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Refuting the Anti-Cantor Cranks

  1. May 6, 2012 #1
    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.
  2. jcsd
  3. May 6, 2012 #2
  4. May 6, 2012 #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!).
  5. May 7, 2012 #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.
  6. May 7, 2012 #5
    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?
  7. May 7, 2012 #6


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    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 :)
  8. May 7, 2012 #7

    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.

  9. May 7, 2012 #8


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    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?
  10. May 7, 2012 #9
    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.
  11. May 7, 2012 #10


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    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"?
  12. May 7, 2012 #11
    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?
  13. May 7, 2012 #12

    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.

  14. May 7, 2012 #13
    Of course, otherwise they wouldn't be against Cantor in the first place, would they?
    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.
  15. May 7, 2012 #14

    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.


    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.

  16. May 8, 2012 #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.
  17. May 8, 2012 #16


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    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.

  18. May 8, 2012 #17


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    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.
  19. May 8, 2012 #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.
    Last edited: May 8, 2012
  20. May 8, 2012 #19


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    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.
  21. May 8, 2012 #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.
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