by E. T. Bell
in which the c’s are any given algebraic numbers (as defined above), is itself an algebraic number. For example, according to this theorem, all roots of
are algebraic numbers, since the coefficients are. (The first coefficient satisfies x2 −2x + 10 = 0, the second, x2 - 4x− 421 =0, the third, x3 −90 = 0, of the respective degrees 2, 2, 3.)
Imagine (if you can) the set of all algebraic numbers. Among these will be all the positive rational integers 1, 2, 3, . . ., since any one of them, say n, satisfies an algebraic equation, x—n = 0, in which the coefficients (l, and —n) are rational integers. But in addition to these the set of all algebraic numbers will include all roots of all quadratic equations with rational integer coefficients, and all roots of all cubic equations with rational integer coefficients, and so on, indefinitely. Is it not intuitively evident that the set of all algebraic numbers will contain infinitely more members than its sub-set of the rational integers 1, 2, 3, . . .? It might indeed be so, but it happens to be false.
Cantor proved that the set of all rational integers 1, 2, 3, . . . contains precisely as many members as the “infinitely more inclusive” set of all algebraic numbers.
A proof of this paradoxical statement cannot be given here, but the kind of device—that of “one-to-one correspondence”—upon which the proof is based can easily be made intelligible. This should induce in the philosophical mind an understanding of what a cardinal number is. Before describing this simple but somewhat elusive concept it will be helpful to glance at an expression of opinion on this and other definitions of Cantor’s theory which emphasizes a distinction between the attitudes of some mathematicians and many philosophers toward all questions regarding “number” or “magnitude.”
“A mathematician never defines magnitudes in themselves, as a philosopher would be tempted to do; he defines their equality, their sum and their product, and these definitions determine, or rather constitute, all the mathematical properties of magnitudes. In a yet more abstract and more formal manner he lays down symbols and at the same time prescribes the rules according to which they must be combined; these rules suffice to characterize these symbols and to give them a mathematical value. Briefly, he creates mathematical entities by means of arbitrary conventions, in the same way that the several chessmen are defined by the conventions which govern their moves and the relations between them.”II Not all schools of mathe matical thought would subscribe to these opinions, but they suggest at least one “philosophy” responsible for the following definition of cardinal numbers.
Note that the initial stage in the definition is the description of “same cardinal number,” in the spirit of Couturat’s opening remarks; “cardinal number” then arises phoenix-like from the ashes of its “sameness.” It is all a matter of relations between concepts not explicitly defined.
Two sets are said to have the same cardinal number when all the things in the sets can be paired off one-to-one. After the pairing there are to be no unpaired things in either set.
Some examples will clarify this esoteric definition. It is one of those trivially obvious and fecund nothings which are so profound that they are overlooked for thousands of years. The sets (x, y, z), (a, b, c) have the same cardinal number (we shall not commit the blunder of saying “Of course! Each contains three letters”) because we can pair off the things x, y, z in the first set with those, a, b, c in the second as follows, x with a, y with b, z with c, and having done so, find that none remain unpaired in either set. Obviously there are other ways for effecting the pairing. Again, in a Christian community practising technical monogamy, if twenty married couples sit down together to dinner, the set of husbands will have the same cardinal number as the set of wives.
As another instance of this “obvious” sameness, we recall Galileo’s example of the set of all squares of positive integers and the set of all positive integers:
12, 22, 32, 42, . . . , n2, . . .
1, 2, 3, 4, . . ., n, . . .
The “paradoxical” distinction between this and the preceding examples is apparent. If all the wives retire to the drawing room, leaving their spouses to sip port and tell stories, there will be precisely twenty human beings sitting at the table, just half as many as there were before. But if all the squares desert the natural numbers, there are just as many left as there were before. Dislike it or not as we may (we should not, if we are rational animals), the crude miracle stares us in the face that a part of a set may have the same cardinal number as the entire set. If anyone dislikes the “pairing” definition of “same cardinal number,” he may be challenged to produce a comelier. Intuition (male, female, or mathematical) has been greatly overrated. Intuition is the root of all superstition.
Notice at this stage that a difficulty of the first magnitude has been glossed. What is a set, or a class? “That,” in the words of Hamlet, is “the question.” We shall return to it, but we shall not answer it. Whoever succceeds in answering that innocent question to the entire satisfaction of Cantor’s critics will quite likely dispose of the more serious objections against his ingenious theory of the infinite and at the same time establish mathematical analysis on a non-emotional basis. To see that the difficulty is not trivial, try to imagine the set of all positive rational integers 1, 2, S, . . . , and ask yourself whether, with Cantor, you can hold this totality—which is a “class”—in your mind as a definite object of thought, as easily apprehended as the class x, y, z of three letters. Cantor requires us to do just this thing in order to reach the transfinite numbers which he created.
Proceeding now to the definition of “cardinal number,” we introduce a convenient technical term: two sets or classes whose members can be paired off one-to-one (as in the examples given previously) are said to be similar. How many things are there in the set (or class) x, y,f z? Obviously three. But what is “three”? An answer is contained in the following definition: “The number of things in a given class is the class of all classes that are similar to the given class.”
