Volume 1 / Part 3 — SECTION C. FINITE AND INFINITE.
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Summary of Section C.
The distinction of finite and infinite is not required, as appears from Section B, for the definition of the arithmetical operations or for the proof of their formal laws. There are, however, many important respects in which finite cardinals and classes differ respectively from infinite cardinals and classes, and these differences must now be investigated.
There are two different ways in which we may define the finite and the infinite, and these two ways cannot (so far as is known at present) be shown to be equivalent except by assuming the multiplicative axiom. As there seems no good reason for regarding one of these ways as giving more exactly than the other what is usually meant by the words "finite" and "infinite," we shall, to avoid confusion, give other names than these to each of the two ways of dividing classes and cardinals. The division effected by the first method of definition we shall call the division into inductive and non-inductive; that effected by the second method we shall call the division into non-reflexive and reflexive.
The division into inductive and non-inductive, which is treated in *120, is defined as follows. An inductive cardinal is one which can be reached from 0 by successive additions of 1; that is, an inductive cardinal is one which has to 0 the relation , where (by *38·02) is the relation of to , and the subscript asterisk has the meaning defined in *90. Hence we put By applying the definition of *90, this gives This proposition may be regarded as stating that an inductive cardinal is one which obeys mathematical induction starting from 0, i.e. it is one which possesses every property possessed by 0 and by the numbers obtained by adding 1 to numbers possessing the property. In elementary mathematics, it is customary to regard mathematical induction, as applied to the series of natural numbers, as a principle rather than a definition, but according to the above procedure it becomes a definition rather than a principle. This procedure is unavoidable as soon as it is perceived that there are cardinals which do not obey mathematical induction starting from 0. (This only holds on the assumption that the total number of objects in any one type is not one of the inductive cardinals. This assumption, in a slightly different form, is introduced below as the "axiom of infinity.") Thus for example , and . Hence if is any inductive cardinal, . But we know that , the first of Cantor's transfinite cardinals[6], satisfies . Thus mathematical induction starting from 0 cannot be validly applied to prove properties of . It follows that the inductive cardinals as above defined are only some among cardinals; nor does it appear that there is any way of defining them except as those that obey mathematical induction starting from 0. It follows that mathematical induction is not a principle, to be either proved or assumed as an axiom, but is merely a characteristic defining a certain class of cardinals, namely the class of inductive cardinals.
By a syllogism in Barbara, it is evident that 0 is an inductive cardinal; hence by the definition 1 is an inductive cardinal, and hence 2, 3, ... are inductive cardinals. Thus any given cardinal in the series of natural numbers can be shown to be an inductive cardinal. The usual elementary properties of inductive cardinals, such as the uniqueness of subtraction and division, are easily proved by mathematical induction.
We define an inductive class as a class the number of whose terms is an inductive cardinal. More simply, we put It is then easily shown that an inductive class is one which can be reached from by successive additions of single members. That is, if we put
Thus we have We might equally well have begun by defining inductive classes, and proceeded to define inductive cardinals as the cardinals of inductive classes; in that case, we should have used the above relation to define inductive classes.
Some of the properties which we expect inductive cardinals to possess, such for example as , can only be proved by assuming that no inductive cardinal is null, i.e. that This amounts to the assumption that, in any fixed type, a class can be found having any assigned inductive number of terms. If this were false, there would have to be some definite member of the series of natural numbers which gave the total number of objects of the type in question. Thus suppose there were exactly n individuals in the universe, and no more, where n is an inductive cardinal. We should then have classes, classes of classes, and so on. In that case, in the type of individuals we should have , , etc. Hence we should have In the type of classes, we should get similar results for , and so on. It is plain (though not demonstrable except in each particular case) that if the assumption fails in any one type, it fails in any other type in the same hierarchy, and if it holds in any one, it holds in any other; for if n be the total number of individuals, then if is an inductive cardinal, the total number of any other type is an inductive cardinal, while if is not an inductive cardinal, no more is the total number of any other type. Hence the assumption is either true in any type or false in any type in one hierarchy. We shall call it the "axiom of infinity," putting This assumption, like the multiplicative axiom, will be adduced as a hypothesis whenever it is relevant. It seems plain that there is nothing in logic to necessitate its truth or falsehood, and that it can only be legitimately believed or disbelieved on empirical grounds. When we wish to use a typically definite form of the axiom, we shall employ the definition which asserts that, if is any inductive cardinal, there are at least terms of the same type as .
It is important to observe that, although the axiom of infinity cannot (so far as appears) be proved a priori, we can prove that any given inductive cardinal exists in a sufficiently high type. For if the total number of individuals be , the numbers of objects in succeeding types are , , etc., and these numbers grow beyond any assigned inductive cardinal. Owing, however, to the fact that we cannot add together an infinite number of classes whose types increase without limit, we cannot hence show that there is a type in which every inductive cardinal exists, though we can show of every inductive cardinal that there is a type in which it exists. i.e. if is any inductive cardinal, there must be a type for such that is true; but there need not be a type for such that if is any inductive cardinal, is true.
