Introduction to Mathematical Philosophy

is a number" becomes: "The term to which a given member of the domain has the relation in question is again a member of the domain." This is proved as follows: By the definition, every member of the domain is a member of the posterity of the first term; hence the successor of a member of the domain must be a member of the posterity of the first term (because the posterity of a term always contains its own successors, by the general definition of posterity), and therefore a member of the domain, because by the definition the posterity of the first term is the same as the domain. (3) "No two numbers have the same successor." This is only to say that the relation is one-many, which it is by definition (being one-one). [Pg 82] (4) "0 is not the successor of any number" becomes: "The first term is not a member of the converse domain," which is again an immediate result of the definition. (5) This is mathematical induction, and becomes: "Every member of the domain belongs to the posterity of the first term," which was part of our definition. Thus progressions as we have defined them have the five formal properties from which Peano deduces arithmetic. It is easy to show that two progessions are "similar" in the sense defined for similarity of relations in Chapter VI. We can, of course, derive a relation which is serial from the one-one relation by which we define a progression: the method used is that explained in Chapter IV., and the relation is that of a term to a member of its proper posterity with respect to the original one-one relation. Two transitive asymmetrical relations which generate progressions are similar, for the same reasons for which the corresponding one-one relations are similar. The class of all such transitive generators of progressions is a "serial number" in the sense of Chapter VI.; it is in fact the smallest of infinite serial numbers, the number to which Cantor has given the name , by which he has made it famous. But we are concerned, for the moment, with cardinal numbers. Since two progressions are similar relations, it follows that their domains (or their fields, which are the same as their domains) are similar classes. The domains of progressions form a cardinal number, since every class which is similar to the domain of a progression is easily shown to be itself the domain of a progression. This cardinal number is the smallest of the infinite cardinal numbers; it is the one to which Cantor has appropriated the Hebrew Aleph with the suffix 0, to distinguish it from larger infinite cardinals, which have other suffixes. Thus the name of the smallest of infinite cardinals is . To say that a class has terms is the same thing as to say that it is a member of , and this is the same thing as to say [Pg 83] that the members of the class can be arranged in a progression. It is obvious that any progression remains a progression if we omit a finite number of terms from it, or every other term, or all except every tenth term or every hundredth term. These methods of thinning out a progression do not make it cease to be a progression, and therefore do not diminish the number of its terms, which remains . In fact, any selection from a progression is a progression if it has no last term, however sparsely it may be distributed. Take (say) inductive numbers of the form , or . Such numbers grow very rare in the higher parts of the number series, and yet there are just as many of them as there are inductive numbers altogether, namely, . Conversely, we can add terms to the inductive numbers without increasing their number. Take, for example, ratios. One might be inclined to think that there must be many more ratios than integers, since ratios whose denominator is 1 correspond to the integers, and seem to be only an infinitesimal proportion of ratios. But in actual fact the number of ratios (or fractions) is exactly the same as the number of inductive numbers, namely, . This is easily seen by arranging ratios in a series on the following plan: If the sum of numerator and denominator in one is less than in the other, put the one before the other; if the sum is equal in the two, put first the one with the smaller numerator. This gives us the series This series is a progression, and all ratios occur in it sooner or later. Hence we can arrange all ratios in a progression, and their number is therefore . It is not the case, however, that all infinite collections have terms. The number of real numbers, for example, is greater than ; it is, in fact, , and it is not hard to prove that is greater than even when is infinite. The easiest way of proving this is to prove, first, that if a class has members, it contains sub-classes—in other words, that there are ways [Pg 84] of selecting some of its members (including the extreme cases where we select all or none); and secondly, that the number of sub-classes contained in a class is always greater than the number of members of the class. Of these two propositions, the first is familiar in the case of finite numbers, and is not hard to extend to infinite numbers. The proof of the second is so simple and so instructive that we shall give it: In the first place, it is clear that the number of sub-classes of a given class (say ) is at least as great as the number of members, since each member constitutes a sub-class, and we thus have a correlation of all the members with some of the sub-classes. Hence it follows that, if the number of sub-classes is not equal to the number of members, it must be greater. Now it is easy to prove that the number is not equal, by showing that, given any one-one relation whose domain is the members and whose converse domain is contained among the set of sub-classes, there must be at least one sub-class