For every natural number N, there's a Cantor Crank C(n)

More crankery? of course! What kind? What else? Cantor crankery!

It’s amazing that so many people are so obsessed with Cantor. Cantor just gets under peoples’ skin, because it feels wrong. How can there be more than one infinity? How can it possibly make sense?

As usual in math, it all comes down to the axioms. In most math, we’re working from a form of set theory – and the result of the axioms of set theory are quite clear: the way that we define numbers, the way that we define sizes, this is the way it is.

Today’s crackpot doesn’t understand this. But interestingly, the focus of his problem with Cantor isn’t the diagonalization. He thinks Cantor went wrong way before that: Cantor showed that the set of even natural numbers and the set of all natural numbers are the same size!

Unfortunately, his original piece is written in Portuguese, and I don’t speak Portuguese, so I’m going from a translation, here.

The Brazilian philosopher Olavo de Carvalho has written a philosophical “refutation” of Cantor’s theorem in his book “O Jardim das Aflições” (“The Garden of Afflictions”). Since the book has only been published in Portuguese, I’m translating the main points here. The enunciation of his thesis is:

Georg Cantor believed to have been able to refute Euclid’s fifth common notion (that the whole is greater than its parts). To achieve this, he uses the argument that the set of even numbers can be arranged in biunivocal correspondence with the set of integers, so that both sets would have the same number of elements and, thus, the part would be equal to the whole.

And his main arguments are:

It is true that if we represent the integers each by a different sign (or figure), we will have a (infinite) set of signs; and if, in that set, we wish to highlight with special signs, the numbers that represent evens, then we will have a “second” set that will be part of the first; and, being infinite, both sets will have the same number of elements, confirming Cantor’s argument. But he is confusing numbers with their mere signs, making an unjustifiable abstraction of mathematical properties that define and differentiate the numbers from each other.

The series of even numbers is composed of evens only because it is counted in twos, i.e., skipping one unit every two numbers; if that series were not counted this way, the numbers would not be considered even. It is hopeless here to appeal to the artifice of saying that Cantor is just referring to the “set” and not to the “ordered series”; for the set of even numbers would not be comprised of evens if its elements could not be ordered in twos in an increasing series that progresses by increments of 2, never of 1; and no number would be considered even if it could be freely swapped in the series of integeres.

He makes two arguments, but they both ultimately come down to: “Cantor contradicts Euclid, and his argument just can’t possibly make sense, so it must be wrong”.

The problem here is: Euclid, in “The Elements”, wrote severaldifferent collections of axioms as a part of his axioms. One of them was the following five rules:

  1. Things which are equal to the same thing are also equal to one another.
  2. If equals be added to equals, the wholes are equal.
  3. If equals be subtracted from equals, the remainders are equal.
  4. Things which coincide with one another are equal to one another.
  5. The whole is greater that the part.

The problem that our subject has is that Euclid’s axiom isn’t an axiom of mathematics. Euclid proposed it, but it doesn’t work in number theory as we formulate it. When we do math, the axioms that we start with do not include this axiom of Euclid.

In fact, Euclid’s axioms aren’t what modern math considers axioms at all. These aren’t really primitive ground statements. Most of them are statements that are provable from the actual axioms of math. For example, the second and third axioms are provable using the axioms of Peano arithmetic. The fourth one doesn’t appear to be a statement about numbers at all; it’s a statement about geometry. And in modern terms, the fifth one is either a statement about geometry, or a statement about measure theory.

The first argument is based on some strange notion of signs distinct from numbers. I can’t help but wonder if this is an error in translation, because the argument is so ridiculously shallow. Basically, it concedes that Cantor is right if we’re considering the representations of numbers, but then goes on to draw a distinction between representations (“signs”) and the numbers themselves, and argues that for the numbers, the argument doesn’t work. That’s the beginning of an interesting argument: numbers and the representations of numbers are different things. It’s definitely possible to make profound mistakes by confusing the two. You can prove things about representations of numbers that aren’t true about the numbers themselves. Only he doesn’t actually bother to make an argument beyond simply asserting that Cantor’s proof only works for the representations.

That’s particularly silly because Cantor’s proof that the even naturals and the naturals have the same cardinality doesn’t talk about representation at all. It shows that there’s a 1 to 1 mapping between the even naturals and the naturals. Period. No “signs”, no representations.

