[page 131]
The axiom of infinity is an assumption which may be enunciated as follows:—
“If n be any inductive cardinal number, there is at least one class of individuals having n terms.”
If this is true, it follows, of course, that there are many classes of individuals having n 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 n+1 terms, from which it follows that there are many classes of n terms and that n is not the number of individuals in the world. Since n 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 ℵ0 individuals, unless we assume the multiplicative axiom. But we do know that there are at least ℵ0 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 m+1=n+1 unless m=n, if m and n 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 n is an inductive cardinal, [page 132] n is not equal to n+1. It might be thought that this could be proved. We can prove that, if α is an inductive class, and n is the number of members of α, then n is not equal to n+1. 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 α. What it does imply is this: If n is an inductive cardinal such that there is at least one class having n members, then n is not equal to n+1. The axiom of infinity assures us (whether truly or falsely) that there are classes having n members, and thus enables us to assert that n is not equal to n+1. But without this axiom we should be left with the possibility that n and n+1 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 9+1) would be the null-class. It will be remembered that n+1 may be defined as follows: n+1 is the collection of all those classes which have a term x such that, when x is taken away, there remains a class of n terms. Now applying this definition, we see that, in the case supposed, 9+1 is a class consisting of no classes, i.e. it is the null-class. The same will be true of 9+2, or generally of 9+n, unless n 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, [page 133] multiplication, 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 n, the number of classes of individuals will be 2n. This is in virtue of the general proposition mentioned in Chapter VIII. that the number of classes contained in a class which has n members is 2n. Now 2n is always greater than n. 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 29, 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 2512, 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 10152. 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.1
As regards ratios, we have a very similar state of affairs. If a ratio μ/ν 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 μ/ν, without the axiom of [page 134] 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 μ objects, whatever inductive number μ 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 ℵ0, 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 n, where n 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
n+2n+22n … ad inf.,
which is ℵ0. Thus taking all kinds of objects together, and not [page 135] 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 [page 136] 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? If it is, it is one of those classes that are not members of themselves, i.e. it is not a member of itself. If it is not, it is not one of those classes that are not members of themselves, i.e. it is a member of itself. Thus of the two hypotheses—that it is, and that it is not, a member of itself—each implies its contradictory. This is a contradiction.
There is no difficulty in manufacturing similar contradictions ad lib. The solution of such contradictions by the theory of types is set forth fully in Principia Mathematica,2 and also, more briefly, in articles by the present author in the American Journal [page 137] of Mathematics3 and in the Revue de Métaphysique et de Morale.4 For the present an outline of the solution must suffice.
The fallacy consists in the formation of what we may call “impure” classes, i.e. classes which are not pure as to “type.” As we shall see in a later chapter, classes are logical fictions, and a statement which appears to be about a class will only be significant if it is capable of translation into a form in which no mention is made of the class. This places a limitation upon the ways in which what are nominally, though not really, names for classes can occur significantly: a sentence or set of symbols in which such pseudo-names occur in wrong ways is not false, but strictly devoid of meaning. The supposition that a class is, or that it is not, a member of itself is meaningless in just this way. And more generally, to suppose that one class of individuals is a member, or is not a member, of another class of individuals will be to suppose nonsense; and to construct symbolically any class whose members are not all of the same grade in the logical hierarchy is to use symbols in a way which makes them no longer symbolise anything.
Thus if there are n individuals in the world, and 2n classes of individuals, we cannot form a new class, consisting of both individuals and classes and having n+2n members. In this way the attempt to escape from the need for the axiom of infinity breaks down. I do not pretend to have explained the doctrine of types, or done more than indicate, in rough outline, why there is need of such a doctrine. I have aimed only at saying just so much as was required in order to show that we cannot prove the existence of infinite numbers and classes by such conjurer’s methods as we have been examining. There remain, however, certain other possible methods which must be considered.
Various arguments professing to prove the existence of infinite classes are given in the Principles of Mathematics, §339 (p. 357). [page 138] In so far as these arguments assume that, if n is an inductive cardinal, n is not equal to n+1, they have been already dealt with. There is an argument, suggested by a passage in Plato’s Parmenides, to the effect that, if there is such a number as 1, then 1 has being; but 1 is not identical with being, and therefore 1 and being are two, and therefore there is such a number as 2, and 2 together with 1 and being gives a class of three terms, and so on. This argument is fallacious, partly because “being” is not a term having any definite meaning, and still more because, if a definite meaning were invented for it, it would be found that numbers do not have being—they are, in fact, what are called “logical fictions,” as we shall see when we come to consider the definition of classes.
