[page 97]
The conception of a “limit” is one of which the importance in mathematics has been found continually greater than had been thought. The whole of the differential and integral calculus, indeed practically everything in higher mathematics, depends upon limits. Formerly, it was supposed that infinitesimals were involved in the foundations of these subjects, but Weierstrass showed that this is an error: wherever infinitesimals were thought to occur, what really occurs is a set of finite quantities having zero for their lower limit. It used to be thought that “limit” was an essentially quantitative notion, namely, the notion of a quantity to which others approached nearer and nearer, so that among those others there would be some differing by less than any assigned quantity. But in fact the notion of “limit” is a purely ordinal notion, not involving quantity at all (except by accident when the series concerned happens to be quantitative). A given point on a line may be the limit of a set of points on the line, without its being necessary to bring in co-ordinates or measurement or anything quantitative. The cardinal number ℵ0 is the limit (in the order of magnitude) of the cardinal numbers 1, 2, 3, … n, …, although the numerical difference between ℵ0 and a finite cardinal is constant and infinite: from a quantitative point of view, finite numbers get no nearer to ℵ0 as they grow larger. What makes ℵ0 the limit of the finite numbers is the fact that, in the series, it comes immediately after them, which is an ordinal fact, not a quantitative fact.
[page 98]
There are various forms of the notion of “limit,” of increasing complexity. The simplest and most fundamental form, from which the rest are derived, has been already defined, but we will here repeat the definitions which lead to it, in a general form in which they do not demand that the relation concerned shall be serial. The definitions are as follows:—
The “minima” of a class α with respect to a relation P are those members of α and the field of P (if any) to which no member of α has the relation P.
The “maxima” with respect to P are the minima with respect to the converse of P.
The “sequents” of a class α with respect to a relation P are the minima of the “successors” of α, and the “successors” of α are those members of the field of P to which every member of the common part of α and the field of P has the relation P.
The “precedents” with respect to P are the sequents with respect to the converse of P.
The “upper limits” of α with respect to P are the sequents provided α has no maximum; but if α has a maximum, it has no upper limits.
The “lower limits” with respect to P are the upper limits with respect to the converse of P.
Whenever P has connexity, a class can have at most one maximum, one minimum, one sequent, etc. Thus, in the cases we are concerned with in practice, we can speak of “the limit” (if any).
When P is a serial relation, we can greatly simplify the above definition of a limit. We can, in that case, define first the “boundary” of a class α, i.e. its limit or maximum, and then proceed to distinguish the case where the boundary is the limit from the case where it is a maximum. For this purpose it is best to use the notion of “segment.”
We will speak of the “segment of P defined by a class α” as all those terms that have the relation P to some one or more of the members of α. This will be a segment in the sense defined [page 99] in Chapter VII.; indeed, every segment in the sense there defined is the segment defined by some class α. If P is serial, the segment defined by α consists of all the terms that precede some term or other of α. If α has a maximum, the segment will be all the predecessors of the maximum. But if α has no maximum, every member of α precedes some other member of α, and the whole of α is therefore included in the segment defined by α. Take, for example, the class consisting of the fractions
1/2, 3/4, 7/8, 15/16, …,
i.e. of all fractions of the form 1−1/2n for different finite values of n. This series of fractions has no maximum, and it is clear that the segment which it defines (in the whole series of fractions in order of magnitude) is the class of all proper fractions. Or, again, consider the prime numbers, considered as a selection from the cardinals (finite and infinite) in order of magnitude. In this case the segment defined consists of all finite integers.
Assuming that P is serial, the “boundary” of a class α will be the term x (if it exists) whose predecessors are the segment defined by α.
A “maximum” of α is a boundary which is a member of α.
An “upper limit” of α is a boundary which is not a member of α.
If a class has no boundary, it has neither maximum nor limit. This is the case of an “irrational” Dedekind cut, or of what is called a “gap.”
Thus the “upper limit” of a set of terms α with respect to a series P is that term x (if it exists) which comes after all the α’s, but is such that every earlier term comes before some of the α’s.
We may define all the “upper limiting-points” of a set of terms β as all those that are the upper limits of sets of terms chosen out of β. We shall, of course, have to distinguish upper limiting-points from lower limiting-points. If we consider, for example, the series of ordinal numbers:
1, 2, 3, … ω, ω+1, … 2ω, 2ω+1, … 3ω, … ω2, … ω3, …,
[page 100]
the upper limiting-points of the field of this series are those that have no immediate predecessors, i.e.
