17. Sections of the real numbers

In §§ 4-7 we considered ‘sections’ of the rational numbers, i.e. modes of division of the rational numbers (or of the positive rational numbers only) into two classes $L$ and $R$ possessing the following characteristic properties:

(i) that every number of the type considered belongs to one and only one of the two classes;

(ii) that both classes exist;

(iii) that any member of $L$ is less than any member of $R$.

It is plainly possible to apply the same idea to the aggregate of all real numbers, and the process is, as the reader will find in later chapters, of very great importance.

Let us then suppose¹ that $P$ and $Q$ are two properties which are mutually exclusive, and one of which is possessed by every real number. Further let us suppose that any number which possesses $P$ is less than any which possesses $Q$. We call the numbers which possess $P$ the lower or left-hand class $L$, and those which possess $Q$ the upper or right-hand class $R$.

Thus $P$ might be $x \leq \sqrt{2}$ and $Q$ be $x > \sqrt{2}$. It is important to observe that a pair of properties which suffice to define a section of the rational numbers may not suffice to define one of the real numbers. This is so, for example, with the pair ‘$x < \sqrt{2}$’ and ‘$x > \sqrt{2}$’ or (if we confine ourselves to positive numbers) with ‘$x^{2} < 2$’ and ‘$x^{2} > 2$’. Every rational number possesses one or other of the properties, but not every real number, since in either case $\sqrt{2}$ escapes classification.

There are now two possibilities.² Either $L$ has a greatest member $l$, or $R$ has a least member $r$. Both of these events cannot occur. For if $L$ had a greatest member $l$, and $R$ a least member $r$, the number $\frac{1}{2}(l + r)$ would be greater than all members of $L$ and less than all members of $R$, and so could not belong to either class. On the other hand one event must occur.³

For let $L_{1}$ and $R_{1}$ denote the classes formed from $L$ and $R$ by taking only the rational members of $L$ and $R$. Then the classes $L_{1}$ and $R_{1}$ form a section of the rational numbers. There are now two cases to distinguish.

It may happen that $L_{1}$ has a greatest member $\alpha$. In this case $\alpha$ must be also the greatest member of $L$. For if not, we could find a greater, say $\beta$. There are rational numbers lying between $\alpha$ and $\beta$, and these, being less than $\beta$, belong to $L$, and therefore to $L_{1}$; and this is plainly a contradiction. Hence $\alpha$ is the greatest member of $L$.

On the other hand it may happen that $L_{1}$ has no greatest member. In this case the section of the rational numbers formed by $L_{1}$ and $R_{1}$ is a real number $\alpha$. This number $\alpha$ must belong to $L$ or to $R$. If it belongs to $L$ we can show, precisely as before, that it is the greatest member of $L$, and similarly, if it belongs to $R$, it is the least member of $R$.

Thus in any case either $L$ has a greatest member or $R$ a least. Any section of the real numbers therefore ‘corresponds’ to a real number in the sense in which a section of the rational numbers sometimes, but not always, corresponds to a rational number. This conclusion is of very great importance; for it shows that the consideration of sections of all the real numbers does not lead to any further generalisation of our idea of number. Starting from the rational numbers, we found that the idea of a section of the rational numbers led us to a new conception of a number, that of a real number, more general than that of a rational number; and it might have been expected that the idea of a section of the real numbers would have led us to a conception more general still. The discussion which precedes shows that this is not the case, and that the aggregate of real numbers, or the continuum, has a kind of completeness which the aggregate of the rational numbers lacked, a completeness which is expressed in technical language by saying that the continuum is closed.

The result which we have just proved may be stated as follows:

Dedekind’s Theorem. If the real numbers are divided into two classes $L$ and $R$ in such a way that

(i) every number belongs to one or other of the two classes,

(ii) each class contains at least one number,

(iii) any member of $L$ is less than any member of $R$,
then there is a number $\alpha$, which has the property that all the numbers less than it belong to $L$ and all the numbers greater than it to $R$. The number $\alpha$ itself may belong to either class.

In applications we have often to consider sections not of all numbers but of all those contained in an interval ${[\beta, \gamma]}$, that is to say of all numbers $x$ such that $\beta \leq x \leq \gamma$. A ‘section’ of such numbers is of course a division of them into two classes possessing the properties (i), (ii), and (iii). Such a section may be converted into a section of all numbers by adding to $L$ all numbers less than $\beta$ and to $R$ all numbers greater than $\gamma$. It is clear that the conclusion stated in Dedekind’s Theorem still holds if we substitute ‘the real numbers of the interval ${[\beta, \gamma]}$’ for ‘the real numbers’, and that the number $\alpha$ in this case satisfies the inequalities $\beta \leq \alpha \leq \gamma$.

The discussion which follows is in many ways similar to that of § 6. We have not attempted to avoid a certain amount of repetition. The idea of a ‘section,’ first brought into prominence in Dedekind’s famous pamphlet Stetigkeit und irrationale Zahlen, is one which can, and indeed must, be grasped by every reader of this book, even if he be one of those who prefer to omit the discussion of the notion of an irrational number contained in §§ 6-12.↩︎
There were three in § 6.↩︎
This was not the case in § 6.↩︎

$\leftarrow$ 16. The continuous real variable

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18. Points of accumulation $\rightarrow$