Is Cantor's theorem based on a fallacy?

Question

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.

Answer 1

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.

Answer 2

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.

Answer 3

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 logic/mathematics. Mathematics is a philosophical exercise, it is a pure mental manipulation of concepts, except it comes out of the scientific tradition rather than the humanistic one. Carvalho's difficulty with the problem is just a mature and well-spoken misunderstanding which in the mathematical tradition is only a minor stumbling block to set theory for newcomers.

Answer 4

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.

Answer 5

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!

Answer 6

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.