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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.

In short, his point is that it doesn’t make sense to talk about the set of even numbers and the set of all integers as if they were two separate sets. They are essentially the same set, only with different names, or rather, the same set as seen from different points of view. According to him, this is the fallacy upon which Cantor’s proof is based. Is he right? What is the mathematical response to Carvalho’s argument?

Edit: From the answers and comments, I've realized that the term "greater" is troublesome. We should use, instead, the more precise terms "subset" and "cardinality". But there is still the problem of whether the set of integers and the set of evens are the same or not. Carvalho argues that they are the same, and rests his argument on the distinction between numbers ("quantities", in the Aristotelian tradition) and the symbols used to represent those numbers. I would appreciate if this point was addressed in more detail.

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migrated from math.stackexchange.com Dec 19 '11 at 0:46

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If I really understand the philosopher point of view, he's confusing set with class of equipotent sets. –  Pacciu Dec 19 '11 at 0:45
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"...and, being infinite, both sets will have the same number of elements..." This is one of the signs that Carvalho does not understand the argument he is criticizing. But Cantor's theory of the infinite doesn't refute Euclid either, so I disagree with the whole premise. –  Jonas Meyer Dec 19 '11 at 1:05
    
So, what is that business about signs vs. numbers? Math generally deals with numbers, not representations, and Cantor's proof deals specifically with numbers. –  David Thornley May 24 '13 at 3:35

8 Answers 8

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First of all, I'm surprised this was migrated from mathematics; it seems to me to be a much better fit there than here in philosophy. That being said:

In short, his point is that it doesn’t make sense to talk about the set of even numbers and the set of all integers as if they were two separate sets. They are essentially the same set, only with different names, or rather, the same set as seen from different points of view.

That's not a refutation of Cantor, that's a restatement of it. If the set of even numbers is the same size as the set of integers, Cantor prevails.

There is an enormous cottage industry of Cantor cranks-- amateurs without a substantial background in mathematics who think they have a way to refute Cantor. I am not familiar with Olavo de Carvalho, but a perusal of his Wikipedia entry suggests that he falls firmly into this category.

EDIT:

Since the question was updated to draw attention to a specific part of de Carvalho's argument, let's look at that in more detail.

In short, his point is that it doesn’t make sense to talk about the set of even numbers and the set of all integers as if they were two separate sets. They are essentially the same set, only with different names, or rather, the same set as seen from different points of view. According to him, this is the fallacy upon which Cantor’s proof is based. Is he right?

Is he right? No. What is is involved in is mere wordplay. The set of even numbers is a set, and it has the same cardinality as the set of integers. The set of prime numbers is also a set, with the same cardinality, and this is clearly not a case of "counting by twos". His objection is nonsensical.

Again: de Carvalho has all the earmarks of a crank. Until his "refutation" is published in a peer-reviewed mathematical journal (and if it were a valid refutation, the math journals would be fighting over the publication rights) I see no reason to take it seriously.

If you are interested in Cantor crankery in general, you'll find some amusing reference cases here and here.

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Page Not Found! for the second link. –  Vladimir Putin Nov 2 '13 at 8:33

This is not a mathematical argument, so no mathematical response is necessary. Using the standard axioms of set theory and the standard mathematical definition of "cardinality", it is an absolutely true statement that the cardinality of the even numbers is the same as the cardinality of the integers. One can argue about whether the notion of cardinality used in mathematics corresponds to the intuitive notion of "size", but this argument does not affect the truth of Cantor's theorem as interpreted by mathematicians.

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Euclid and Cantor, whatever they are saying, are using a refined and stipulative vocabulary: 'set', 'whole' 'greater' 'part' mean something very exact to them ('greater' has a different meaning between them). Euclid uses 'greater' like the more modern (actually Cantor's) definition of 'subset'. Cantor interprets 'greater' not as the subset relation, but as the 'cardinalilty' relation. (actually, Euclid's notion is somewhat underspecified, but like most of his mathematics, is not wrong at all, just in need of the slightest clarification, which is presumptuous to say since it took more than two thousand years to get that clarification via Cantor (for sets) and Hilbert (for pure axiomaticity).

Carvalho, from your explanation, seems to be interpreting these words non-mathematically, or with an interpretation that doesn't match what either Euclid or Cantor meant. The set of natural numbers can be labeled one way, N, the set of evens another, E. Yes, E is a subset of N (there are things in N that are not in E), but also |E| = |N| because of the mapping n in N -> f(n) = 2n, and so f(n) is in E (and you can go back directly, that is there is a one-to-one onto correspondence between N and E, which defines 'same cardinality'.

Carvalho says Cantor is 'confusing numbers with their mere signs'. I think Carvalho is misunderstanding how labeling of mathematical objects works; he is confounding the proof of cardinality with the construction of the objects at hand. The ordinal nature of the naturals (and the evens) is irrelevant to the proof.

