[page 117]
In this chapter we have to consider an axiom which can be enunciated, but not proved, in terms of logic, and which is convenient, though not indispensable, in certain portions of mathematics. It is convenient, in the sense that many interesting propositions, which it seems natural to suppose true, cannot be proved without its help; but it is not indispensable, because even without those propositions the subjects in which they occur still exist, though in a somewhat mutilated form.
Before enunciating the multiplicative axiom, we must first explain the theory of selections, and the definition of multiplication when the number of factors may be infinite.
In defining the arithmetical operations, the only correct procedure is to construct an actual class (or relation, in the case of relation-numbers) having the required number of terms. This sometimes demands a certain amount of ingenuity, but it is essential in order to prove the existence of the number defined. Take, as the simplest example, the case of addition. Suppose we are given a cardinal number μ, and a class α which has μ terms. How shall we define μ+μ? For this purpose we must have two classes having μ terms, and they must not overlap. We can construct such classes from α in various ways, of which the following is perhaps the simplest: Form first all the ordered couples whose first term is a class consisting of a single member of α, and whose second term is the null-class; then, secondly, form all the ordered couples whose first term is [page 118] the null-class and whose second term is a class consisting of a single member of α. These two classes of couples have no member in common, and the logical sum of the two classes will have μ+μ terms. Exactly analogously we can define μ+ν, given that μ is the number of some class α and ν is the number of some class β.
Such definitions, as a rule, are merely a question of a suitable technical device. But in the case of multiplication, where the number of factors may be infinite, important problems arise out of the definition.
Multiplication when the number of factors is finite offers no difficulty. Given two classes α and β, of which the first has μ terms and the second ν terms, we can define μ×ν as the number of ordered couples that can be formed by choosing the first term out of α and the second out of β. It will be seen that this definition does not require that α and β should not overlap; it even remains adequate when α and β are identical. For example, let α be the class whose members are x1, x2, x3. Then the class which is used to define the product μ×μ is the class of couples:
(x1, x1), (x1, x2), (x1, x3); (x2, x1), (x2, x2), (x2, x3); (x3, x1), (x3, x2), (x3, x3).
This definition remains applicable when μ or ν or both are infinite, and it can be extended step by step to three or four or any finite number of factors. No difficulty arises as regards this definition, except that it cannot be extended to an infinite number of factors.
The problem of multiplication when the number of factors may be infinite arises in this way: Suppose we have a class κ consisting of classes; suppose the number of terms in each of these classes is given. How shall we define the product of all these numbers? If we can frame our definition generally, it will be applicable whether κ is finite or infinite. It is to be observed that the problem is to be able to deal with the case when κ is infinite, not with the case when its members are. If [page 119] κ is not infinite, the method defined above is just as applicable when its members are infinite as when they are finite. It is the case when κ is infinite, even though its members may be finite, that we have to find a way of dealing with.
The following method of defining multiplication generally is due to Dr Whitehead. It is explained and treated at length in Principia Mathematica, vol. i. *80ff., and vol. ii. *114.
Let us suppose to begin with that κ is a class of classes no two of which overlap—say the constituencies in a country where there is no plural voting, each constituency being considered as a class of voters. Let us now set to work to choose one term out of each class to be its representative, as constituencies do when they elect members of Parliament, assuming that by law each constituency has to elect a man who is a voter in that constituency. We thus arrive at a class of representatives, who make up our Parliament, one being selected out of each constituency. How many different possible ways of choosing a Parliament are there? Each constituency can select any one of its voters, and therefore if there are μ voters in a constituency, it can make μ choices. The choices of the different constituencies are independent; thus it is obvious that, when the total number of constituencies is finite, the number of possible Parliaments is obtained by multiplying together the numbers of voters in the various constituencies. When we do not know whether the number of constituencies is finite or infinite, we may take the number of possible Parliaments as defining the product of the numbers of the separate constituencies. This is the method by which infinite products are defined. We must now drop our illustration, and proceed to exact statements.
Let κ be a class of classes, and let us assume to begin with that no two members of κ overlap, i.e. that if α and β are two different members of κ, then no member of the one is a member of the other. We shall call a class a “selection” from κ when it consists of just one term from each member of κ; i.e. μ is a “selection” from κ if every member of μ belongs to some member [page 120] of κ, and if α be any member of κ, μ and α have exactly one term in common. The class of all “selections” from κ we shall call the “multiplicative class” of κ. The number of terms in the multiplicative class of κ, i.e. the number of possible selections from κ, is defined as the product of the numbers of the members of κ. This definition is equally applicable whether κ is finite or infinite.
