to get it. If you argue with an anti-Cantor crank, realize you're doing it for your own amusement. It makes no difference to them.
I have actually seen a few anti-Cantor cranks over the years see the light after reams of discussion. They often have peculiar misconceptions, like a belief that infinite sets of numbers must have infinitely large numbers, but if you break their arguments down and get to the heart of their confusion, Cantor's proof may suddenly click for them.

In any case, what do you think would the best response in the dialogue I wrote above? How would you argue with someone who claims that the contradiction derived in Cantor's proof comes not from the assumption that the reals are countable, but rather from the definition of the number constructed from the diagonal of the list?
 
  • #6
Hey lugita15.

Even though I'm not arguing with (or going to) about the Cantor diagonalization, I'm grateful for your FAQ so just remember it's useful to have these kinds of things even for the people that aren't out for an argument per se.

I would follow Don Antonio's advice in that stop wasting your energy on people that just want to argue rather than to converse (which is a two way thing and not a one way like an argument). It drains energy and it's just not worth it in my mind.

Arguments are 'in it to win it' and not for conversing or learning so let them feed off someone else rather than yourself.

Again thanks for your FAQ :)
 
  • #7
lugita15 said:
DonAntonio, a lot of them may be beyond saving, but there are some people who can otherwise reason quite well who just don't quite grasp this argument. So I'd like to know what a convincing response would be in the typical dialogue above, at least if you're dealing with a somewhat open-minded crank (they do exist!).



Somebody actually grasping something and being ready to listen and think is NOT, by definition, a crank. In my book, a crank

is someone so deeply stupid/ignorant/annoying-on-purpose that has this inner feeling that he's infallible and knows everything about some

part of mathematics without having studied mathematics ever (beyond H.S., i.e.: actual mathematics), so that ANYTHING you tell

them falls in free fall in that awesome void between their two ears. In short, it is not merely somebody incapable to grasp Cantor's Theory,

and in particular his Diagonal trick theorem, but somebody 100% convinced he's right and ALL the rest wrong.

DonAntonio
 
  • #8
It seems like they're quarreling over the definition of what a real number is. Are these people interested in trying to understand the analytical concept of completeness and how completeness is necessary for the real numbers to behave as a true continuum?
 
  • #9
mbs said:
It seems like they're quarreling over the definition of what a real number is. Are these people interested in trying to understand the analytical concept of completeness and how completeness is necessary for the real numbers to behave as a true continuum?
They cranks I discuss in the dialogue above accept the real number system and its properties, including completeness. But they question whether the real number you construct in the proof is well defined, because if it were well-defined then it would be somewhere on the list (since they're assuming the list is complete), say the 57th number on the list. But the definition says "let it differ from the Nth number in the Nth place", so they're saying the definition is contradictory because it requires the number to differ in the 57th place from the 57th number, i.e. from itself which is impossible. So they're saying the problem is with a contradictory definition, not their countability assumption.
 
  • #10
lugita15 said:
They cranks I discuss in the dialogue above accept the real number system and its properties, including completeness. But they question whether the real number you construct in the proof is well defined, because if it were well-defined then it would be somewhere on the list (since they're assuming the list is complete), say the 57th number on the list. But the definition says "let it differ from the Nth number in the Nth place", so they're saying the definition is contradictory because it requires the number to differ in the 57th place from the 57th number, i.e. from itself which is impossible. So they're saying the problem is with a contradictory definition, not their countability assumption.

I don't quite follow that line of reasoning because the diagonalization procedure for constructing a real number not in the list does not depend on the assumption that the list is complete. The procedure works for any list of real numbers. Do these people accept the claim "given any list of real numbers, there exists a real number not in the list"?
 
  • #11
mbs said:
I don't quite follow that line of reasoning because the diagonalization procedure for constructing a real number not in the list does not depend on the assumption that the list is complete. The procedure works for any list of real numbers. Do these people accept the claim "given any list of real numbers, there exists a real number not in the list"?
No, they believe that you can have a complete list of real numbers. They're saying that given a list of real numbers, it is not well-defined to construct x as a number that differs in the Nth place from the Nth number on the list, unless you've first proven that x is not on the list. Because if x IS on the list, then it's impossible for x to differ from itself, so the definition of x is self-contradictory. So how would you rebut them?
 
  • #12
lugita15 said:
No, they believe that you can have a complete list of real numbers. They're saying that given a list of real numbers, it is not well-defined to construct x as a number that differs in the Nth place from the Nth number on the list, unless you've first proven that x is not on the list. Because if x IS on the list, then it's impossible for x to differ from itself, so the definition of x is self-contradictory. So how would you rebut them?