The second argument is, if anything, even worse. It’s almost the rhetorical equivalent of sticking his fingers in his ears and shouting “la la la la la”. Basically – he says that when you’re producing the set of even naturals, you’re skipping things. And if you’re skipping things, those things can’t possible be in the set that doesn’t include the skipped things. And if there are things that got skipped and left out, well that means that it’s ridiculous to say that the set that included the left out stuff is the same size as the set that omitted the left out stuff, because, well, stuff got left out!!!.

Here’s the point. Math isn’t about intuition. The properties of infinitely large sets don’t make intuitive sense. That doesn’t mean that they’re wrong. Things in math are about formal reasoning: starting with a valid inference system and a set of axioms, and then using the inference to reason. If we look at set theory, we use the axioms of ZFC. And using the axioms of ZFC, we define the size (or, technically, the cardinality) of sets. Using that definition, two sets have the same cardinality if and only if there is a one-to-one mapping between the elements of the two sets. If there is, then they’re the same size. Period. End of discussion. That’s what the math says.

Cantor showed, quite simply, that there is such a mapping:

{ (i rightarrow itimes 2) | i in N }

There it is. It exists. It’s simple. It works, by the axioms of Peano arithmetic and the axiom of comprehension from ZFC. It doesn’t matter whether it fits your notion of “the whole is greater than the part”. The entire proof is that set comprehension. It exists. Therefore the two sets have the same size.

38 thoughts on “For every natural number N, there's a Cantor Crank C(n)

  1. Comradde PhysioProffe

    I am no expert mathematician, but the different scale of the infinities of the integers and the real numbers seems very intuitively obvious, as does the identical scale of the infinities of the integers and the even (or odd) integers.

    Reply
    1. Yiab

      The different sizes of the integers and the real numbers becomes a lot less intuitively obvious when you take into account the fact that the integers and the algebraic numbers (roots of polynomials with integer coefficients) are sets of the same size (here size=cardinality).

      Reply
  2. Sophie

    Let’s play a round of rescue the crank. The rules are simple: Make a true and sensible statement out of what the crank said, although he obviously did not say it.

    One could argue that he says that Cantors proof works well in the category of sets, where elements are so called “signs”. But to talk about the even numbers as “numbers” we need to look at the category of integral domains without one, because in SET the whole point of the even numbers being even is not visible. Now in this category we have indeed that the even numbers are a true subobject of the whole numbers and the reason for this is that the whole numbers have the number one, which is a unit, while the even numbers don’t have such an element and skipped it.

    Reply
  3. Nick Johnson

    I like to imagine that cantor, while writing up his proof, was giggling to himself and muttering “this is totally going to blow their minds”.

    Regarding the title, are you sure? There seem to be uncountably many cantor cranks.

    Reply
    1. Snoof

      It wouldn’t be hard to find out. We could just hold a convention at Hilbert’s Hotel. If there’s anyone who can’t get a room, then there must be an uncountable number of cranks.

      Reply
  4. Invincible Irony Man

    What’s the problem, that Cantor doesn’t conform to our naive intuitions enough so it must be wrong? Since when was the concept of infinity intuitive in the first place? It seems to me that if you want to know its properties, then do the math! Why assume that your unmathematical concept of infinity – ie. trying to intuit it’s properties without doing the math – is informed by anything other than prejudice?

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  5. Invincible Irony Man

    Carvalho cites Euclid’s fifth common notion “that the whole is greater than its parts”.

    You cite it as “The whole is greater than the part.”

    (You are right)

    It doesn’t take recourse to mathematics to see what is wrong with that!

    Carvalho seems to be confusing Euclid’s fifth common notion with the phrase “the whole is greater than the sum of its parts”, which may sound similar to Euclid, but clearly means something different. Surely Euclid was talking about “part” (singular), not “parts” (plural).

    The phrase “the whole is greater than the sum of its parts” is difficult to source definitively. I can find it attributed to John Stuart Mill (poorly) and Aristotle (for no reason at all, as far as I can tell), or it may be a mis-quote of Kurt Koffka’s phrase, “The whole is other than the sum of the parts” from gestalt psychology.* One thing I am pretty sure of though, it fails pretty miserably as an axiom of mathematics, which is probably why Euclid never said it.