Before we can be wholly satisfied with these definitions, we must remove the restriction that no two members of κ are to overlap. For this purpose, instead of defining first a class called a “selection,” we will define first a relation which we will call a “selector.” A relation R will be called a “selector” from κ if, from every member of κ, it picks out one term as the representative of that member, i.e. if, given any member α of κ, there is just one term x which is a member of α and has the relation R to α; and this is to be all that R does. The formal definition is:
A “selector” from a class of classes κ is a one-many relation, having κ for its converse domain, and such that, if x has the relation to α, then x is a member of α.
If R is a selector from κ, and α is a member of κ, and x is the term which has the relation R to α, we call x the “representative” of α in respect of the relation R.
A “selection” from κ will now be defined as the domain of a selector; and the multiplicative class, as before, will be the class of selections.
But when the members of κ overlap, there may be more selectors than selections, since a term x which belongs to two classes α and β may be selected once to represent α and once to represent β, giving rise to different selectors in the two cases, but to the same selection. For purposes of defining multiplication, it is the selectors we require rather than the selections. Thus we define:
“The product of the numbers of the members of a class of classes κ” is the number of selectors from κ.
We can define exponentiation by an adaptation of the above [page 121] plan. We might, of course, define μν as the number of selectors from ν classes, each of which has μ terms. But there are objections to this definition, derived from the fact that the multiplicative axiom (of which we shall speak shortly) is unnecessarily involved if it is adopted. We adopt instead the following construction:—
Let α be a class having μ terms, and β a class having ν terms. Let y be a member of β, and form the class of all ordered couples that have y for their second term and a member of α for their first term. There will be μ such couples for a given y, since any member of α may be chosen for the first term, and α has μ members. If we now form all the classes of this sort that result from varying y, we obtain altogether ν classes, since y may be any member of β, and β has ν members. These ν classes are each of them a class of couples, namely, all the couples that can be formed of a variable member of α and a fixed member of β. We define μν as the number of selectors from the class consisting of these ν classes. Or we may equally well define μν as the number of selections, for, since our classes of couples are mutually exclusive, the number of selectors is the same as the number of selections. A selection from our class of classes will be a set of ordered couples, of which there will be exactly one having any given member of β for its second term, and the first term may be any member of α. Thus μν is defined by the selectors from a certain set of ν classes each having μ terms, but the set is one having a certain structure and a more manageable composition than is the case in general. The relevance of this to the multiplicative axiom will appear shortly.
What applies to exponentiation applies also to the product of two cardinals. We might define “μ×ν” as the sum of the numbers of ν classes each having μ terms, but we prefer to define it as the number of ordered couples to be formed consisting of a member of α followed by a member of β, where α has μ terms and β has ν terms. This definition, also, is designed to evade the necessity of assuming the multiplicative axiom.
[page 122]
With our definitions, we can prove the usual formal laws of multiplication and exponentiation. But there is one thing we cannot prove: we cannot prove that a product is only zero when one of its factors is zero. We can prove this when the number of factors is finite, but not when it is infinite. In other words, we cannot prove that, given a class of classes none of which is null, there must be selectors from them; or that, given a class of mutually exclusive classes, there must be at least one class consisting of one term out of each of the given classes. These things cannot be proved; and although, at first sight, they seem obviously true, yet reflection brings gradually increasing doubt, until at last we become content to register the assumption and its consequences, as we register the axiom of parallels, without assuming that we can know whether it is true or false. The assumption, loosely worded, is that selectors and selections exist when we should expect them. There are many equivalent ways of stating it precisely. We may begin with the following:—
“Given any class of mutually exclusive classes, of which none is null, there is at least one class which has exactly one term in common with each of the given classes.”
This proposition we will call the “multiplicative axiom.”1 We will first give various equivalent forms of the proposition, and then consider certain ways in which its truth or falsehood is of interest to mathematics.
The multiplicative axiom is equivalent to the proposition that a product is only zero when at least one of its factors is zero; i.e. that, if any number of cardinal numbers be multiplied together, the result cannot be 0 unless one of the numbers concerned is 0.
The multiplicative axiom is equivalent to the proposition that, if R be any relation, and κ any class contained in the converse domain of R, then there is at least one one-many relation implying R and having κ for its converse domain.
The multiplicative axiom is equivalent to the assumption that if α be any class, and κ all the sub-classes of α with the exception [page 123] of the null-class, then there is at least one selector from κ. This is the form in which the axiom was first brought to the notice of the learned world by Zermelo, in his “Beweis, dass jede Menge wohlgeordnet werden kann.”2 Zermelo regards the axiom as an unquestionable truth. It must be confessed that, until he made it explicit, mathematicians had used it without a qualm; but it would seem that they had done so unconsciously. And the credit due to Zermelo for having made it explicit is entirely independent of the question whether it is true or false.
