Introduction to Mathematical Philosophy

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 is the limit (in the order of magnitude) of the cardinal numbers 1, 2, 3, ... , ..., although the numerical difference between and a finite cardinal is constant and infinite: from a quantitative point of view, finite numbers get no nearer to as they grow larger. What makes 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. [Pg 97] 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 with respect to a relation are those members of and the field of (if any) to which no member of has the relation . The "maxima" with respect to are the minima with respect to the converse of . The "sequents" of a class with respect to a relation are the minima of the "successors" of , and the "successors" of are those members of the field of to which every member of the common part of and the field of has the relation . The "precedents" with respect to are the sequents with respect to the converse of . The "upper limits" of with respect to are the sequents provided has no maximum; but if has a maximum, it has no upper limits. The "lower limits" with respect to are the upper limits with respect to the converse of . Whenever 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 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 , i.e. its limits 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 defined by a class " as all those terms that have the relation to some one or more of the members of . This will be a segment in the sense defined [Pg 98] in Chapter VII.; indeed, every segment in the sense there defined is the segment defined by some class . If is serial, the segment defined by consists of all the terms that precede some term or other of . If has a maximum, the segment will be all the predecessors of the maximum. But if has no maximum, every member of precedes some other member of , and the whole of is therefore included in the segment defined by . Take, for example, the class consisting of the fractions i.e. of all fractions of the form for different finite values of . 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 is serial, the "boundary" of a class will be the term (if it exists) whose predecessors are the segment defined by . A "maximum" of is a boundary which is a member of . An "upper limit" of is a boundary which is not a member of . 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 with respect to a series is that term (if it exists) which comes after all the 's, but is such that every earlier term comes before some of the 's. We may define all the "upper limiting-points" of a set of terms as all those that are the upper limits of sets of terms chosen out of . We shall, of course, have to distinguish upper limiting-points from lower limiting-points. If we consider, for example, the series of ordinal numbers: [Pg 99] the upper limiting-points of the field of this series are those that have no immediate predecessors, i.e. The upper limiting-points of the field of this new series will be 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, [Pg 100] 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 [Pg 101] 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). [Pg 102] We come now to the second property by which perfection was