Introduction to Mathematical Philosophy

THE axiom of infinity is an assumption which may be enunciated as follows:— "If be any inductive cardinal number, there is at least one class of individuals having terms." If this is true, it follows, of course, that there are many classes of individuals having terms, and that the total number of individuals in the world is not an inductive number. For, by the axiom, there is at least one class having terms, from which it follows that there are many classes of terms and that is not the number of individuals in the world. Since is any inductive number, it follows that the number of individuals in the world must (if our axiom be true) exceed any inductive number. In view of what we found in the preceding chapter, about the possibility of cardinals which are neither inductive nor reflexive, we cannot infer from our axiom that there are at least individuals, unless we assume the multiplicative axiom. But we do know that there are at least classes of classes, since the inductive cardinals are classes of classes, and form a progression if our axiom is true. The way in which the need for this axiom arises may be explained as follows:—One of Peano's assumptions is that no two inductive cardinals have the same successor, i.e. that we shall not have unless , if and are inductive cardinals. In Chapter VIII. we had occasion to use what is virtually the same as the above assumption of Peano's, namely, that, if is an inductive cardinal, [Pg 131] is not equal to . It might be thought that this could be proved. We can prove that, if is an inductive class, and is the number of members of , then is not equal to . This proposition is easily proved by induction, and might be thought to imply the other. But in fact it does not, since there might be no such class as . What it does imply is this: If is an inductive cardinal such that there is at least one class having members, then is not equal to . The axiom of infinity assures us (whether truly or falsely) that there are classes having members, and thus enables us to assert that is not equal to . But without this axiom we should be left with the possibility that and might both be the null-class. Let us illustrate this possibility by an example: Suppose there were exactly nine individuals in the world. (As to what is meant by the word "individual," I must ask the reader to be patient.) Then the inductive cardinals from 0 up to 9 would be such as we expect, but 10 (defined as ) would be the null-class. It will be remembered that may be defined as follows: is the collection of all those classes which have a term such that, when is taken away, there remains a class of terms. Now applying this definition, we see that, in the case supposed, is a class consisting of no classes, i.e. it is the null-class. The same will be true of , or generally of , unless is zero. Thus 10 and all subsequent inductive cardinals will all be identical, since they will all be the null-class. In such a case the inductive cardinals will not form a progression, nor will it be true that no two have the same successor, for 9 and 10 will both be succeeded by the null-class (10 being itself the null-class). It is in order to prevent such arithmetical catastrophes that we require the axiom of infinity. As a matter of fact, so long as we are content with the arithmetic of finite integers, and do not introduce either infinite integers or infinite classes or series of finite integers or ratios, it is possible to obtain all desired results without the axiom of infinity. That is to say, we can deal with the addition, multiplication, [Pg 132] and exponentiation of finite integers and of ratios, but we cannot deal with infinite integers or with irrationals. Thus the theory of the transfinite and the theory of real numbers fails us. How these various results come about must now be explained. Assuming that the number of individuals in the world is , the number of classes of individuals will be . This is in virtue of the general proposition mentioned in Chapter VIII. that the number of classes contained in a class which has members is . Now is always greater than . Hence the number of classes in the world is greater than the number of individuals. If, now, we suppose the number of individuals to be 9, as we did just now, the number of classes will be , i.e. 512. Thus if we take our numbers as being applied to the counting of classes instead of to the counting of individuals, our arithmetic will be normal until we reach 512: the first number to be null will be 513. And if we advance to classes of classes we shall do still better: the number of them will be , a number which is so large as to stagger imagination, since it has about 153 digits. And if we advance to classes of classes of classes, we shall obtain a number represented by 2 raised to a power which has about 153 digits; the number of digits in this number will be about three times . In a time of paper shortage it is undesirable to write out this number, and if we want larger ones we can obtain them by travelling further along the logical hierarchy. In this way any assigned inductive cardinal can be made to find its place among numbers which are not null, merely by travelling along the hierarchy for a sufficient distance.