1, ω, 2ω, 3ω, … ω2, ω2+ω, … 2ω2, … ω3 …
The upper limiting-points of the field of this new series will be
1, ω2, 2ω2, … ω3, ω3+ω2 …
On the other hand, the series of ordinals—and indeed every well-ordered series—has no lower limiting-points, because there are no terms except the last that have no immediate successors. But if we consider such a series as the series of ratios, every member of this series is both an upper and a lower limiting-point for suitably chosen sets. If we consider the series of real numbers, and select out of it the rational real numbers, this set (the rationals) will have all the real numbers as upper and lower limiting-points. The limiting-points of a set are called its “first derivative,” and the limiting-points of the first derivative are called the second derivative, and so on.
With regard to limits, we may distinguish various grades of what may be called “continuity” in a series. The word “continuity” had been used for a long time, but had remained without any precise definition until the time of Dedekind and Cantor. Each of these two men gave a precise significance to the term, but Cantor’s definition is narrower than Dedekind’s: a series which has Cantorian continuity must have Dedekindian continuity, but the converse does not hold.
The first definition that would naturally occur to a man seeking a precise meaning for the continuity of series would be to define it as consisting in what we have called “compactness,” i.e. in the fact that between any two terms of the series there are others. But this would be an inadequate definition, because of the existence of “gaps” in series such as the series of ratios. We saw in Chapter VII. that there are innumerable ways in which the series of ratios can be divided into two parts, of which one wholly precedes the other, and of which the first has no last term, [page 101] while the second has no first term. Such a state of affairs seems contrary to the vague feeling we have as to what should characterise “continuity,” and, what is more, it shows that the series of ratios is not the sort of series that is needed for many mathematical purposes. Take geometry, for example: we wish to be able to say that when two straight lines cross each other they have a point in common, but if the series of points on a line were similar to the series of ratios, the two lines might cross in a “gap” and have no point in common. This is a crude example, but many others might be given to show that compactness is inadequate as a mathematical definition of continuity.
It was the needs of geometry, as much as anything, that led to the definition of “Dedekindian” continuity. It will be remembered that we defined a series as Dedekindian when every sub-class of the field has a boundary. (It is sufficient to assume that there is always an upper boundary, or that there is always a lower boundary. If one of these is assumed, the other can be deduced.) That is to say, a series is Dedekindian when there are no gaps. The absence of gaps may arise either through terms having successors, or through the existence of limits in the absence of maxima. Thus a finite series or a well-ordered series is Dedekindian, and so is the series of real numbers. The former sort of Dedekindian series is excluded by assuming that our series is compact; in that case our series must have a property which may, for many purposes, be fittingly called continuity. Thus we are led to the definition:
A series has “Dedekindian continuity” when it is Dedekindian and compact.
But this definition is still too wide for many purposes. Suppose, for example, that we desire to be able to assign such properties to geometrical space as shall make it certain that every point can be specified by means of co-ordinates which are real numbers: this is not insured by Dedekindian continuity alone. We want to be sure that every point which cannot be specified by rational co-ordinates can be specified as the limit of a progression of points [page 102] whose co-ordinates are rational, and this is a further property which our definition does not enable us to deduce.
We are thus led to a closer investigation of series with respect to limits. This investigation was made by Cantor and formed the basis of his definition of continuity, although, in its simplest form, this definition somewhat conceals the considerations which have given rise to it. We shall, therefore, first travel through some of Cantor’s conceptions in this subject before giving his definition of continuity.
Cantor defines a series as “perfect” when all its points are limiting-points and all its limiting-points belong to it. But this definition does not express quite accurately what he means. There is no correction required so far as concerns the property that all its points are to be limiting-points; this is a property belonging to compact series, and to no others if all points are to be upper limiting- or all lower limiting-points. But if it is only assumed that they are limiting-points one way, without specifying which, there will be other series that will have the property in question—for example, the series of decimals in which a decimal ending in a recurring 9 is distinguished from the corresponding terminating decimal and placed immediately before it. Such a series is very nearly compact, but has exceptional terms which are consecutive, and of which the first has no immediate predecessor, while the second has no immediate successor. Apart from such series, the series in which every point is a limiting-point are compact series; and this holds without qualification if it is specified that every point is to be an upper limiting-point (or that every point is to be a lower limiting-point).
Although Cantor does not explicitly consider the matter, we must distinguish different kinds of limiting-points according to the nature of the smallest sub-series by which they can be defined. Cantor assumes that they are to be defined by progressions, or by regressions (which are the converses of progressions). When every member of our series is the limit of a progression or regression, Cantor calls our series “condensed in itself” (insichdicht).