Whoever says that is an ignorant of the basics in logic and mathematics or, if you pardon my saying so, an idiot.

IF the number x WAS in the list then it would be impossible to carry out the diagonal "trick" (be careful here: some cranks are prone to

see in this a Houdini-like thing) in Cantor's proof.

Say, the number x appears in the n-th position in the list. Then, when we're to build th n-th digit of our number (x) we'll get that we can't

do this as it is ALREADY there.

Of course, the above is algebraic mumbo-jumbo that one sometime's is pushed to get into by battling vs cranks (and

this is gratifying for most of them): the construction in Cantor's proof is not pre-assigned on certain number. We do actually build

the number as to be sure it is NOT in the list.

DonAntonio
 
  • #13
DonAntonio said:
Whoever says that is an ignorant of the basics in logic and mathematics or, if you pardon my saying so, an idiot.
Of course, otherwise they wouldn't be against Cantor in the first place, would they?
IF the number x WAS in the list then it would be impossible to carry out the diagonal "trick" (be careful here: some cranks are prone to

see in this a Houdini-like thing) in Cantor's proof.

Say, the number x appears in the n-th position in the list. Then, when we're to build th n-th digit of our number (x) we'll get that we can't

do this as it is ALREADY there.
Exactly, that is precisely the argument they use to argue that the x is ill-defined because its construction is self-contradictory. They say that you can only demonstrate that the construction is well-defined if you first demonstrate that the number being constructed is not on the list.
this is gratifying for most of them): the construction in Cantor's proof is not pre-assigned on certain number. We do actually build

the number as to be sure it is NOT in the list.
But they're saying that the construction is impossible if the list contains all real numbers. I know, it's hard to make sense of what does not make sense, but I want to come up with a good rebuttal that can make at least some of them see the light.
 
  • #14
lugita15 said:
Of course, otherwise they wouldn't be against Cantor in the first place, would they?


No. Otherwise they wouldn't crank or troll about it. There are actual mathematicians who go out against Cantor and his ideas, and

though some of them are as bad-blooded and cruel as Kroenecker was in their time against Cantor, many of them try to base mathematically

their disagreement with these ideas. Not the cranks, no. These only babble huge nonsenses devoid of almost any mathematical

content, in a whimsical, idiotic fashion.

DonAntonio


Exactly, that is precisely the argument they use to argue that the x is ill-defined because its construction is self-contradictory. They say that you can only demonstrate that the construction is well-defined if you first demonstrate that the number being constructed is not on the list. But they're saying that the construction is impossible if the list contains all real numbers. I know, it's hard to make sense of what does not make sense, but I want to come up with a good rebuttal that can make at least some of them see the light.


If you want to make them see the light advice them to look up straight at the sun, during an eclipse if possible. You won't succeed and the most

you can wish for, as somebody else already pointed out, is to have some fun during your leisure time, nothing more.

DonAntonio
 
  • #15
Well, as far as I know , there are other ways to prove the uncountability of real numbers. So, if they are not so happy with Cantor diagonal argument , other proofs may convince them. One other possibility is to prove that a perfect set in Rk is uncountable. Hence the reals are uncountable.
However, if your goal is to force them to be happy with Cantor's argument, then you will have to waste a lot of your time creating more and more items in your FAQ.
 
  • #16
lugita15 said:
No, they believe that you can have a complete list of real numbers. They're saying that given a list of real numbers, it is not well-defined to construct x as a number that differs in the Nth place from the Nth number on the list, unless you've first proven that x is not on the list. Because if x IS on the list, then it's impossible for x to differ from itself, so the definition of x is self-contradictory. So how would you rebut them?

I suppose it is possible (but extremely tedious) to make the cantor diagonalization argument into a completely formal proof, relying on the ZFC axioms and formal logical rules alone.

Maybe this will help.

http://us.metamath.org/mpegif/mmcomplex.html#uncountable
 
  • #17
DonAntonio said:
No. Otherwise they wouldn't crank or troll about it. There are actual mathematicians who go out against Cantor and his ideas, and

though some of them are as bad-blooded and cruel as Kroenecker was in their time against Cantor, many of them try to base mathematically

their disagreement with these ideas. Not the cranks, no. These only babble huge nonsenses devoid of almost any mathematical

content, in a whimsical, idiotic fashion.