[26] [26]On this subject see Principia Mathematica, vol. II. * 120 ff. On the corresponding problems as regards ratio, see ibid., vol. III. * 303 ff. As regards ratios, we have a very similar state of affairs. If a ratio is to have the expected properties, there must be enough objects of whatever sort is being counted to insure that the null-class does not suddenly obtrude itself. But this can be insured, for any given ratio , without the axiom of [Pg 133] infinity, by merely travelling up the hierarchy a sufficient distance. If we cannot succeed by counting individuals, we can try counting classes of individuals; if we still do not succeed, we can try classes of classes, and so on. Ultimately, however few individuals there may be in the world, we shall reach a stage where there are many more than objects, whatever inductive number may be. Even if there were no individuals at all, this would still be true, for there would then be one class, namely, the null-class, 2 classes of classes (namely, the null-class of classes and the class whose only member is the null-class of individuals), 4 classes of classes of classes, 16 at the next stage, 65,536 at the next stage, and so on. Thus no such assumption as the axiom of infinity is required in order to reach any given ratio or any given inductive cardinal. It is when we wish to deal with the whole class or series of inductive cardinals or of ratios that the axiom is required. We need the whole class of inductive cardinals in order to establish the existence of , and the whole series in order to establish the existence of progressions: for these results, it is necessary that we should be able to make a single class or series in which no inductive cardinal is null. We need the whole series of ratios in order of magnitude in order to define real numbers as segments: this definition will not give the desired result unless the series of ratios is compact, which it cannot be if the total number of ratios, at the stage concerned, is finite. It would be natural to suppose—as I supposed myself in former days—that, by means of constructions such as we have been considering, the axiom of infinity could be proved. It may be said: Let us assume that the number of individuals is , where may be 0 without spoiling our argument; then if we form the complete set of individuals, classes, classes of classes, etc., all taken together, the number of terms in our whole set will be which is . Thus taking all kinds of objects together, and not [Pg 134] confining ourselves to objects of any one type, we shall certainly obtain an infinite class, and shall therefore not need the axiom of infinity. So it might be said. Now, before going into this argument, the first thing to observe is that there is an air of hocus-pocus about it: something reminds one of the conjurer who brings things out of the hat. The man who has lent his hat is quite sure there wasn't a live rabbit in it before, but he is at a loss to say how the rabbit got there. So the reader, if he has a robust sense of reality, will feel convinced that it is impossible to manufacture an infinite collection out of a finite collection of individuals, though he may be unable to say where the flaw is in the above construction. It would be a mistake to lay too much stress on such feelings of hocus-pocus; like other emotions, they may easily lead us astray. But they afford a prima facie ground for scrutinising very closely any argument which arouses them. And when the above argument is scrutinised it will, in my opinion, be found to be fallacious, though the fallacy is a subtle one and by no means easy to avoid consistently. The fallacy involved is the fallacy which may be called "confusion of types." To explain the subject of "types" fully would require a whole volume; moreover, it is the purpose of this book to avoid those parts of the subjects which are still obscure and controversial, isolating, for the convenience of beginners, those parts which can be accepted as embodying mathematically ascertained truths. Now the theory of types emphatically does not belong to the finished and certain part of our subject: much of this theory is still inchoate, confused, and obscure. But the need of some doctrine of types is less doubtful than the precise form the doctrine should take; and in connection with the axiom of infinity it is particularly easy to see the necessity of some such doctrine. This necessity results, for example, from the "contradiction of the greatest cardinal." We saw in Chapter VIII. that the number of classes contained in a given class is always greater than the [Pg 135] number of members of the class, and we inferred that there is no greatest cardinal number. But if we could, as we suggested a moment ago, add together into one class the individuals, classes of individuals, classes of classes of individuals, etc., we should obtain a class of which its own sub-classes would be members. The class consisting of all objects that can be counted, of whatever sort, must, if there be such a class, have a cardinal number which is the greatest possible. Since all its sub-classes will be members of it, there cannot be more of them than there are members. Hence we arrive at a contradiction. When I first came upon this contradiction, in the year 1901, I attempted to discover some flaw in Cantor's proof that there is no greatest cardinal, which we gave in Chapter VIII. Applying this proof to the supposed class of all imaginable objects, I was led to a new and simpler contradiction, namely, the following:— The comprehensive class we are considering, which is to embrace everything, must embrace itself as one of its members. In other words, if there is such a thing as "everything," then "everything" is something, and is a member of the class "everything." But normally a class is not a member of itself. Mankind, for example, is not a man. Form now the assemblage of all classes which are not members of themselves. This is a class: is it a member of itself or not?