The argument that the number of numbers from 0 to n (both inclusive) is n+1 depends upon the assumption that up to and including n no number is equal to its successor, which, as we have seen, will not be always true if the axiom of infinity is false. It must be understood that the equation n=n+1, which might be true for a finite n if n exceeded the total number of individuals in the world, is quite different from the same equation as applied to a reflexive number. As applied to a reflexive number, it means that, given a class of n terms, this class is “similar” to that obtained by adding another term. But as applied to a number which is too great for the actual world, it merely means that there is no class of n individuals, and no class of n+1 individuals; it does not mean that, if we mount the hierarchy of types sufficiently far to secure the existence of a class of n terms, we shall then find this class “similar” to one of n+1 terms, for if n is inductive this will not be the case, quite independently of the truth or falsehood of the axiom of infinity.
There is an argument employed by both Bolzano5 and Dedekind6 to prove the existence of reflexive classes. The argument, in brief, is this: An object is not identical with the idea of the [page 139] object, but there is (at least in the realm of being) an idea of any object. The relation of an object to the idea of it is one-one, and ideas are only some among objects. Hence the relation “idea of” constitutes a reflexion of the whole class of objects into a part of itself, namely, into that part which consists of ideas. Accordingly, the class of objects and the class of ideas are both infinite. This argument is interesting, not only on its own account, but because the mistakes in it (or what I judge to be mistakes) are of a kind which it is instructive to note. The main error consists in assuming that there is an idea of every object. It is, of course, exceedingly difficult to decide what is meant by an “idea”; but let us assume that we know. We are then to suppose that, starting (say) with Socrates, there is the idea of Socrates, and then the idea of the idea of Socrates, and so on ad inf. Now it is plain that this is not the case in the sense that all these ideas have actual empirical existence in people’s minds. Beyond the third or fourth stage they become mythical. If the argument is to be upheld, the “ideas” intended must be Platonic ideas laid up in heaven, for certainly they are not on earth. But then it at once becomes doubtful whether there are such ideas. If we are to know that there are, it must be on the basis of some logical theory, proving that it is necessary to a thing that there should be an idea of it. We certainly cannot obtain this result empirically, or apply it, as Dedekind does, to “meine Gedankenwelt”—the world of my thoughts.
If we were concerned to examine fully the relation of idea and object, we should have to enter upon a number of psychological and logical inquiries, which are not relevant to our main purpose. But a few further points should be noted. If “idea” is to be understood logically, it may be identical with the object, or it may stand for a description (in the sense to be explained in a subsequent chapter). In the former case the argument fails, because it was essential to the proof of reflexiveness that object and idea should be distinct. In the second case the argument also fails, because the relation of object and description is not [page 140] one-one: there are innumerable correct descriptions of any given object. Socrates (e.g.) may be described as “the master of Plato,” or as “the philosopher who drank the hemlock,” or as “the husband of Xantippe.” If—to take up the remaining hypothesis—“idea” is to be interpreted psychologically, it must be maintained that there is not any one definite psychological entity which could be called the idea of the object: there are innumerable beliefs and attitudes, each of which could be called an idea of the object in the sense in which we might say “my idea of Socrates is quite different from yours,” but there is not any central entity (except Socrates himself) to bind together various “ideas of Socrates,” and thus there is not any such one-one relation of idea and object as the argument supposes. Nor, of course, as we have already noted, is it true psychologically that there are ideas (in however extended a sense) of more than a tiny proportion of the things in the world. For all these reasons, the above argument in favour of the logical existence of reflexive classes must be rejected.
It might be thought that, whatever may be said of logical arguments, the empirical arguments derivable from space and time, the diversity of colours, etc., are quite sufficient to prove the actual existence of an infinite number of particulars. I do not believe this. We have no reason except prejudice for believing in the infinite extent of space and time, at any rate in the sense in which space and time are physical facts, not mathematical fictions. We naturally regard space and time as continuous, or, at least, as compact; but this again is mainly prejudice. The theory of “quanta” in physics, whether true or false, illustrates the fact that physics can never afford proof of continuity, though it might quite possibly afford disproof. The senses are not sufficiently exact to distinguish between continuous motion and rapid discrete succession, as anyone may discover in a cinema. A world in which all motion consisted of a series of small finite jerks would be empirically indistinguishable from one in which motion was continuous. It would take up too much space to [page 141] defend these theses adequately; for the present I am merely suggesting them for the reader’s consideration. If they are valid, it follows that there is no empirical reason for believing the number of particulars in the world to be infinite, and that there never can be; also that there is at present no empirical reason to believe the number to be finite, though it is theoretically conceivable that some day there might be evidence pointing, though not conclusively, in that direction.