DonAntonio


If you want to make them see the light advice them to look up straight at the sun, during an eclipse if possible. You won't succeed and the most

you can wish for, as somebody else already pointed out, is to have some fun during your leisure time, nothing more.

DonAntonio

I don't see cranks as totally useless. Finding the errors in crank arguments can be enlightening for someone who hasn't critically examined the foundations of an argument or theory. So long as you don't think your goal is to convince them that they are wrong. Even if you do get them to question it's not like they're going to admit it as it's all just a game to them.
 
  • #18
A lot of the more naive arguments stem from a basic misunderstanding of proofs by contradiction.

The more serious challenges to Cantor's argument always stem from a disagreement over what we are allowed to take as axioms. These disputes cannot really be resolved. One either accepts the axioms and what follows or one doesn't accept the underlying axioms and is left with a different mathematics...

I would hesitate to call people who question the proof cranks. Some do not understand the argument and others are merely attempting to challenge it. It is healthy and admirable to attempt to challenge a proof.

There are some people who simply fail to grasp the argument -- in spite of repeated discussion --and are incapable of overcoming their moral convictions about the statement. These people likely won't succeed in mathematics, and I wouldn't worry about trying to convince them.
 
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  • #19
nucl34rgg said:
A lot of the more naive arguments stem from a basic misunderstanding of proofs by contradiction.

The more serious challenges to Cantor's argument always stem from a disagreement over what we are allowed to take as axioms. These disputes cannot really be resolved. One either accepts the axioms and what follows or one doesn't accept the underlying axioms and is left with a different mathematics...

I would hesitate to call people who question the proof cranks. Some do not understand the argument and others are merely attempting to challenge it. It is healthy and admirable to attempt to challenge a proof.

There are some people who simply fail to grasp the argument -- in spite of repeated discussion --and are incapable of overcoming their moral convictions about the statement. These people likely won't succeed in mathematics, and I wouldn't worry about trying to convince them.

I feel the bolded part is what gets people labeled as "cranks". It seems to be an ego thing. If I'm feeling a little miffed by something I at least have the humility to admit that I might be missing something, or it might be over my head, at least at my current level of understanding. Questioning is a good thing, but you also have to have the ability to question your own understanding of a concept to truly learn. A part of intelligence is being able to discern your own knowledge and level of understanding. In short, you must first know what you don't know in order to learn.
 
  • #20
I think an important thing in math and science is to be skeptical of yourself. If you recognize your own fallibility along with the fallibility of other people, you come to accept counter-intuitive ideas more readily when they are proven or evidence is given that favors them.

The above post reminds me of a quote:

“He who knows not and knows not that he knows not is a fool—shun him. He who knows not and knows that he knows not is a child—teach him. He who knows and knows not that he knows is asleep—wake him. He who knows and knows that he knows is wise—follow him.”

I'm not sure if I agree that "shunning him" is the best action, but I think that quote fits well.
 
  • #21
nucl34rgg said:
I think an important thing in math and science is to be skeptical of yourself. If you recognize your own fallibility along with the fallibility of other people, you come to accept counter-intuitive ideas more readily when they are proven or evidence is given that favors them.

The above post reminds me of a quote:

“He who knows not and knows not that he knows not is a fool—shun him. He who knows not and knows that he knows not is a child—teach him. He who knows and knows not that he knows is asleep—wake him. He who knows and knows that he knows is wise—follow him.”

I'm not sure if I agree that "shunning him" is the best action, but I think that quote fits well.

I think it's a bit hypocritical and counterproductive to respond to the stubborn convictions of a "crank" with arrogance and condescension yourself, even if it's hard to resist doing so in shear frustration. Constantly being told one is a rambling idiot and full of **** doesn't exactly encourage self-examination. It just puts them on the defensive and makes them dig their heels in and become more hardened and stubborn.
 
  • #22
mbs said:
I think it's a bit hypocritical and counterproductive to respond to the stubborn convictions of a "crank" with arrogance and condescension yourself, even if it's hard to resist doing so in shear frustration. Constantly being told one is a rambling idiot and full of **** doesn't exactly encourage self-examination. It just puts them on the defensive and makes them dig their heels in and become more hardened and stubborn.

I know that most people don't want to be told that they are 'inferior' (or at least interpret this), but I think it's important for people to face criticism of any kind. Usually if it is a huge personal attack with no substance, then this will be easy to see.