The multiplicative axiom has been shown by Zermelo, in the above-mentioned proof, to be equivalent to the proposition that every class can be well-ordered, i.e. can be arranged in a series in which every sub-class has a first term (except, of course, the null-class). The full proof of this proposition is difficult, but it is not difficult to see the general principle upon which it proceeds. It uses the form which we call “Zermelo’s axiom,” i.e. it assumes that, given any class α, there is at least one one-many relation R whose converse domain consists of all existent sub-classes of α and which is such that, if x has the relation R to ξ, then x is a member of ξ. Such a relation picks out a “representative” from each sub-class; of course, it will often happen that two sub-classes have the same representative. What Zermelo does, in effect, is to count off the members of α, one by one, by means of R and transfinite induction. We put first the representative of α; call it x1. Then take the representative of the class consisting of all of α except x1; call it x2. It must be different from x1, because every representative is a member of its class, and x1 is shut out from this class. Proceed similarly to take away x2, and let x3 be the representative of what is left. In this way we first obtain a progression x1, x2, … xn, …, assuming that α is not finite. We then take away the whole progression; let xω be the representative of what is left of α. In this way we can go on until nothing is left. The successive representatives will form a [page 124] well-ordered series containing all the members of α. (The above is, of course, only a hint of the general lines of the proof.) This proposition is called “Zermelo’s theorem.”
The multiplicative axiom is also equivalent to the assumption that of any two cardinals which are not equal, one must be the greater. If the axiom is false, there will be cardinals μ and ν such that μ is neither less than, equal to, nor greater than ν. We have seen that ℵ1 and 2ℵ0 possibly form an instance of such a pair.
Many other forms of the axiom might be given, but the above are the most important of the forms known at present. As to the truth or falsehood of the axiom in any of its forms, nothing is known at present.
The propositions that depend upon the axiom, without being known to be equivalent to it, are numerous and important. Take first the connection of addition and multiplication. We naturally think that the sum of ν mutually exclusive classes, each having μ terms, must have μ×ν terms. When ν is finite, this can be proved. But when ν is infinite, it cannot be proved without the multiplicative axiom, except where, owing to some special circumstance, the existence of certain selectors can be proved. The way the multiplicative axiom enters in is as follows: Suppose we have two sets of ν mutually exclusive classes, each having μ terms, and we wish to prove that the sum of one set has as many terms as the sum of the other. In order to prove this, we must establish a one-one relation. Now, since there are in each case ν classes, there is some one-one relation between the two sets of classes; but what we want is a one-one relation between their terms. Let us consider some one-one relation S between the classes. Then if κ and λ are the two sets of classes, and α is some member of κ, there will be a member β of λ which will be the correlate of α with respect to S. Now α and β each have μ terms, and are therefore similar. There are, accordingly, one-one correlations of α and β. The trouble is that there are so many. In order to obtain a one-one correlation of the sum of κ with the sum of λ, we have to pick out one correlator of α with β, and similarly for every other pair. This requires a selection from a set of classes [page 125] of correlators, one class of the set being all the one-one correlators of α with β. If κ and λ are infinite, we cannot in general know that such a selection exists, unless we can know that the multiplicative axiom is true. Hence we cannot establish the usual kind of connection between addition and multiplication.
This fact has various curious consequences. To begin with, we know that ℵ02=ℵ0×ℵ0=ℵ0. It is commonly inferred from this that the sum of ℵ0 classes each having ℵ0 members must itself have ℵ0 members, but this inference is fallacious, since we do not know that the number of terms in such a sum is ℵ0×ℵ0, nor consequently that it is ℵ0. This has a bearing upon the theory of transfinite ordinals. It is easy to prove that an ordinal which has ℵ0 predecessors must be one of what Cantor calls the “second class,” i.e. such that a series having this ordinal number will have ℵ0 terms in its field. It is also easy to see that, if we take any progression of ordinals of the second class, the predecessors of their limit form at most the sum of ℵ0 classes each having ℵ0 terms. It is inferred thence—fallaciously, unless the multiplicative axiom is true—that the predecessors of the limit are ℵ0 in number, and therefore that the limit is a number of the “second class.” That is to say, it is supposed to be proved that any progression of ordinals of the second class has a limit which is again an ordinal of the second class. This proposition, with the corollary that ω1 (the smallest ordinal of the third class) is not the limit of any progression, is involved in most of the recognised theory of ordinals of the second class. In view of the way in which the multiplicative axiom is involved, the proposition and its corollary cannot be regarded as proved. They may be true, or they may not. All that can be said at present is that we do not know. Thus the greater part of the theory of ordinals of the second class must be regarded as unproved.
