Introduction to Mathematical Philosophy

AN "infinite series" may be defined as a series of which the field is an infinite class. We have already had occasion to consider one kind of infinite series, namely, progressions. In this chapter we shall consider the subject more generally. The most noteworthy characteristic of an infinite series is that its serial number can be altered by merely re-arranging its terms. In this respect there is a certain oppositeness between cardinal and serial numbers. It is possible to keep the cardinal number of a reflexive class unchanged in spite of adding terms to it; on the other hand, it is possible to change the serial number of a series without adding or taking away any terms, by mere re-arrangement. At the same time, in the case of any infinite series it is also possible, as with cardinals, to add terms without altering the serial number: everything depends upon the way in which they are added. In order to make matters clear, it will be best to begin with examples. Let us first consider various different kinds of series which can be made out of the inductive numbers arranged on various plans. We start with the series which, as we have already seen, represents the smallest of infinite serial numbers, the sort that Cantor calls . Let us proceed to thin out this series by repeatedly performing the [Pg 89] operation of removing to the end the first even number that occurs. We thus obtain in succession the various series: and so on. If we imagine this process carried on as long as possible, we finally reach the series in which we have first all the odd numbers and then all the even numbers. The serial numbers of these various series are . Each of these numbers is "greater" than any of its predecessors, in the following sense:— One serial number is said to be "greater" than another if any series having the first number contains a part having the second number, but no series having the second number contains a part having the first number. If we compare the two series we see that the first is similar to the part of the second which omits the last term, namely, the number 2, but the second is not similar to any part of the first. (This is obvious, but is easily demonstrated.) Thus the second series has a greater serial number than the first, according to the definition—i.e. is greater than . But if we add a term at the beginning of a progression instead of the end, we still have a progression. Thus . Thus is not equal to . This is characteristic of relation-arithmetic generally: if and are two relation-numbers, the general rule is that is not equal to . The case of finite ordinals, in which there is equality, is quite exceptional. The series we finally reached just now consisted of first all the odd numbers and then all the even numbers, and its serial [Pg 90] number is . This number is greater than or , where is finite. It is to be observed that, in accordance with the general definition of order, each of these arrangements of integers is to be regarded as resulting from some definite relation. E.g. the one which merely removes 2 to the end will be defined by the following relation: " and are finite integers, and either is 2 and is not 2, or neither is 2 and is less than ." The one which puts first all the odd numbers and then all the even ones will be defined by: " and are finite integers, and either is odd and is even or is less than and both are odd or both are even." We shall not trouble, as a rule, to give these formulæ in future; but the fact that they could be given is essential. The number which we have called , namely, the number of a series consisting of two progressions, is sometimes called . Multiplication, like addition, depends upon the order of the factors: a progression of couples gives a series such as which is itself a progression; but a couple of progressions gives a series which is twice as long as a progression. It is therefore necessary to distinguish between and . Usage is variable; we shall use for a couple of progressions and for a progression of couples, and this decision of course governs our general interpretation of "" when and are relation-numbers: "" will have to stand for a suitably constructed sum of a relations each having terms. We can proceed indefinitely with the process of thinning out the inductive numbers. For example, we can place first the odd numbers, then their doubles, then the doubles of these, and so on. We thus obtain the series of which the number is , since it is a progression of progressions. Any one of the progressions in this new series can of course be [Pg 91] thinned out as we thinned out our original progression. We can proceed to , , ... , and so on; however far we have gone, we can always go further. The series of all the ordinals that can be obtained in this way, i.e. all that can be obtained by thinning out a progression, is itself longer than any series that can be obtained by re-arranging the terms of a progression. (This is not difficult to prove.) The cardinal number of the class of such ordinals can be shown to be greater than ; it is the number which Cantor calls . The ordinal number of the series of all ordinals that can be made out of an , taken in order of magnitude, is called . Thus a series whose ordinal number is has a field whose cardinal number is . We can proceed from and to and by a process exactly analogous to that by which we advanced from and to and . And there is nothing to prevent us from advancing indefinitely in this way to new cardinals and new ordinals. It is not known whether is equal to any of the cardinals in the series of Alephs. It is not even known whether it is comparable with them in magnitude; for aught we know, it may be neither equal to nor greater nor