The thing is that it is dangerous to have people that think they know absolutely everything because these kinds of people end up creating scenarios of total destruction when you get the kind of 'Messianic' personalities.

Also dealing with criticism in a constructive way is a sign of real security and not insecurity. Insecure people need to realize that they can be wrong and that it's ok to be wrong just like everyone else.

If someone can't deal with an opposing point of view and seriously consider it in a way that isn't 'seen as ego-damaging', then I wouldn't want to waste my time either with them and I don't imagine lots of other people would: it would be an unnecessary draining experience and a waste of time.
 
  • #23
Can we discuss the substance? How would you refute the bizarre belief of the cranks that the fault of the contradiction lies not in the countability assumption, but rather in the diagonal construction, which they believe is self-contradictory if the number being constructed were already on the list?
 
  • #24
lugita15 said:
Can we discuss the substance? How would you refute the bizarre belief of the cranks that the fault of the contradiction lies not in the countability assumption, but rather in the diagonal construction, which they believe is self-contradictory if the number being constructed were already on the list?



It begins to worry me that after all that has been written in this thread you still insist on this...I, for one, shall stop participating

here if this is the mode you want to impose on your thread.

DonAntonio
 
  • #25
DonAntonio said:
It begins to worry me that after all that has been written in this thread you still insist on this...I, for one, shall stop participating

here if this is the mode you want to impose on your thread.

DonAntonio
By all means feel free to continue the discussion as it is, DonAntonio! I was just asking whether anyone could come up with a good rebuttal I can use in my (perhaps futile) future battles with cranks.
 
  • #26
I'm seeing this proof for the first time so I can't be a crank by definition. Consider me a laboratory for the thread.

I accept the conclusion of the proof; its almost obvious. But I disagree that you have given a construction procedure for the list. The only element you can "give" is the first one, the one that is "all zeros". The second one can't be constructed let alone the nth one.

The next number after all zeros, which would correspond to the natural number "2" would be an infinite string of zeros followed by a single "1", itself an impossible construct.

So how you go beyond this impasse to construct the next N entries is beyond me.

The proof of course circles back to this very fact to elicit the "obvious" conclusion. But it's a fact that the list itself couldn't be validly constructed during the proof procedure except for the first element. The contradiction itself seems to be an axiom of the proof.
 
  • #27
Antiphon said:
I'm seeing this proof for the first time so I can't be a crank by definition. Consider me a laboratory for the thread.

I accept the conclusion of the proof; its almost obvious. But I disagree that you have given a construction procedure for the list. The only element you can "give" is the first one, the one that is "all zeros". The second one can't be constructed let alone the nth one.

The next number after all zeros, which would correspond to the natural number "2" would be an infinite string of zeros followed by a single "1", itself an impossible construct.

So how you go beyond this impasse to construct the next N entries is beyond me.

The proof of course circles back to this very fact to elicit the "obvious" conclusion. But it's a fact that the list itself couldn't be validly constructed during the proof procedure except for the first element. The contradiction itself seems to be an axiom of the proof.
I'm not sure what you're talking about. I'm not constructing a list, I'm saying "take any list of real numbers". Then I am constructing a real number not on the list, by making it differ in the first decimal place from the first number on the list, in the second place from the second number on the list, etc. Do you understand that?
 
  • #28
lugita15 said:
I'm not sure what you're talking about. I'm not constructing a list, I'm saying "take any list of real numbers". Then I am constructing a real number not on the list, by making it differ in the first decimal place from the first number on the list, in the second place from the second number on the list, etc. Do you understand that?

Sure. Let me rephrase what I'm hearing you say.

Here's an initial list of real numbers along with a correspondence to the first few natural numbers.

1 0.000
2 0.100
3 0.200

You then give a prescription, an algorithm actually, for creating a number which can't be in the list above, as follows:

0.111, a number which differs as you describe, in diagonal fashion.

So your modified list is now:

1 0.000
2 0.100
3 0.111
4 0.200

I assert that you have expanded the list by one member and extended the range of the one-to-one correspondence between the natural numbers and the real numbers.

Your turn.
 
  • #29
Antiphon said:
Sure. Let me rephrase what I'm hearing you say.

Here's an initial list of real numbers along with a correspondence to the first few natural numbers.

1 0.000
2 0.100
3 0.200

You then give a prescription, an algorithm actually, for creating a number which can't be in the list above, as follows:

0.111, a number which differs as you describe, in diagonal fashion.