    * One person had it, “the hull is greater than the sum of its ports” – Mr. Spock

    Reply
    1. John

      Carvalho is confused, but not in this. He’s comparing the whole (the integers) with one part (the even integers).
      The problem is that “the whole is greater than the part” comes with the unstated assumption that the part is finite. That’s how Euclid always uses it. It is not necessarily true for infinite objects, but Euclid isn’t interested in them.
      I’m sure I’ve seen this used as a definition of infinite sets: a set is infinite if is bijective with some proper subset of itself.

      Reply
      1. Reinier Post

        In geometry, it is true even for infinite objects, but I think you’re right that Euclid was considering them. In any case, I think you hit the nail on the head. By the way, Euclid’s notion of ‘greater than’ is captured pretty exactly in mathematics by the subset relationship,by which the set of whole numbers is indeed greater than the set of even numbers. However,t it doesn’t follow that the *number* of whole numbers is greater than the *number* of even numbers, because, being infinite sets, they can’t be numbered with natural numbers, and what Cantor showed is that you can in fact extend the notion of numbers to number infinite sets by using the existence of bijections, but at the expense of having to assigning the same number to sets and many of their subsets (e.g. the whiole and even numbers). I think it’s a stretch to call Carvalho a ‘crank’: nobody seems to have explained these missing steps, andhe’s trying to make up his own explanation, which boils down to: by comparing the sets with a bijection, you’re only putting the *denotations* (not: signs) of the numbers into correspondence, and you’re ignoring their magnitudes, and you can’t do that. The problem with that explanation is that he doesn’t succeed in explaining why you can’t ignore them, and that is no surprise, because his feeling that you can’t is based on a confusion between the magnitude of a set and the magnitudes of its elements.

        Reply
  6. Joseph McCauley

    My high school students loved finding out about different infinities. (This is sooo cool!) Still, none of them succumbed to new-age ideas that, therefore, anything is possible. I had some sharp students. No cranks.

    Reply
  7. David

    “For every natural number N, there’s a Cantor Crank C(n)”

    I would have thought that you would need IRRATIONAL numbers to count the Cantor Cranks. Putting them in correspondence with rationals seems to be inviting trouble.

    Reply
    1. delosgatos

      Nope, If N_k denotes crank k corresponding to natural number N, the mapping N_k -> 1 + 2 + … + (N-1) + k is a bijection between the cranks and the natural numbers.

      Reply
  8. Davi

    The refutation seems pretty clear and well funded to me.

    Of course there is a mapping. Simply because the sets are the same! That is the refutation. The set of integers and the set of evens are the same set. Where’s the cranckery in this?

    The refutation simply says that Cantor’s error is not in the mapping, but in the sets. For Cantor, the set of integers is ONE set and the set of evens is ANOTHER set (different from the ONE).

    The counter argument to the refutation only addresses some notions of Euclid which seem hardly to be the main case or the foundation of it.

    The fact that Cantor did not address representations or signs, is EXACTLY the reason why he failed to understand his mistake. When he dealt with the set of even numbers, he did not realize that this set is the same as the set of integers, BUT represented in a different way.

    Maybe I’m wrong. But the refutation has nothing to do with math, simply because the error is not mathematical per se. It’s simply a confusion between signs and quantities. I.e., the sign “4” is not the double of the sign “2”, but the quantity “Four” is the double of quantity “Two”. Cantor is mixing both and simply stating that the symbol “4” (which sometimes is used to indicate the quantity “four”) is the actual double of the symbol “2” (which sometimes is used to indicate the quantity “two”). Put in simpler terms, it would be as if Cantor is saying that “D” is the double of “B”, and when finding out that the number of letters is the same (have same cardinality), he deduced some strange notions for the set that contains “B” and the set that contains “D” as if they were different.

    Reply
    1. delosgatos

      He’s talking about sets, not signs or quantities. It’s not that you’re wrong, it’s that you’re not *even* wrong.

      Reply
  9. Vicki

    Davi,

    If “four” is a quantity and “4” is an arbitrary sign, we can do this mapping by spelling out those quantities using the Latin alphabet: one, two, three, four, five….

    The mapping gives us one → two
    two → four
    three → six

    It’s more tedious to do this way than with the conventional Arabic numerals, but it’s the same valid mapping of the natural numbers onto the even numbers.

    This is the same sort of error as asserting that a proof is valid in English but not in Portuguese, or vice versa: that’s a sign of a flawed translation, not evidence that the math is different in Rio than in New York.

