Georg Cantor Refuted
The celebrated Georg Cantor believed he could refute Euclid’s 5th principle (that the whole is greater than the part) by arguing that the set of even numbers, although being part of the set of whole numbers, can be put in one-to-one correspondence with it, so that the two sets would have the same number of elements and thus the part would be equal to the whole:
1, 2, 3, 4… n
2, 4, 6, 8… 2n = n
With this demonstration, Cantor and his epigones believed that, along with a principle of ancient geometry, he was also breaking down an established belief of common sense and one of the pillars of classical logic, thus opening up the horizons of a new era of human thought.
This reasoning is based on the assumption that both the set of integers and pairs are actual infinite sets, and it can therefore be rejected by those who believe, with Aristotle, that the quantitative infinite is only potential, never actual. But, even accepting the assumption of the present infinities, Cantor’s demonstration is only a play on words, and not very ingenious at heart.
First, it is true that, if we represent the whole numbers each by a sign (or cipher), we will have there an (infinite) set of signs or ciphers; and if, in this set, we want to highlight the numbers that represent pairs by special signs or numbers, then we will have a “second” set that will be part of the first; and, both being infinite, the two sets will have the same number of elements, confirming Cantor’s argument. But that is to confuse numbers with their mere signs, making an unjustified abstraction of the mathematical properties that define and differentiate numbers from one another and, therefore, implicitly abolishing the same distinction between even and odd, on which the alleged argument is based. “4” is a sign, “2” is a sign, but it is not the sign “4” which is twice as much as 2, but the quantity 4, whether it is represented by this sign or by four little dots. The set of whole numbers may contain more numerical signs than the set of even numbers — since it includes both odd and even signs — but not a greater number of units than that contained in the even series. Cantor’s thesis slips out of this obviousness through the expedient of playing with a double meaning of the word “number”, now using it to designate a defined quantity with certain properties (among which to occupy a certain place in the series of numbers and that of being even or odd), now to designate the mere sign of number, that is, the cipher.
The series of even numbers is only made up of pairs because it is counted two by two, that is, by skipping one unit between each two numbers; if it were not counted like this, the numbers would not be even. It is useless here to resort to the subterfuge that Cantor refers to the mere “ensemble” and not to the “ordered series”; for the set of even numbers would not be of pairs if its elements could not be ordered two by two in an uninterrupted ascending series that progresses by adding 2, never 1; and no number could be considered even if it could freely change places with any other in the series of integers. “Parity” and “place in the series” are inseparable concepts: if n is even, it is because both n + 1 and n — 1 are odd. In this sense, it is only the implicit sum of the units not mentioned that makes the series of pairs, pairs. So — and here is Cantor’s fallacy — there are not two series of numbers here, but one, counted in two ways: the series of even numbers is not really part of the series of whole numbers, but it is the series of numbers itself whole, counted or named in a certain way. The notion of “ensemble” is that, detached abusively from the notion of “series”, it produces all this crazy mess, giving the appearance that even numbers can constitute a “ensemble” regardless of the place of each one in the series, when the fact is that, abstracted the position in the series, there is no more parity or impairment. If the series of whole numbers can be represented by two sets of signs, one only of pairs, the other of pairs plus odds, this does not mean that these are really two distinct series. The confusion that exists there is between “element” and “unity”. A set of x units certainly contains the same number of “elements” as a set of x pairs, but not the same number of units.
What Cantor does is, in essence, substantiate or even hypostise the notion of “pair” or “parity”, assuming that any number can be pair “in itself”, regardless of its place in the series and its relationship with all other numbers (including, of course, with their own half), and that the pairs can be counted as things and not as mere interleaved positions in the whole number series.
In his “argument”, this is not a real distinction between the whole and the part, but a merely verbal comparison between a whole and the same whole, differently called. Since this is not a true whole and a true part, then we cannot speak of an equality of elements between the whole and part, nor, therefore, of a refutation of Euclid’s 5th principle. Cantor misses the target by many meters.
Metaphysics, science of universal possibility, has no limits, and even pure mathematical formalism, when exploring merely imaginary possibilities, does not escape the realm of the conceivable imaginable, as metaphysical as any other. Second, Cantor’s argument about the “two infinities” is that, when presented as a valid refutation of Euclid’s 5th principle, he enters the metaphysical domain, since this principle — the whole is greater than the part — it is obviously metaphysical, since it aims not only to imagine a possible space but to describe a property of real space. Therefore, of the two one: either Cantor’s argument has a metaphysical scope, and can therefore be contested on the metaphysical plane, or it is not a metaphysical argument and it is not valid against Euclid’s 5th principle (nor indeed against anything). Tertium non datur.
That such crude sophisms could pass as serious threats to the foundations of classical geometry and even to the principles of civilization that we inherited from the Greco-Roman tradition, is only the sign of the impotent revolt of the mathematical imagination exacerbated against the real order of things, that that tradition, with all its defects and limitations, incarnates in an exemplary way.