So your modified list is now:

1 0.000
2 0.100
3 0.111
4 0.200

I assert that you have expanded the list by one member and extended the range of the one-to-one correspondence between the natural numbers and the real numbers.

Your turn.
First of all, there are obviously infinitely many real numbers, so in Cantor's proof we start with an arbitrary infinite list of real numbers, not a finite list.
 
  • #30
lugita15 said:
First of all, there are obviously infinitely many real numbers, so in Cantor's proof we start with an arbitrary infinite list of real numbers, not a finite list.

Ok. I'll arrive at the situation you describe by repeating the diagonal algorithm over and over. This procedure, taken in the infinite limit, approches your construction as closely as desired.

It also reinforces the one-to-one correspondence between the natural and real numbers. Each time I extend the list with a new real member, the natural numbers also go up by one.
 
  • #31
Antiphon said:
Ok. I'll arrive at the situation you describe by repeating the diagonal algorithm over and over. This procedure, taken in the infinite limit, approches your construction as closely as desired.


You begin here to enter deep waters: what exactly is "this procedure, taken in the infinite limit"? Whose "infinite limit"?

The construction is plainly simple: you suppose you're given any list (=countable infinite set) of real numbers, preferably in

the interval (0,1) to make things less messier, and then a number is constructed as to make it impossible to belong to that

list, thus showing no list can contain ALL the real numbers.

Now, stick to the above and tell us how whatever doesn't go well...


It also reinforces the one-to-one correspondence between the natural and real numbers. Each time I extend the list with a new real member, the natural numbers also go up by one.


Unless you give a reasonable, mathematical definition of what you mean by "extending by one an infinite list" (see def. above), what you

wrote is just meaningless (not to mention that weird "the naturals numbers go up by one" thingy...)

DonAntonio
 
  • #32
Antiphon said:
Sure. Let me rephrase what I'm hearing you say.

Here's an initial list of real numbers along with a correspondence to the first few natural numbers.

1 0.000
2 0.100
3 0.200

You then give a prescription, an algorithm actually, for creating a number which can't be in the list above, as follows:

0.111, a number which differs as you describe, in diagonal fashion.

So your modified list is now:

1 0.000
2 0.100
3 0.111
4 0.200

I assert that you have expanded the list by one member and extended the range of the one-to-one correspondence between the natural numbers and the real numbers.

Your turn.
That violates the basic idea of the Cantor proof that we already have a list that includes all real numbers- that is, that the set of all real numbers (between 0 and 1) is countable. You cannot add new numbers.
 
  • #33
DonAntonio said:
You begin here to enter deep waters: what exactly is "this procedure, taken in the infinite limit"? Whose "infinite limit"?

The construction is plainly simple: you suppose you're given any list (=countable infinite set) of real numbers, preferably in

the interval (0,1) to make things less messier, and then a number is constructed as to make it impossible to belong to that

list, thus showing no list can contain ALL the real numbers.

Now, stick to the above and tell us how whatever doesn't go well...





Unless you give a reasonable, mathematical definition of what you mean by "extending by one an infinite list" (see def. above), what you

wrote is just meaningless (not to mention that weird "the naturals numbers go up by one" thingy...)

DonAntonio

I was trying to lead the horse to water. The infinite limit I presented is an algorithmic construction that in principle is no different from the way one might sum an infinite series. The point of it was to show that one simply cannot arrive at Cantor's result without starting off with unbounded sets as you pointed out. That you cannot approach the result the way you might show that .999... is the same as 1.0. But let's move on because this doesn't invalidate the Cantor result.

A couple of posts back I did exactly give you as requested "any list of real numbers". It was in fact the first three entries of an infinite countable list exactly as you specified, and between 0 and 1 to keep things from getting messy.

But that's not good enough apparently. So it looks like you need to tighten up your specs before we can go to the next step.

Is your objection to my list the fact that it excluded 0.05? Because perhaps that was the very Cantor number that would be added by diagonolization.

Next you're going to say that I'm not getting it, that the first entry corresponding to natural number 1 is all zeros (say) and the second entry which corresponds to natural number 2 is "the next real number after 0.000..." so that I haven't skipped any.

Which brings us full circle to my very first post. It seems the point of the constructed proof is to arrive at the contradiction that by adding one more real, I wasn't able to enumerate them all after all. But one doesn't need to add the diagonalized real to arrive at this conclusion. You arrived at it YOURSELF when you kept saying that my finite, truncated, first few entries of the countably infinite list I provided aren't good enough because why? Because they didn't contain all the real numbers!