[page 103]
We come now to the second property by which perfection was to be defined, namely, the property which Cantor calls that of being “closed” (abgeschlossen). This, as we saw, was first defined as consisting in the fact that all the limiting-points of a series belong to it. But this only has any effective significance if our series is given as contained in some other larger series (as is the case, e.g., with a selection of real numbers), and limiting-points are taken in relation to the larger series. Otherwise, if a series is considered simply on its own account, it cannot fail to contain its limiting-points. What Cantor means is not exactly what he says; indeed, on other occasions he says something rather different, which is what he means. What he really means is that every subordinate series which is of the sort that might be expected to have a limit does have a limit within the given series; i.e. every subordinate series which has no maximum has a limit, i.e. every subordinate series has a boundary. But Cantor does not state this for every subordinate series, but only for progressions and regressions. (It is not clear how far he recognises that this is a limitation.) Thus, finally, we find that the definition we want is the following:—
A series is said to be “closed” (abgeschlossen) when every progression or regression contained in the series has a limit in the series.
We then have the further definition:—
A series is “perfect” when it is condensed in itself and closed, i.e. when every term is the limit of a progression or regression, and every progression or regression contained in the series has a limit in the series.
In seeking a definition of continuity, what Cantor has in mind is the search for a definition which shall apply to the series of real numbers and to any series similar to that, but to no others. For this purpose we have to add a further property. Among the real numbers some are rational, some are irrational; although the number of irrationals is greater than the number of rationals, yet there are rationals between any two real numbers, however [page 104] little the two may differ. The number of rationals, as we saw, is ℵ0. This gives a further property which suffices to characterise continuity completely, namely, the property of containing a class of ℵ0 members in such a way that some of this class occur between any two terms of our series, however near together. This property, added to perfection, suffices to define a class of series which are all similar and are in fact a serial number. This class Cantor defines as that of continuous series.
We may slightly simplify his definition. To begin with, we say:
A “median class” of a series is a sub-class of the field such that members of it are to be found between any two terms of the series.
Thus the rationals are a median class in the series of real numbers. It is obvious that there cannot be median classes except in compact series.
We then find that Cantor’s definition is equivalent to the following:—
A series is “continuous” when (1) it is Dedekindian, (2) it contains a median class having ℵ0 terms.
To avoid confusion, we shall speak of this kind as “Cantorian continuity.” It will be seen that it implies Dedekindian continuity, but the converse is not the case. All series having Cantorian continuity are similar, but not all series having Dedekindian continuity.
The notions of limit and continuity which we have been defining must not be confounded with the notions of the limit of a function for approaches to a given argument, or the continuity of a function in the neighbourhood of a given argument. These are different notions, very important, but derivative from the above and more complicated. The continuity of motion (if motion is continuous) is an instance of the continuity of a function; on the other hand, the continuity of space and time (if they are continuous) is an instance of the continuity of series, or (to speak more cautiously) of a kind of continuity which can, by sufficient mathematical [page 105] manipulation, be reduced to the continuity of series. In view of the fundamental importance of motion in applied mathematics, as well as for other reasons, it will be well to deal briefly with the notions of limits and continuity as applied to functions; but this subject will be best reserved for a separate chapter.
The definitions of continuity which we have been considering, namely, those of Dedekind and Cantor, do not correspond very closely to the vague idea which is associated with the word in the mind of the man in the street or the philosopher. They conceive continuity rather as absence of separateness, the sort of general obliteration of distinctions which characterises a thick fog. A fog gives an impression of vastness without definite multiplicity or division. It is this sort of thing that a metaphysician means by “continuity,” declaring it, very truly, to be characteristic of his mental life and of that of children and animals.
The general idea vaguely indicated by the word “continuity” when so employed, or by the word “flux,” is one which is certainly quite different from that which we have been defining. Take, for example, the series of real numbers. Each is what it is, quite definitely and uncompromisingly; it does not pass over by imperceptible degrees into another; it is a hard, separate unit, and its distance from every other unit is finite, though it can be made less than any given finite amount assigned in advance. The question of the relation between the kind of continuity existing among the real numbers and the kind exhibited, e.g. by what we see at a given time, is a difficult and intricate one. It is not to be maintained that the two kinds are simply identical, but it may, I think, be very well maintained that the mathematical conception which we have been considering in this chapter gives the abstract logical scheme to which it must be possible to bring empirical material by suitable manipulation, if that material is to be called “continuous” in any precisely definable sense. It would be quite impossible [page 106] to justify this thesis within the limits of the present volume. The reader who is interested may read an attempt to justify it as regards time in particular by the present author in the Monist for 1914–5, as well as in parts of Our Knowledge of the External World. With these indications, we must leave this problem, interesting as it is, in order to return to topics more closely connected with mathematics.