The axiom of infinity suffices to prove the existence, in appropriate types, of , , , ... , , ...[7]. It does not suffice, so far as we know, to prove the existence of or any Aleph with a greater suffix than , because the existences of , , ... are proved in successively rising types, and no meaning can be found for a type whose order is infinite.
The other definition of finite and infinite is of less importance in practice than the definition by induction. It is dealt with in *124. According to this definition, we call a class reflexive when it contains a proper part similar to itself, i.e. we put or, what comes to the same thing, We call a cardinal reflexive when it is the homogeneous cardinal of a reflexive class, i.e. we put It is easy to show that We find that inductive classes and cardinals are non-reflexive, and reflexive classes and cardinals are non-inductive. We find also that reflexive cardinals are those that are equal to or greater than , while inductive cardinals are those that are less than . By assuming the multiplicative axiom, we can show that every cardinal is equal to, greater than, or less than , whence it follows that every cardinal is either reflexive or inductive, thus identifying the two definitions of finite and infinite. But so long as we refrain from assuming either the multiplicative axiom or some special axiom ad hoc, it remains possible (so far as is known at present) that there may be cardinals neither greater than, nor equal to, nor less than . Such cardinals, if they exist, are neither inductive nor reflexive: they are infinite if we define infinity by the negation of induction, but finite if we define infinity by reflexiveness. It is possible that further investigation may either prove or disprove the existence of such cardinals; for the present, their existence must remain an open question, except for those who regard the multiplicative axiom as a self-evident truth.
In *121 we shall consider intervals in a discrete series; i.e. in a series generated by a one-one relation between consecutive terms. If be the generating relation of such a series, and and be two members of the series, of which is the later, the terms which lie between and are the terms for which we have where has the meaning defined in *91. Hence we put where "" means "the -interval between and ." We want also symbols for the interval together with one or both of its end-points. For these we put [8] Thus, for example, if and be inductive cardinals, and be the relation of to , and , will be the numbers greater than and less than , while will be these numbers together with , will be these numbers together with , and will be these numbers together with both and . By means of intervals, we define a class of relations (where is any inductive cardinal), where "" means that we can pass from to in steps. In order to fit the case in which and are identical, and to insure that no relation such as shall hold between terms which do not both belong to the field of , we put Then, provided , , and if further , then , , etc. If is a transitive serial relation, is the relation "immediately preceding," which has great importance in well-ordered series. In this case, . If is a transitive serial relation generating a finite series or a progression or a series of the type of the negative and positive integers in order of magnitude, we have
In *121 we shall only consider in the case where and generally we shall have the further hypothesis . We can then prove that the interval between and is always an inductive class (it will be null unless ); this proposition is useful in its application to the number-series and to progressions generally.
When , the class of such relations as (where is an inductive cardinal) is identical with , the class of powers of (cf. *91 seq.). This identification (which does not hold in general without the above hypothesis) leads to many useful propositions. In *91 seq., we treated powers of a relation without the use of numbers, i.e. without defining the th power of . When the powers of are the class of such relations as , we can of course take as the th power of . The general definition of the th power of (where is an inductive cardinal) will be given later, in *301; we shall denote it by , thereby including the notation already defined.
In *122 we shall deal with progressions, i.e. with series of the type of the series of natural numbers. In this number, we shall deal with such series as generated by one-one relations; they will be dealt with at a later stage (*263) as generated by transitive relations. We define a progression as a one-one relation whose domain is the posterity of its first term, i.e. According to this definition, there must be a first term ; will be , i.e. , which is contained in , i.e. in ; since , every term of the field of has a successor, so that there is no end to the series; since , every term of the series can be reached from the beginning by successive steps. These characteristics suffice to define progressions.
In *123 we proceed to the definition and discussion of , the smallest of reflexive cardinals. This is the cardinal number of any class whose terms can be arranged in a progression; hence it is the class of domains of progressions, i.e. we may put With this definition, remembering that is a cardinal, we can prove that is a cardinal; but to prove that is an existent cardinal, we need the axiom of infinity. The existence-theorem for is then derived from the inductive cardinals, which, if no one of them is null, form a progression when arranged in order of magnitude. It should be observed that this existence-theorem is for a higher type than that for which the axiom of infinity is assumed. In order to get an existence-theorem for the same type, we need the multiplicative axiom as well.
After a number on reflexive classes and cardinals (*124) and a number on the axiom of infinity (*125), the Section ends with a number (*126) on "typically indefinite inductive cardinals." The constant inductive cardinals are the typically ambiguous symbols 0, 1, 2, ...; thus we want to define the class of inductive cardinals in such a way that a variable member of the class shall be typically ambiguous. This is not possible without a sacrifice of rigour, but in *126 it is shown how to minimize the sacrifice of rigour, and how to obviate the resulting logical dangers. A variable whose values are typically ambiguous is said to be "typically indefinite."
A proof that all inductive cardinals exist has often been derived from *120·57 (below). But according to the doctrine of types, this proof is invalid, since "" in *120·57 is necessarily of higher type than "."