Another illustration may help to make the point clearer. We know that 2×ℵ0=ℵ0. Hence we might suppose that the sum of ℵ0 pairs must have ℵ0 terms. But this, though we can prove that it is sometimes the case, cannot be proved to happen always [page 126] unless we assume the multiplicative axiom. This is illustrated by the millionaire who bought a pair of socks whenever he bought a pair of boots, and never at any other time, and who had such a passion for buying both that at last he had ℵ0 pairs of boots and ℵ0 pairs of socks. The problem is: How many boots had he, and how many socks? One would naturally suppose that he had twice as many boots and twice as many socks as he had pairs of each, and that therefore he had ℵ0 of each, since that number is not increased by doubling. But this is an instance of the difficulty, already noted, of connecting the sum of ν classes each having μ terms with μ×ν. Sometimes this can be done, sometimes it cannot. In our case it can be done with the boots, but not with the socks, except by some very artificial device. The reason for the difference is this: Among boots we can distinguish right and left, and therefore we can make a selection of one out of each pair, namely, we can choose all the right boots or all the left boots; but with socks no such principle of selection suggests itself, and we cannot be sure, unless we assume the multiplicative axiom, that there is any class consisting of one sock out of each pair. Hence the problem.
We may put the matter in another way. To prove that a class has ℵ0 terms, it is necessary and sufficient to find some way of arranging its terms in a progression. There is no difficulty in doing this with the boots. The pairs are given as forming an ℵ0, and therefore as the field of a progression. Within each pair, take the left boot first and the right second, keeping the order of the pair unchanged; in this way we obtain a progression of all the boots. But with the socks we shall have to choose arbitrarily, with each pair, which to put first; and an infinite number of arbitrary choices is an impossibility. Unless we can find a rule for selecting, i.e. a relation which is a selector, we do not know that a selection is even theoretically possible. Of course, in the case of objects in space, like socks, we always can find some principle of selection. For example, take the centres of mass of the socks: there will be points p in space such that, with any [page 127] pair, the centres of mass of the two socks are not both at exactly the same distance from p; thus we can choose, from each pair, that sock which has its centre of mass nearer to p. But there is no theoretical reason why a method of selection such as this should always be possible, and the case of the socks, with a little goodwill on the part of the reader, may serve to show how a selection might be impossible.
It is to be observed that, if it were impossible to select one out of each pair of socks, it would follow that the socks could not be arranged in a progression, and therefore that there were not ℵ0 of them. This case illustrates that, if μ is an infinite number, one set of μ pairs may not contain the same number of terms as another set of μ pairs; for, given ℵ0 pairs of boots, there are certainly ℵ0 boots, but we cannot be sure of this in the case of the socks unless we assume the multiplicative axiom or fall back upon some fortuitous geometrical method of selection such as the above.
Another important problem involving the multiplicative axiom is the relation of reflexiveness to non-inductiveness. It will be remembered that in Chapter VIII. we pointed out that a reflexive number must be non-inductive, but that the converse (so far as is known at present) can only be proved if we assume the multiplicative axiom. The way in which this comes about is as follows:—
It is easy to prove that a reflexive class is one which contains sub-classes having ℵ0 terms. (The class may, of course, itself have ℵ0 terms.) Thus we have to prove, if we can, that, given any non-inductive class, it is possible to choose a progression out of its terms. Now there is no difficulty in showing that a non-inductive class must contain more terms than any inductive class, or, what comes to the same thing, that if α is a non-inductive class and ν is any inductive number, there are sub-classes of α that have ν terms. Thus we can form sets of finite sub-classes of α: First one class having no terms, then classes having 1 term (as many as there are members of α), then classes having [page 128] 2 terms, and so on. We thus get a progression of sets of sub-classes, each set consisting of all those that have a certain given finite number of terms. So far we have not used the multiplicative axiom, but we have only proved that the number of collections of sub-classes of α is a reflexive number, i.e. that, if μ is the number of members of α, so that 2μ is the number of sub-classes of α and 22μ is the number of collections of sub-classes, then, provided μ is not inductive, 22μ must be reflexive. But this is a long way from what we set out to prove.