not belonging to the converse domain. The proof is as follows:[21] When a one-one correlation is established between all the members of and some of the sub-classes, it may happen that a given member is correlated with a sub-class of which it is a member; or, again, it may happen that is correlated with a sub-class of which it is not a member. Let us form the whole class, say, of those members which are correlated with sub-classes of which they are not members. This is a sub-class of , and it is not correlated with any member of . For, taking first the members of , each of them is (by the definition of ) correlated with some sub-class of which it is not a member, and is therefore not correlated with . Taking next the terms which are not members of , each of them (by the definition of ) is correlated with some sub-class of which it is a member, and therefore again is not correlated with . Thus no member of is correlated with . Since was any one-one correlation of all members [Pg 85] with some sub-classes, it follows that there is no correlation of all members with all sub-classes. It does not matter to the proof if has no members: all that happens in that case is that the sub-class which is shown to be omitted is the null-class. Hence in any case the number of sub-classes is not equal to the number of members, and therefore, by what was said earlier, it is greater. Combining this with the proposition that, if is the number of members, is the number of sub-classes, we have the theorem that is always greater than , even when is infinite. [21]This proof is taken from Cantor, with some simplifications: see Jahresbericht der deutschen Mathematiker-Vereinigung, I. (1892), p. 77. It follows from this proposition that there is no maximum to the infinite cardinal numbers. However great an infinite number may be, will be still greater. The arithmetic of infinite numbers is somewhat surprising until one becomes accustomed to it. We have, for example, (This follows from the case of the ratios, for, since a ratio is determined by a pair of inductive numbers, it is easy to see that the number of ratios is the square of the number of inductive numbers, i.e. it is ; but we saw that it is also .) But In fact, as we shall see later, is a very important number, namely, the number of terms in a series which has "continuity" in the sense in which this word is used by Cantor. Assuming space and time to be continuous in this sense (as we commonly do in analytical geometry and kinematics), this will be the number of points in space or of instants in time; it will also be the number of points in any finite portion of space, whether [Pg 86] line, area, or volume. After , is the most important and interesting of infinite cardinal numbers. Although addition and multiplication are always possible with infinite cardinals, subtraction and division no longer give definite results, and cannot therefore be employed as they are employed in elementary arithmetic. Take subtraction to begin with: so long as the number subtracted is finite, all goes well; if the other number is reflexive, it remains unchanged. Thus , if is finite; so far, subtraction gives a perfectly definite result. But it is otherwise when we subtract from itself; we may then get any result, from 0 up to . This is easily seen by examples. From the inductive, numbers, take away the following collections of terms:— (1) All the inductive numbers—remainder, zero. (2) All the inductive numbers from onwards—remainder, the numbers from 0 to , numbering terms in all. (3) All the odd numbers—remainder, all the even numbers, numbering terms. All these are different ways of subtracting from , and all give different results. As regards division, very similar results follow from the fact that is unchanged when multiplied by 2 or 3 or any finite number or by . It follows that divided by may have any value from 1 up to . From the ambiguity of subtraction and division it results that negative numbers and ratios cannot be extended to infinite numbers. Addition, multiplication, and exponentiation proceed quite satisfactorily, but the inverse operations—subtraction, division, and extraction of roots—are ambiguous, and the notions that depend upon them fail when infinite numbers are concerned. The characteristic by which we defined finitude was mathematical induction, i.e. we defined a number as finite when it obeys mathematical induction starting from 0, and a class as finite when its number is finite. This definition yields the sort of result that a definition ought to yield, namely, that the finite [Pg 87] numbers are those that occur in the ordinary number-series 0, 1, 2, 3, ... But in the present chapter, the infinite numbers we have discussed have not merely been non-inductive: they have also been reflexive. Cantor used reflexiveness as the definition of the infinite, and believes that it is equivalent to non-inductiveness; that is to say, he believes that every class and every cardinal is either inductive or reflexive. This may be true, and may very possibly be capable of proof; but the proofs hitherto offered by Cantor and others (including the present author in former days) are fallacious, for reasons which will be explained when we come to consider the "multiplicative axiom." At present, it is not known whether there are classes and cardinals which are neither reflexive nor inductive. If were such a cardinal, we should not have , but would not be one of the "natural numbers," and would be lacking in some of the inductive properties. All known infinite classes and cardinals are reflexive; but for the present it is well to preserve an open mind as to whether there are instances, hitherto unknown, of classes and cardinals which are neither reflexive nor inductive. Meanwhile, we adopt