This definition gains nothing from attempted explanation; it must be grasped as it is. It was proposed in 1879 by Gottlob Frege, and again (independently) by Bertrand Russell in 1901. One advantage which it has over other definitions of “cardinal number of a class” is its applicability to both finite and infinite classes. Those who believe the definition too mystical for mathematics can avoid it by following Couturat’s advice and not attempting to define “cardinal number.” However, that way also leads to difficulties.
Cantor’s spectacular result that the class of all algebraic numbers is similar (in the technical sense defined above) to its sub-class of all the positive rational integers was but the first of many wholly unexpected properties of infinite classes. Granting for the moment that his reasoning in reaching these properties is sound, or, if not unobjectionable in the form in which Cantor left it, that it can be made rigorous, we must admit its power.
Consider for example the “existence” of transcendental numbers. In an earlier chapter we saw what a tremendous effort it cost Hermite to prove the transcendence of a particular number of this kind. Even today there is no general method known whereby the transcendence of any number which we suspect is transcendental can be proved; each new type requires the invention of special and ingenious methods. It is suspected, for example, that the number (it is a constant, although it looks as if it might be a variable from its definition) which is defined as the limit of
as n tends to infinity, is transcendental, but we cannot prove that it is. What is required is to show that this constant is not a root of any algebraic equation with rational integer coefficients.
All this suggests the question “How many transcendental numbers are there?” Are they more numerous than the integers, or the rationals, or the algebraic numbers as a whole, or are they less numerous? Since (by Cantor’s theorem) the integers, the rationals, and all algebraic numbers are equally numerous, the question amounts to this: can the transcendental numbers be counted off 1, 2, 3, . . .
? Is the class of all transcendental numbers similar to the class of all positive rational integers? The answer is no; the transcendentals are infinitely more numerous than the integers.
Here we begin to get into the controversial aspects of the theory of sets. The conclusion just stated was like a challenge to a man of Kronecker’s temperament. Discussing Lindemann’s proof that π is transcendental (see Chapter 24), Kronecker asked, “Of what use is your beautiful investigation regarding π? Why study such problems, since irrational [and hence transcendental] numbers do not exist?” We can imagine the effect on such a skepticism of Cantor’s proof that the transcendentals are infinitely more numerous than the integers 1, 2, 3, . . . which, according to Kronecker, are the noblest work of God and the only numbers that do “exist.”
Even a summary of Cantor’s proof is out of the question here, but something of the kind of reasoning he used can be seen from the following simple considerations. If a class is similar (in the above technical sense) to the class of all positive rational integers, the class is said to be denumerable. The things in a denumerable class can be counted off 1, 2, 3, . . .; the things in a non-denumerable class can not be counted off 1, 2, 3, . . . : there will be more things in a non-denumerable class than in a denumerable class. Do non-denumerable classes exist? Cantor proved that they do. In fact the class of all points on any line-segment, no matter how small the segment is (provided it is more than a single point), is non-denumerable.
From this we see a hint of why the transcendentals are non-denumerable. In the chapter on Gauss we saw that any root of any algebraic equation is representable by a point on the plane of Cartesian geometry. All these roots constitute the set of all algebraic numbers, which Cantor proved to be denumerable. But if the points on a mere line-segment are non-denumerable, it follows that all the points on the Cartesian plane are likewise non-denumerable. The algebraic numbers are spotted over the plane like stars against a black sky; the dense blackness is the firmament of the transcendentals.
The most remarkable thing about Cantor’s proof is that it provides no means whereby a single one of the transcendentals can be constructed. To Kronecker any such proof was sheer nonsense. Much milder instances of “existence proofs” roused his wrath. One of these in particular is of interest as it prophesied Brouwer’s objection to the full use of classical (Aristotelian) logic in reasoning about infinite sets.
A polynomial axn + bxn–1 + . . . + l, in which the coefficients a, b, . . . l are rational numbers is said to be irreducible if it cannot be factored into a product of two polynomials both of which have rational number coefficients. Now, it is a meaningful statement to most human beings to assert, as Aristotle would, that a given polynomial either is irreducible or is not irreducible.
Not so for Kronecker. Until some definite process, capable of being carried out in a finite number of nontentative steps, is provided whereby we can settle the reducibility of any given polynomial, we have no logical right, according to Kronecker, to use the concept of irreducibility in our mathematical proofs. To do otherwise, according to him, is to court inconsistencies in our conclusions and, at best, the use of “irreducibility” without the process described, can give us only a Scotch verdict of “not proven.” All such non-constructive reasoning is—according to Kronecker—illegitimate.
* * *
As Cantor’s reasoning in his theory of infinite classes is largely non-constructive, Kronecker regarded it as a dangerous type of mathematical insanity. Seeing mathematics headed for the madhouse under Cantor’s leadership, and being passionately devoted to what he considered the truth of mathematics, Kronecker attacked “the positive theory of infinity” and its hypersensitive author vigorously and viciously with every weapon that came to his hand, and the tragic outcome was that not the theory of sets went to the asylum, but Cantor. Kronecker’s attack broke the creator of the theory.