In order to advance beyond this point, we must employ the multiplicative axiom. From each set of sub-classes let us choose out one, omitting the sub-class consisting of the null-class alone. That is to say, we select one sub-class containing one term, α1, say; one containing two terms, α2, say; one containing three, α3, say; and so on. (We can do this if the multiplicative axiom is assumed; otherwise, we do not know whether we can always do it or not.) We have now a progression α1, α2, α3, … of sub-classes of α, instead of a progression of collections of sub-classes; thus we are one step nearer to our goal. We now know that, assuming the multiplicative axiom, if μ is a non-inductive number, 2μ must be a reflexive number.
The next step is to notice that, although we cannot be sure that new members of α come in at any one specified stage in the progression α1, α2, α3, … we can be sure that new members keep on coming in from time to time. Let us illustrate. The class α1, which consists of one term, is a new beginning; let the one term be x1. The class α2, consisting of two terms, may or may not contain x1; if it does, it introduces one new term; and if it does not, it must introduce two new terms, say x2, x3. In this case it is possible that α3 consists of x1, x2, x3, and so introduces no new terms, but in that case α4 must introduce a new term. The first ν classes α1, α2, α3, … αν contain, at the very most, 1+2+3+ … +ν terms, i.e. ν(ν+1)/2 terms; thus it would be possible, if there were no repetitions in the first ν classes, to go on with only repetitions from the (ν+1)th [page 129] class to the ν(ν+1)/2th class. But by that time the old terms would no longer be sufficiently numerous to form a next class with the right number of members, i.e. ν(ν+1)/2+1, therefore new terms must come in at this point if not sooner. It follows that, if we omit from our progression α1, α2, α3, … all those classes that are composed entirely of members that have occurred in previous classes, we shall still have a progression. Let our new progression be called β1, β2, β3 … (We shall have α1=β1 and α2=β2, because α1 and α2 must introduce new terms. We may or may not have α3=β3, but, speaking generally, βμ will be αν, where ν is some number greater than μ; i.e. the β’s are some of the α’s.) Now these β’s are such that any one of them, say βμ, contains members which have not occurred in any of the previous β’s. Let γμ be the part of βμ which consists of new members. Thus we get a new progression γ1, γ2, γ3, … (Again γ1 will be identical with β1 and with α1; if α2 does not contain the one member of α1, we shall have γ2=β2=α2, but if α2 does contain this one member, γ2 will consist of the other member of α2.) This new progression of γ’s consists of mutually exclusive classes. Hence a selection from them will be a progression; i.e. if x1 is the member of γ1, x2 is a member of γ2, x3 is a member of γ3, and so on; then x1, x2, x3, … is a progression, and is a sub-class of α. Assuming the multiplicative axiom, such a selection can be made. Thus by twice using this axiom we can prove that, if the axiom is true, every non-inductive cardinal must be reflexive. This could also be deduced from Zermelo’s theorem, that, if the axiom is true, every class can be well-ordered; for a well-ordered series must have either a finite or a reflexive number of terms in its field.
There is one advantage in the above direct argument, as against deduction from Zermelo’s theorem, that the above argument does not demand the universal truth of the multiplicative axiom, but only its truth as applied to a set of ℵ0 classes. It may happen that the axiom holds for ℵ0 classes, though not for larger numbers of classes. For this reason it is better, when [page 130] it is possible, to content ourselves with the more restricted assumption. The assumption made in the above direct argument is that a product of ℵ0 factors is never zero unless one of the factors is zero. We may state this assumption in the form: “ℵ0 is a multipliable number,” where a number ν is defined as “multipliable” when a product of ν factors is never zero unless one of the factors is zero. We can prove that a finite number is always multipliable, but we cannot prove that any infinite number is so. The multiplicative axiom is equivalent to the assumption that all cardinal numbers are multipliable. But in order to identify the reflexive with the non-inductive, or to deal with the problem of the boots and socks, or to show that any progression of numbers of the second class is of the second class, we only need the very much smaller assumption that ℵ0 is multipliable.
It is not improbable that there is much to be discovered in regard to the topics discussed in the present chapter. Cases may be found where propositions which seem to involve the multiplicative axiom can be proved without it. It is conceivable that the multiplicative axiom in its general form may be shown to be false. From this point of view, Zermelo’s theorem offers the best hope: the continuum or some still more dense series might be proved to be incapable of having its terms well-ordered, which would prove the multiplicative axiom false, in virtue of Zermelo’s theorem. But so far, no method of obtaining such results has been discovered, and the subject remains wrapped in obscurity.
[Chapter XII notes]
1. See Principia Mathematica, vol. i. *88. Also vol. iii. *257–258.
2. Mathematische Annalen, vol. lix. pp. 514–6. In this form we shall speak of it as Zermelo’s axiom.