There is a question as to why this problem is even being discussed under philosophy when it is a matter of elementary mathematics. Mathematics is a philosophical exercise, it is pure mental manipulation of concepts, except it comes out the symbolic tradition rather than the humanistic one. Carvalho's difficulty with the problem is just a mature and well-spoken misunderstanding that in the mathematical tradition is a only stumbling block for newcomers to set theory.

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Let me remark that when Euclid wrote that the whole is greater than its parts, he didn't refer to cardinality of sets. Indeed, the idea of cardinality did not exist at the time, and what Euclid had in mind would in modern terminology be the idea of a "measure". When he applies this postulate, it is to conclude that a subset of a line segment has smaller length, or that a figure drawn inside of another figure has a smaller area, etc.

Note also that this still holds today: if A is a subset of B, both measurable, then μ(A) ≤ μ(B).

It is possible that Cantor believed that he had refuted Euclid, but then Cantor was reading Euclid in an ahistorical way.

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It needs to be stressed that in (modern) set theory we have to construct sets and don't get something for nothing.

That is, {1,2,3,...} {2,4,6,...} {2,3,5,7,11,...} are all different sets regardless of how you look at them (I am guessing your thinking of sets naively as something you just 'generate formally' and so a 'set' formed by the symbols {1,2,3,4,...} will be the same - in your opinion - as {2,4,6,8,...} because you don't attach any meaning to these symbols?).

It happens that if we consider the class of equipotent sets that we do indeed have a class defined by an equivalence relation (which is what you perhaps meant).

However, we don't attach 'the same' 'equals' 'not different' to two sets simply because they're of the same cardinilty.

It's even frowned upon to do it when they're isomorphic!

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3 is in the set of integers but not in the set of even numbers. I honestly don't think anything else is needed to refute the argument.

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Quoting another passage of his argument: "'parity' and 'place in the sequence' are inseparable concepts: if n is even, it is because both n + 1 and n - 1 are odd." In other words, the set of evens would not exist if the set of odds didn't exist as well. To count {2, 4, ...} you have to skip the number 3. It's as though he is saying that the number 3 is in the sequence of even numbers, albeit "hidden". –  Otavio Macedo Dec 19 '11 at 15:36
    
I don't see how you know when to stop with this argument. Why not then argue then that there really is just one set -- the "set of all things" and that when I talk about my left index finger, everything else is just "hidden"? –  David Schwartz Dec 19 '11 at 16:07
    
"why not then argue that there is really just one set -- the set of all things" Because of Russell's paradox. [But I agree with your main point] –  Seamus Dec 20 '11 at 13:33

But there is still the problem of whether the set of integers and the set of evens are the same or not

You can't simply rename a set of symbols without renaming them in their related sets because otherwise these symbols couldn't have any meaning at all. For instance, I can't take an equation x = y + 1 and then rename x to y and say y = y + 1.

The set of even numbers is by definition a (proper) subset of the integers no matter which symbols you use, and this alone makes it a different set.

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I want to move from 1-space to 3-space, by talking about the three-dimensional version of Cantor's proof: the Banach–Tarski paradox:

Given a solid ball in 3‑dimensional space, there exists a decomposition of the ball into a finite number of non-overlapping pieces (i.e., disjoint subsets), which can then be put back together in a different way to yield two identical copies of the original ball. The reassembly process involves only moving the pieces around and rotating them, without changing their shape. However, the pieces themselves are not "solids" in the usual sense, but infinite scatterings of points.

Crucially, this paradox requires the axiom of choice, about which there exists skepticism. Here is a modified informal definition:

Informally put, the axiom of choice says that given any collection of bins, each containing at least one object, it is possible to make a selection of exactly one object from each bin. In many cases such a selection can be made without invoking the axiom of choice; this is in particular the case if the number of bins is finite, or if a selection rule is available: a distinguishing property that happens to hold for exactly one object in each bin. To give an informal example, for any (even infinite) collection of pairs of shoes, one can pick out the left shoe from each pair to obtain an appropriate selection, but for an infinite collection of pairs of socks (assumed to have no distinguishing features), such a selection can be obtained only by invoking the axiom of choice.

I claim there is a problem with Cantor's use of a well-ordered set: when it comes to real objects, well-ordering often does not exist. For example, there can exist a set of bosons which are indistinguishable by quantum numbers, which allow awesome forms of matter such as Bose-Einstein condensates. But it also means that we must utilize the axiom of choice if we want to 'pick' them. Now, there is a bit of a finity-infinity problem when it comes to talking about quantized matter, but the point behind Cantor's proof is that he assumes real infinities, so we should allow him that assumption.

The fact that Cantor doesn't need the axiom of choice for one dimension is perhaps just an artifact. After all, we really live in 3+1 dimensions. I say that this axiom of choice is suspicious, and is the intuition behind the suspicion that Cantor's proof is not so clearly valid as one might have thought.

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