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.