In the spring of 1884, in his fortieth year, Cantor experienced the first of those complete breakdowns which were to recur with varying intensity throughout the rest of his long life and drive him from society to the shelter of a mental clinic. His explosive temper aggravated his difficulty. Profound fits of depression humbled himself in his own eyes and he came to doubt the soundness of his work. During one lucid interval he begged the authorities at Halle to transfer him from his professorship of mathematics to a chair of philosophy. Some of his best work on the positive theory of the infinite was done in the intervals between one attack and the next. On recovering from a seizure he noticed that his mind became extraordinarily clear.
Kronecker perhaps has been blamed too severely for Cantor’s tragedy; his attack was but one of many contributing causes. Lack of recognition embittered the man who believed he had taken the first—and last—steps toward a rational theory of the infinite and he brooded himself into melancholia and irrationality. Kronecker however does appear to have been largely responsible for Cantor’s failure to obtain the position he craved in Berlin. It is usually considered not quite sporting for one scientist to deliver a savage attack on the work of a contemporary to his students. The disagreement can be handled objectively in scientific papers. Kronecker laid himself out in 1891 to criticize Cantor’s work to his students at Berlin, and it became obvious that there was no room for both under one roof. As Kronecker was already in possession, Cantor resigned himself to staying out in the cold.
However, he was not without some comfort. The sympathetic Mittag-Leffler not only published some of Cantor’s work in his journal (Acta Mathematica) but comforted Cantor in his fight against Kronecker. In one year alone Mittag-Leffler received no less than fifty two letters from the suffering Cantor. Of those who believed in Cantor’s theories, the genial Hermite was one of the most enthusiastic. His cordial acceptance of the new doctrine warmed Cantor’s modest heart: “The praises which Hermite pours out to me in this letter . . . on the subject of the theory of sets are so high in my eyes, so unmerited, that I should not care to publish them lest I incur the reproach of being dazzled by them.”
* * *
With the opening of the new century Cantor’s work gradually came to be accepted as a fundamental contribution to all mathematics and particularly to the foundations of analysis. But unfortunately for the theory itself the paradoxes and antinomies which still infect it began to appear simultaneously. These may in the end be the greatest contribution which Cantor’s theory is destined to make to mathematics, for their unsuspected existence in the very rudiments of logical and mathematical reasoning about the infinite was the direct inspiration of the present critical movement in all deductive reasoning. Out of this we hope to derive a mathematics which is both richer and “truer”—freer from inconsistency—than the mathematics of the pre-Cantor era.
Cantor’s most striking results were obtained in the theory of non-denumerable sets, the simplest example of which is the set of all points on a line-segment. Only one of the simplest of his conclusions can be stated here. Contrary to what intuition would predict, two unequal line-segments contain the same number of points. Remembering that two sets contain the same number of things if, and only if, the things in them can be paired off one-to-one, we easily see the reasonableness of Cantor’s conclusion. Place the unequal segments AB, CD as in the figure. The line OPQ cuts CD in the point P, and AB in Q; P and Q are thus paired off. As OPQ rotates about 0, the point P traverses CD, while Q simultaneously traverses AB, and each point of CD has one, and only one, “paired” point of AB.
An even more unexpected result can be proved. Any line-segment, no matter how small, contains as many points as an infinite straight line. Further, the segment contains as many points as there are in an entire plane, or in the whole of three-dimensional space, or in the whole of space of n dimensions (where n is any integer greater than zero) or, finally, in a space of a denumerably infinite number of dimensions.
In all this we have not yet attempted to define a class or a set. Possibly (as Russell held in 1912) it is not necessary to do so in
order to have a clear conception of Cantor’s theory or for that theory to be consistent with itself—which is enough to demand of any mathematical theory. Nevertheless present disputes seem to require that some clear, self-consistent definition be given. The following used to be thought satisfactory.
A set is characterized by three qualities: it contains all things to which a certain definite property (say redness, or volume, or taste) belongs; no thing not having this property belongs to the set; each thing in the set is recognizable as the same thing and as different from all other things in the set—briefly, each thing in the set has a permanently recognizable individuality. The set itself is to be grasped as a whole. This definition may be too drastic for use. Consider, for example, what happens to Cantor’s set of all transcendental numbers under the third demand.
At this point we may glance back over the whole history of mathematics—or as much of it as is revealed by the treatises of the master mathematicians in their purely technical works—and note two modes of expression which recur constantly in nearly all mathematical exposition. The reader perhaps has been irritated by the repetitious use of phrases such as “we can find a whole number greater than 2,” or “we can choose a number less than n and greater than n −2.” The choice of such phraseology is not merely stereotyped pedantry. There is a reason for its use, and careful writers mean exactly what they say when they assert that “we can find, etc” They mean that they can do what they say.
In sharp distinction to this is the other phrase which is reiterated over and over again in mathematical writing: “There exists.” For example, some would say “there exists a whole number greater than 2,” or “there exists a number less than n and greater than n— 2.” The use of such phraseology definitely commits its user to the creed which Kronecker held to be untenable, unless, of course, the “existence” is proved by a construction. The existence is not proved for the sets (as defined above) which appear in Cantor’s theory.
