Page:EB1911 - Volume 19.djvu/880

 4. The Natural Scale.—Let $$a$$ be any element of a well-ordered aggregate $$\mbox{A}$$. Then all the elements posterior to $$a$$ form an aggregate $$\mbox{A}'$$, which is a part of $$\mbox{A}$$ and, by definition, has a first element $$a'$$. This element $$a'$$ is different from $$a$$, and immediately succeeds it in the order of $$\mbox{A}$$. (It may happen, of course, that $$a'$$ does not exist; in this case $$a$$ is the last element of $$\mbox{A}$$.) Thus in a well-ordered aggregate every element except the last (if there be a last element) is succeeded by a definite next element. The ingenuity of man has developed a symbolism by means of which every symbol is associated with a definite next succeeding symbol, and in this way we have a set of visible or audible signs 1, 2, 3, &c. (or their verbal equivalents), representing an aggregate in which (1) there is a definite order, (2) there is a first term, (3) each term has one next following, and consequently there is no last term. Counting a set of objects means associating them in order with the first and subsequent members of this conventional aggregate. The process of counting may lead to three different results: (1) the set of objects may be finite in number, so that they are associated with a part of the conventional aggregate which has a last term; (2) the set of objects may have the same power as the conventional aggregate; (3) the set of objects may have a higher power than the conventional aggregate. Examples of (2) and (3) will be found further on. The order-type of 1, 2, 3, &c., and of similar aggregates will be denoted by $$\omega$$; this is the first and simplest member of a set of transfinite ordinal numbers to be considered later on. Any finite number such as 3 is used ordinally as representing the order-type of 1, 2, 3 or any similar aggregate, and cardinally as representing the power of 1, 2, 3 or any equivalent aggregate. For reasons that will appear, $$\omega$$ is only used in an ordinal sense. The aggregate 1, 2, 3, &c., in any of its written or spoken forms, may be called the natural scale, and denoted by $$\mbox{N}$$. It has already been shown that $$\mbox{N}$$ is infinite: this appears in a more elementary way from the fact that $$(1, 2, 3, 4,...)\simeq(2, 3, 4, 5,...)$$, where each element of $$\mbox{N}$$ is made to correspond with the next following. Any aggregate which is equivalent to the natural scale or a part thereof is said to be countable.

5. Arithmetical Operations.—When the natural scale $$\mbox{N}$$ has once been obtained it is comparatively easy, although it requires a long process of induction, to define the arithmetical operations of addition, multiplication and involution, as applied to natural numbers. It can be proved that these operations are free from ambiguity and obey certain formal laws of commutation, &c., which will not be discussed here. Each of the three direct operations leads to an inverse problem which cannot be solved except under certain implied conditions. Let $$a, b$$ denote any two assigned natural numbers: then it is required to find natural numbers, $$x, y, z$$ such that

respectively. The solutions, when they exist, are perfectly definite, and may be denoted by $$a-b,\;a/b$$ and $$\sqrt[b]{a}$$; but they are only possible in the first case when $$a>b$$, in the second when $$a$$ is a multiple of $$b$$, and in the third when $$a$$ is a perfect $$b$$th power. It is found to be possible, by the construction of certain elements, called respectively negative, fractional and irrational numbers, and zero, to remove all these restrictions.

6. There are certain properties, common to the aggregates with which we have next to deal, analogous to those possessed by the natural scale, and consequently justifying us in applying the term number to any one of their elements. They are stated here, once for all, to avoid repetition; the verification, in each case, will be, for the most part, left to the reader. Each of the aggregates in question ($$\mbox{A}$$, suppose) is an ordered aggregate. If $$\alpha, \beta$$ are any two elements of $$\mbox{A}$$, they may be combined by two definite operations, represented by $$+$$ and $$\times$$, so as to produce two definite elements of $$\mbox{A}$$ represented by $$\alpha+\beta$$ and $$\alpha\times\beta$$ (or $$\alpha\beta$$); these operations obey the formal laws satisfied by those of addition and multiplication. The aggregate $$\mbox{A}$$ contains one (and only one) element $$\iota$$, such that if $$\alpha$$ is any element of $$\mbox{A}$$ ($$\iota$$ included), then $$\alpha+\iota>\alpha$$, and $$\alpha\iota>\alpha$$. Thus $$\mbox{A}$$ contains the elements $$\iota,\iota+\iota,\iota+\iota+\iota, \&\mbox{c}.$$, or, as we may write them, $$\iota, 2\iota, 3\iota, ... m\iota ...$$ such that $$m\iota + n\iota = (m+n)\iota$$ and $$m\iota \times n\iota = mn\iota$$; also $$\iota<2\iota<3\iota ...$$ We may express this by saying that $$\mbox{A}$$ contains an image of the natural scale. The element denoted by $$\iota$$ may be called the ground element of $$\mbox{A}$$.