    (You could write it as “map I onto II, map II onto IV, map III onto VI, map IV onto VIII,…” and risk new sources of confusion confusion, but the actual statements wouldn’t change.)

    Reply
    1. Davi

      I’m sorry, but you seem to be getting back to “mapping”. Again, the problem is not in the mapping. The problem is in the sets.

      It’s sort of like this: You have houses. Identical to one another. Side by side. In order to differentiate the houses you decide to put a label on them. The labels are “1”, “2, “3”, “4”…. Is it easy to see that the house labeled as “4” is not the double of the house labeled as “2”? If you change the labels of the houses to “2”, “4”, “6”, “8”… you don’t get the doubles of the houses neither the “even” houses. You get the same houses labeled in a different way.

      Cantor’s mistake is this: He took an infinite set labeled as “1”, “2”, “3”; later on, took the same infinite set labeled as “2”, “4”, “6”, BUT he did not notice that they were the same set. Because they were labeled in a different way, Cantor thought that they were different sets but with the same “size” (cardinality). He failed to grasp that the mapping only occurred because they were the same sets!

      It’s like getting the sets {1,2,3} and {2,4,6} and saying that they are “trans-three-finites” sets.

      Reply
      1. delosgatos

        Don’t you think it’s rather interesting that you can throw out all the odd numbered labels on the houses labeled {1,2,3,4,5…}, and then redistribute all the remaining labels so that every house has a label and there are none left over?

        Note that we didn’t throw out any houses, so clearly the set of houses originally labeled with even labels is different from the set of houses ultimately labeled with even labels.

        Also note that this doesn’t work for any finite set of houses.

        Reply
        1. Davi

          Of course it’s interesting. But only because infinity is interesting (hint: the left over part is where it gets interesting). As it’s also interesting that you can throw out all numbered labels and substitute them for {!,@,#,$,%….} such that each symbol is different from all the previous ones and finding out that all houses have labels and there are none left over.

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      2. John

        You appear to be claiming that the natural numbers are indistinguishable (at least, that’s what appears to be the case from your identical houses). They aren’t. Arithmetic would be impossible otherwise. Students could write ‘7’ as the answer to every problem, and be right. Chaos. But fortunately, numbers have identities independent of the names we give them.
        The labels are arbitrary, but once we’ve chosen an assignment we generally don’t change it. The two sets {1, 2, 3, …} and {2, 4, 6, …}, if they’re using the same assignment of labels to numbers, are different sets. One of them contains the number one, and the other doesn’t. You’re not claiming that the number one and the number two are the same, are you?

        Reply
        1. Davi

          From the refutation point of view, you’re getting back to Cantor’s confusion. You’re talking about quantities and signs and mixing both. The quantity represented by the number “1” is not the same as the quantity represented by the number “2”. But if I’m simply relabelling a house from “1” to “2”, then yes 1 and 2 are the same. They are the same house! Only the label changed.

          Reply
          1. John

            But numbers are not houses, and Cantor-style proofs do not change the identity of the elements of the sets, or rely on (or indeed use) any kind of labelling. And even your houses can be distinguished, by measuring how far they are along the road (which is infinite only in one direction).
            Start with two sets, { 1, 2, 3, … } and { 2, 4, 6, … }. They are different sets, as can be seen from the presence of 1 in the first but not the second. 1 is a number that we can recognise, and distinguish from any other (no other number is the multiplicative identity, for a start).
            Now we define cardinality. Two sets have the same cardinality if we can find a bijection between them. There is no labelling going on. We’re just lining up the elements side by side, and seeing that for each element in one set there is one in the other.
            In this case, we can define the mapping 12, 23, 24, and so on. We haven’t done anything to the elements of the two sets; we’ve just found a partner for each element from the other set. 1 is still 1, 2 is still 2. All we’re saying is that if we put 1 from the first set next to 2 from the second (and similarly with the rest), then every element has a partner. You seem to be misunderstanding this bit.
            Since there is a bijection, the two sets have the same cardinality.

      3. clonus

        Sets are defined by their members, so {1,2,3…} and {2,4,6…} really are different. By {1,2,3..} we literally mean the set containing the natural numbers, not “a set with a bunch things labeled with numbers”.

        Reply
  10. Robin Adams

    Just for historical interest: the discovery that an infinite set can be the same size as a proper subset is older than Cantor. Galileo noted it in his ”Two New Sciences” (1638).