From the fact that the infinite is not self-contradictory, but is also not demonstrable logically, we must conclude that nothing can be known a priori as to whether the number of things in the world is finite or infinite. The conclusion is, therefore, to adopt a Leibnizian phraseology, that some of the possible worlds are finite, some infinite, and we have no means of knowing to which of these two kinds our actual world belongs. The axiom of infinity will be true in some possible worlds and false in others; whether it is true or false in this world, we cannot tell.
Throughout this chapter the synonyms “individual” and “particular” have been used without explanation. It would be impossible to explain them adequately without a longer disquisition on the theory of types than would be appropriate to the present work, but a few words before we leave this topic may do something to diminish the obscurity which would otherwise envelop the meaning of these words.
In an ordinary statement we can distinguish a verb, expressing an attribute or relation, from the substantives which express the subject of the attribute or the terms of the relation. “Cæsar lived” ascribes an attribute to Cæsar; “Brutus killed Cæsar” expresses a relation between Brutus and Cæsar. Using the word “subject” in a generalised sense, we may call both Brutus and Cæsar subjects of this proposition: the fact that Brutus is grammatically subject and Cæsar object is logically irrelevant, since the same occurrence may be expressed in the words “Cæsar was killed by Brutus,” where Cæsar is the grammatical subject. [page 142] Thus in the simpler sort of proposition we shall have an attribute or relation holding of or between one, two or more “subjects” in the extended sense. (A relation may have more than two terms: e.g. “A gives B to C” is a relation of three terms.) Now it often happens that, on a closer scrutiny, the apparent subjects are found to be not really subjects, but to be capable of analysis; the only result of this, however, is that new subjects take their places. It also happens that the verb may grammatically be made subject: e.g. we may say, “Killing is a relation which holds between Brutus and Cæsar.” But in such cases the grammar is misleading, and in a straightforward statement, following the rules that should guide philosophical grammar, Brutus and Cæsar will appear as the subjects and killing as the verb.
We are thus led to the conception of terms which, when they occur in propositions, can only occur as subjects, and never in any other way. This is part of the old scholastic definition of substance; but persistence through time, which belonged to that notion, forms no part of the notion with which we are concerned. We shall define “proper names” as those terms which can only occur as subjects in propositions (using “subject” in the extended sense just explained). We shall further define “individuals” or “particulars” as the objects that can be named by proper names. (It would be better to define them directly, rather than by means of the kind of symbols by which they are symbolised; but in order to do that we should have to plunge deeper into metaphysics than is desirable here.) It is, of course, possible that there is an endless regress: that whatever appears as a particular is really, on closer scrutiny, a class or some kind of complex. If this be the case, the axiom of infinity must of course be true. But if it be not the case, it must be theoretically possible for analysis to reach ultimate subjects, and it is these that give the meaning of “particulars” or “individuals.” It is to the number of these that the axiom of infinity is assumed to apply. If it is true of them, it is true [page 143] of classes of them, and classes of classes of them, and so on; similarly if it is false of them, it is false throughout this hierarchy. Hence it is natural to enunciate the axiom concerning them rather than concerning any other stage in the hierarchy. But whether the axiom is true or false, there seems no known method of discovering.
[Chapter XIII notes]
1. On this subject see Principia Mathematica, vol. ii. *120ff. On the corresponding problems as regards ratio, see ibid., vol. iii. *303ff.
2. Vol. i., Introduction, chap. ii., *12 and *20; vol. ii., Prefatory Statement.
3. “Mathematical Logic as based on the Theory of Types,” vol. xxx., 1908, pp. 222–262.
4. “Les paradoxes de la logique,” 1906, pp. 627–650.
5. Bolzano, Paradoxien des Unendlichen, 13.
6. Dedekind, Was sind und was sollen die Zahlen? No. 66.