Symbolic Logic and the Integer

Consider now a collection of classes which can be put in one to-one correspondence with one another. These can be regarded as associated with a mark which is an ordinary integer if the classes contain a finite number of elements, and which is called a trans finite cardinal, if there are infinitely many elements in each class. The simplest such mark w is that attached to the class of positive integers; co is the mark characteristic of enumerable classes, i.e., of classes which may be put in one-to-one correspondence with the infinite sequence of integers 1, 2, 3. . . . The class of all fractions -"! has also this transfinite cardinal w, for these n fractions may be ordered according to the size of the sum of numerator and denominator : We may now construct another number less than I as follows: Let the first figure after the decimal point be chosen apart from o and the first figure of the first number of S; the second figure be chosen apart from o and the second figure of the second number of S, and so on. There will then be defined a number in decimal form which is obviously distinct from all the numbers of S. Hence the hypothesis that the sequence contains all of the numbers between o and I is incorrect. The conclusion may be stated in the form: The continuum of numbers between o and is not enumerable. By an extension of this method it is possible to define greater and greater transfinite numbers.

In the consideration of the transfinite cardinals various para doxes arise unless some definite logical theory such as the theory of types of Russell and Whitehead is carefully adhered to. More over the precise structure of these transfinite numbers is not known except in the case of the least such number w ; thus it is not known whether or not the transfinite cardinal of the con tinuum is the first exceeding w or not.

Cardinal and Ordinal Numbers.

In the account of number given above no reference has been made to the usual elementary distinction between the cardinal and ordinal integers. Up to the time of the discovery of transfinite number this distinction pos sessed merely elementary pedagogical significance, and the mathematical treatment of number dealt exclusively with the cardinal integers. With the invention of the transfinite numbers, however, it appeared immediately that this distinction, although mathematically barren in the case of ordinary integers, lead to two distinct types of number when applied to infinite classes, namely to the transfinite cardinals and to the transfinite ordinals. The distinction between cardinal and ordinal number may be expressed as follows in the case of finite classes. In the case of the cardinal type of integer we have a mark characteristic of the class without reference to any order of its elements; in the case of the ordinal type of integer the class is ordered and its elements are supposed to be counted in this order, the mark I being attached to the first element, the mark 2 to the second, and so on, to the end of the sequence. This gives an ordinal number for each element. Of course if the last element has the mark n, the cardinal number of the class is also n. Thus the two kinds of integer are very intimately related.

When we deal with infinite classes and attempt to extend the notion of ordinal number, it is necessary first of all to define what is meant by an ordered infinite class. First we have an infinite sequence I, 2, 3, • • • as before. To continue with the ordinal numeration we call the next mark w; after it the suc cessive marks are denoted by co+ 1, co-I-2, • • • , although this is merely by convention. Immediately after this sequence appears the next ordinal, denoted by 0).2, followed by co•2-1-- 1, (.0.2+2, • • • , then co.3, w.3+1, • . Evidently this process of ordinal numeration is now part of a more extensive sequential process which may be written symbolically A first conjecture might be that all infinite classes have the transfinite cardinal co. The following simple reasoning shows, however, that the class of all ordinary numbers between o and has a greater transfinite cardinal than w.

Suppose it possible that the class of these numbers has the transfinite cardinal co. This would signify that all of these numbers could be written in a sequence S just as the integers I, 2, 3, . . . can be written. First let us imagine the numbers of this sequence to be written in decimal form. Now certain rational numbers admit of double representation, as is evident from the fact that 0.879999 . . . = o.880000. . . . Let us agree, in all such cases, to use the mode of representation by g's, so that every number shall have a unique decimal representation.

Thus a series of order types of increasing complexity is formed, and the process may be continued indefinitely. These give the sequence of transfinite ordinals, appropriate to the numeration of well-ordered infinite classes. A class is said to be "well-ordered" if the elements are not only ordered but are such that every ele ment or ordered sequence of elements has an immediately fol lowing element.

The marks for the transfinite ordinals are not the same as for the cardinals. Thus it is immediately evident that the set of transfinite ordinals up to are enumerable; i.e., they have the transfinite cardinal w. It is not as yet known whether or not every class, e.g., the continuum of numbers between o and 1, can be well-ordered or not.

The theory of the transfinite cardinal and ordinal numbers, and their interrelations is of great philosophical as well as mathe matical interest. The subject requires much further development. In particular the underlying basis of symbolic logic has not been agreed upon.

treatment of number: L. L. Conan

t, The Number Concept (New York and London, 1836) ; H. B. Fine, The Number System of Algebra (Boston, 1903) ; G. H. Hardy, A Course of Pure Mathematics (Cambridge, 1914), ch. i., iii.; B. Russell, Intro duction to Mathematical Philosophy (New York, 1919).

General discussion of transfinite number: G. Cantor,

Contributions to the Founding of the Theory of Transfinite Numbers (tr., Chicago and London, 1915) ; E. V. Huntington, The Continuum and Other Types of Serial Order (Cambridge, Mass., 1917) ; E. W. Hobson, Theory of Functions of a Real Variable, ch. i.-iv. (Cambridge, 1927).