    Reply
  11. brunosaboia

    I can translate the portuguese for you, if you want. Very interesting post, anyway, thanks. I myself struggled to grasp it a long time ago, but I understood it with the “bijective” approach.

    Reply
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  13. Pedro Goncalves

    I have browsed through the books’ ideas and i think it’s more of a provocation than anything else. He’s looking at Cantor from a philosophical point of view, not mathematical. Also if we followed Olavo’s method literally anything we can define would be true just because we defined it. Again, a strictly metaphysical notion.

    Reply
  14. Marcelo

    I remember reading two or three “essays” from this guy Olavo de Carvalho some years ago, but didn’t have any math on it… It was all about a worldwide conspiracy trying to disrupt the sanctity of the traditional christian family.

    Crank magnetism?

    Reply
  15. incessable

    I think you are being unfair to Carvalho.

    When we limit our consideration to finite sets, we find the following two principles hold:
    1) Two sets are the same size if there exists a one-to-one correspondence between their members
    2) If A is a proper subset of B, then the size of A is less than the size of B
    Now, when we extend our consideration to infinite sets, we find we cannot maintain both principles without falling into contradiction. So Cantor chose to keep principle (1) and give up principle (2). I think it is fair to ask what would be the results of making the opposite choice to what Cantor did; can one give a coherent theory of infinite sets and cardinals based on that alternative choice? How mathematically useful would that alternative be?

    I don’t know what the mathematical consequences of the alternative choice would be. (I take it that mereology might be that alternative, as Peter Krautzberger mentions.) When we look at these sort of junctures in the history of mathematics, when mathematics began to be developed along one possible line instead of another, it is always interesting to pose the question, to what extent was that choice of direction determined by the inherent nature of mathematics itself (i.e. it may be far easier to develop useful results given one set of axioms than another, which might cause a historical preference for the easier set of axioms), and to what extent these choices are determined by historical accidents, cultural factors, influential individuals, etc. (Similar cases include constructivism vs. non-constructivism, standard vs non-standard analysis, classical vs paraconsistent logic, etc.)

    As a philosopher, Carvalho may believe that there are philosophical reasons to prefer keeping (2) to keeping (1). His reasons may or may not be convincing (but both his position and its opposite are equally in need of philosophical justification.) I think to simply dismiss him as a “crank” is to show a lack of intellectual charity.

    If we accept formalism, then Cantor’s choice of one set of axioms is inherently no more correct than any competing axiom system. Maybe we reject formalism, and argue that Cantor’s axioms are in some objective sense more right than their competitors – well, that’s some interesting philosophical territory you are venturing into then, I’m listening.

    Reply
    1. David Starner

      It’s quite easy to work out the consequences of upholding “If A is a proper subset of B, then the size of A is less than the size of B” and dismissing “Two sets are the same size if there exists a one-to-one correspondence between their members”; there’s no meaningful concept of the idea of size. {1, 2, 3, 4, 5, …} is neither larger nor smaller than {0, 1, 3, 4, 5, …}. It’s conceivable to hedge around that, and other stuff, but not very long if you don’t accept that N={0, 1, 2, 3, 4, 5 …) and P = {x is an element of N|x + 1} = {1, 2, 3, 4, 5 …} are the same size. Are the sets of the even numbers and the odd numbers the same size?

      It’s nice to say Cantor’s system “inherently no more correct”, but Carvalho is saying that Cantor’s system is wrong, and giving no axiomatic alternatives, or detailed system.

      Cantor’s axioms are in some objective sense better then their competitors; they give us a system where every set has a cardinal that represents its size and the cardinals are an ordered set. I’m not sure what competitors you’re talking about, but Carvalho can’t offer that.

      Maybe I was brainwashed with this stuff from an early age, but take Infinity Hotel, which has rooms numbered 1, 2, 3, … . It’s full right now, with one guest per room, but the desk has someone new show up, so they ask all their current guests to move down one room so the new guest can move into room 1. You logically have the same number of full rooms as the number of guests occupying them, so the number of guests before the move and after the move must be the same, despite the addition of one. If you have a convention, with an infinite number of guests show up, you can just have the current guests move from room x to room 2x. Again, the number of rooms hasn’t changed, so the number of guests hasn’t changed. (This example courtesy of someone writing for Isaac Asimov Science Fiction Magazine in the 1980s.)

      Reply

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