7. Negative Numbers.—Let any two natural numbers $$a, b$$ be selected in a definite order $$a, b$$ (to be distinguished from $$b, a$$, in which the order is reversed). In this way we obtain from $$\mbox{N}$$ an aggregate of symbols $$(a, b)$$ which we shall call couples, or more precisely, if necessary, polar couples. This new aggregate may be arranged in order by means of the following rules:—

Two couples $$(a, b), (a', b')$$ are said to be equal if $$a+b' = a'+b$$. In other words $$(a, b), (a', b')$$ are then taken to be equivalent symbols for the same thing.

If $$a+b'>a'+b$$, we write $$(a, b)>(a', b')$$; and if $$a+b' (a, b)$$ and $$(a, b) \times \iota = (2a+b, a+2b) = (a, b)$$. Hence $$\iota$$ is the ground element of $$\bar{\mbox{N}}$$. By definition, $$2\iota = \iota+\iota = (4, 2) = (3, 1)$$: and hence by induction $$m\iota=(m+1, 1)$$, where $$m$$ is any natural integer. Conversely every couple $$(a, b)$$ in which $$a > b$$ can be expressed by the symbol $$(a-b)\iota$$. In the same way, every couple $$(a, b)$$ in which $$b>a$$ can be expressed in the form $$(b-a)\iota'$$, where $$\iota'=(1, 2)$$.

8. It follows as a formal consequence of the definitions that $$\iota+\iota' = (2, 1)+(1, 2) = (3, 3) = (1, 1)$$. It is convenient to denote $$(1, 1)$$ and its equivalent symbols by $$0$$, because hence $$\iota+\iota'=0$$, and we can represent $$\bar{\mbox{N}}$$ by the scheme—

in which each element is obtained from the next before it by the addition of $$\iota$$. With this notation the rules of operation may be written ($$m, n$$, denoting natural numbers)—

with the special rules for zero, that if $$\alpha$$ is any element of $$\bar{\mbox{N}}$$,

To each element, $$\alpha$$, of $$\bar{\mbox{N}}$$ corresponds a definite element $$\alpha'$$ such that $$\alpha+\alpha' = 0$$; if $$\alpha=0$$, then $$\alpha'=0$$, but in every other case $$\alpha, \alpha'$$ are different and may be denoted by $$m\iota, m\iota'$$. The natural number $$m$$ is called the absolute value of $$m\iota$$ and $$m\iota'$$.

9. If $$\alpha, \beta$$ are any two elements of $$\bar{\mbox{N}}$$, the equation $$\xi+\beta = \alpha$$ is satisfied by putting $$\xi = \alpha + \beta'$$. Thus the symbol $$\alpha - \beta$$ is always interpretable as $$\alpha + \beta'$$, and we may say that within $$\bar{\mbox{N}}$$ subtraction is always possible; it is easily proved to be also free from ambiguity. On the other hand, $$\alpha / \beta$$ is intelligible only if the absolute value of $$\alpha$$ is a multiple of the absolute value of $$\beta$$.

The aggregate $$\bar{\mbox{N}}$$ has no first element and no last element. At the same time it is countable, as we see, for instance, by associating the elements $$0, a\iota, b\iota'$$ with the natural numbers $$1, 2a, 2b+1$$ respectively, thus— $(\mbox{N})\, 1, 2, 3, 4, 5, 6, \dots$ $(\bar{\mbox{N}})\, 0, \iota, \iota', 2\iota, 2\iota', 3\iota \dots$|undefined

It is usual to write $$+a$$ (or simply $$a$$) for $$a\iota$$ and $$-a$$ for $$a\iota'$$; that this should be possible without leading to confusion or ambiguity is certainly remarkable.

10. Fractional Numbers.—We will now derive from $$\mbox{N}$$ a different aggregate of couples $$[a, b]$$ subject to the following rules:

The symbols $$[a, b], [a', b'],$$ are equivalent if $$ab' = a'b$$. According as $$ab'$$ is greater or less than $$a'b$$ we regard $$[a, b]$$ as being greater or less than $$[a', b']$$. The formulae for addition and multiplication are

All the couples $$[a, a]$$ are equivalent to $$[1, 1]$$, and if we denote