less than any one of the Alephs. This question is connected with the multiplicative axiom, of which we shall treat later. All the series we have been considering so far in this chapter have been what is called "well-ordered." A well-ordered series is one which has a beginning, and has consecutive terms, and has a term next after any selection of its terms, provided there are any terms after the selection. This excludes, on the one hand, compact series, in which there are terms between any two, and on the other hand series which have no beginning, or in which there are subordinate parts having no beginning. The series of negative integers in order of magnitude, having no beginning, but ending with -1, is not well-ordered; but taken in the reverse order, beginning with -1, it is well-ordered, being in fact a progression. The definition is: [Pg 92] A "well-ordered" series is one in which every sub-class (except, of course, the null-class) has a first term. An "ordinal" number means the relation-number of a well-ordered series. It is thus a species of serial number. Among well-ordered series, a generalised form of mathematical induction applies. A property may be said to be "transfinitely hereditary" if, when it belongs to a certain selection of the terms in a series, it belongs to their immediate successor provided they have one. In a well-ordered series, a transfinitely hereditary property belonging to the first term of the series belongs to the whole series. This makes it possible to prove many propositions concerning well-ordered series which are not true of all series. It is easy to arrange the inductive numbers in series which are not well-ordered, and even to arrange them in compact series. For example, we can adopt the following plan: consider the decimals from .1 (inclusive) to 1 (exclusive), arranged in order of magnitude. These form a compact series; between any two there are always an infinite number of others. Now omit the dot at the beginning of each, and we have a compact series consisting of all finite integers except such as divide by 10. If we wish to include those that divide by 10, there is no difficulty; instead of starting with .1, we will include all decimals less than 1, but when we remove the dot, we will transfer to the right any 0's that occur at the beginning of our decimal. Omitting these, and returning to the ones that have no 0's at the beginning, we can state the rule for the arrangement of our integers as follows: Of two integers that do not begin with the same digit, the one that begins with the smaller digit comes first. Of two that do begin with the same digit, but differ at the second digit, the one with the smaller second digit comes first, but first of all the one with no second digit; and so on. Generally, if two integers agree as regards the first digits, but not as regards the , that one comes first which has either no digit or a smaller one than the other. This rule of arrangement, [Pg 93] as the reader can easily convince himself, gives rise to a compact series containing all the integers not divisible by 10; and, as we saw, there is no difficulty about including those that are divisible by 10. It follows from this example that it is possible to construct compact series having terms. In fact, we have already seen that there are ratios, and ratios in order of magnitude form a compact series; thus we have here another example. We shall resume this topic in the next chapter. Of the usual formal laws of addition, multiplication, and exponentiation, all are obeyed by transfinite cardinals, but only some are obeyed by transfinite ordinals, and those that are obeyed by them are obeyed by all relation-numbers. By the "usual formal laws" we mean the following:— I. The commutative law: II. The associative law: III. The distributive law: When the commutative law does not hold, the above form of the distributive law must be distinguished from As we shall see immediately, one form may be true and the other false. IV. The laws of exponentiation: All these laws hold for cardinals, whether finite or infinite, and for finite ordinals. But when we come to infinite ordinals, or indeed to relation-numbers in general, some hold and some do not. The commutative law does not hold; the associative law does hold; the distributive law (adopting the convention [Pg 94] we have adopted above as regards the order of the factors in a product) holds in the form but not in the form the exponential laws still hold, but not the law which is obviously connected with the commutative law for multiplication. The definitions of multiplication and exponentiation that are assumed in the above propositions are somewhat complicated. The reader who wishes to know what they are and how the above laws are proved must consult the second volume of Principia Mathematica, * 172-176. Ordinal transfinite arithmetic was developed by Cantor at an earlier stage than cardinal transfinite arithmetic, because it has various technical mathematical uses which led him to it. But from the point of view of the philosophy of mathematics it is less important and less fundamental than the theory of transfinite cardinals. Cardinals are essentially simpler than ordinals, and it is a curious historical accident that they first appeared as an abstraction from the latter, and only gradually came to be studied on their own account. This does not apply to Frege's work, in which cardinals, finite and transfinite, were treated in complete independence of ordinals; but it was Cantor's work that made the world aware of the subject, while Frege's remained almost unknown, probably in the main on account of the difficulty of his symbolism. And mathematicians, like other people, have more difficulty in understanding and