That's like saying sin(x) does not equal x+x^3/6+... because I didn't start the series with all the terms already there.
 
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  • #34
Let me further add (because I don't want to fly off on a tangent here) that I understand the result but disagree so far with the soundness of the diagonalization proof as its been presented.

I get it if you say that the terms in a series like sin(x) are given by x^n/n! and clearly you can't "shoehorn in" a new term between the nth and n+1th. Cantor does exactly this, shoehorns in another term thus showing there is not the one to one association as in the series for sin(x). I get it.

The gist of what I'm trying to say (as a non-mathematician) is this: why is it that I can enumerate the first 3 natural numbers in an ordered set of them beginning with 0, but the same cannot be done with the reals beyond the first entry? If Cantor or anyone wants to start there and work toward the idea these sets are fundamentally different in nature, there'd be no Cantor Cranks.

Cantor forms this very list that can't be formed and then a step *later* brings us to a contradiction?

Sorry but that's a little too much like Morpheous jumping from building to building. Telling me to "free my mind" isn't going to cut it.
 
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  • #35
Antiphon said:
Next you're going to say that I'm not getting it, that the first entry corresponding to natural number 1 is all zeros (say) and the second entry which corresponds to natural number 2 is "the next real number after 0.000..." so that I haven't skipped any.
First of all, you should know that in Cantor's proof, the list of real numbers doesn't have to be ordered from least to greatest. In fact, if a list WAS ordered from least to greatest, you don't even need Cantor's proof to show that it's incomplete. You can just take, say, the average of the first and second entry, and it won't be on the list.
 
<h2>1. What is the significance of Cantor's work in mathematics?</h2><p>Cantor's work in mathematics is significant because he introduced the concept of transfinite numbers and developed set theory, which is now a fundamental part of modern mathematics. His work also laid the foundation for important mathematical concepts such as infinity and the continuum hypothesis.</p><h2>2. What is the main argument of the "Anti-Cantor Cranks"?</h2><p>The main argument of the "Anti-Cantor Cranks" is that Cantor's work on transfinite numbers and set theory is flawed and cannot be considered legitimate mathematics. They often claim that Cantor's ideas are illogical or contradictory.</p><h2>3. How do mathematicians refute the arguments of the "Anti-Cantor Cranks"?</h2><p>Mathematicians refute the arguments of the "Anti-Cantor Cranks" by pointing out that their criticisms are based on misunderstandings or misinterpretations of Cantor's work. They also provide rigorous mathematical proofs and examples to demonstrate the validity of Cantor's theories.</p><h2>4. Can Cantor's work on transfinite numbers be applied in real-world situations?</h2><p>Yes, Cantor's work on transfinite numbers has practical applications in various fields such as computer science, physics, and economics. For example, set theory is used in database design and the concept of infinity is crucial in understanding the behavior of complex systems.</p><h2>5. Is it possible to disprove Cantor's theories?</h2><p>No, it is not possible to disprove Cantor's theories as they have been extensively studied and verified by mathematicians for over a century. Any attempt to disprove them would require a fundamental rethinking of mathematics as we know it.</p>

1. What is the significance of Cantor's work in mathematics?

Cantor's work in mathematics is significant because he introduced the concept of transfinite numbers and developed set theory, which is now a fundamental part of modern mathematics. His work also laid the foundation for important mathematical concepts such as infinity and the continuum hypothesis.

2. What is the main argument of the "Anti-Cantor Cranks"?

The main argument of the "Anti-Cantor Cranks" is that Cantor's work on transfinite numbers and set theory is flawed and cannot be considered legitimate mathematics. They often claim that Cantor's ideas are illogical or contradictory.

3. How do mathematicians refute the arguments of the "Anti-Cantor Cranks"?

Mathematicians refute the arguments of the "Anti-Cantor Cranks" by pointing out that their criticisms are based on misunderstandings or misinterpretations of Cantor's work. They also provide rigorous mathematical proofs and examples to demonstrate the validity of Cantor's theories.

4. Can Cantor's work on transfinite numbers be applied in real-world situations?

Yes, Cantor's work on transfinite numbers has practical applications in various fields such as computer science, physics, and economics. For example, set theory is used in database design and the concept of infinity is crucial in understanding the behavior of complex systems.

5. Is it possible to disprove Cantor's theories?

No, it is not possible to disprove Cantor's theories as they have been extensively studied and verified by mathematicians for over a century. Any attempt to disprove them would require a fundamental rethinking of mathematics as we know it.

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