Volume 1 / Part 5 — *211. ON SECTIONS AND SEGMENTS.
Reading edition
*211. ON SECTIONS AND SEGMENTS.
Summary of *211.
The theory of the modes of separation of a series into two classes, one of which wholly precedes the other, and which together make up the whole series, is of fundamental importance. When one out of a pair of such classes is given, the other is the rest of the series; we may therefore, for most purposes, confine our attention to that one of the two classes which comes first in the serial order. Any class which can be the first of such a pair we shall call a section of our series. If is the series, we shall denote the class of its sections by "." If is a section of , we shall call (which is the second class of our pair) the complement of . The class of complements of sections is which is identical with (*211·75).
In order that a class may be a section of , it is necessary and sufficient that it should be contained in and should contain all its own predecessors; thus we put We have also, by *90·23,
Among sections, a specially important class consists of classes which are composed of all the predecessors of some class, i.e. classes of the form , i.e. classes which are members of . Whenever is transitive, ; hence is a section according to the above definition. When is a series, the complement of (when exists and is contained in ) is
The members of are called segments of the series generated by P. In a series in which every sub-class has a maximum or a sequent, (*211·38), i.e. the predecessors of a class are always the predecessors of a single term, namely the maximum of the class if it exists, or the sequent if no maximum exists. But if there are classes which have neither a maximum nor a sequent, the predecessors of such classes are not coextensive with the predecessors of any single term. Thus in general the series of segments will be larger than the original series. For example, if our original series is of the type of the series of rationals in order of magnitude, the series of segments is of the type of the series of real numbers, i.e. the type of the continuum.
Among segments, a specially important class consists of those which have no maximum. In this case, if is such a segment, we have ; and since (provided is transitive) we also have, for all segments, , the segments having no maximum are those for which , i.e. they are the class . In compact series, all segments belong to this latter class, but in general only those segments belong to it which correspond to a "Häufungsstelle." In all cases in which the existence of a limit is not known, the segment fulfils the functions of a limit; that is to say, in those places in the series where a limit might be expected, we have a segment having no limit or maximum, which takes the same place in the series of segments as would be taken by the limit in the original series if the limit existed. Segments having no limit or maximum are limiting points in the series of segments, and every class of segments which has no maximum in the series of segments has a limit in that series.
We have thus three classes to deal with, namely Of these the second is contained in the first when is transitive (*211·15), and the third is contained in the first and second (*211·14). The second consists of those members of the first which have either a sequent or no maximum (*211·32); the third consists of those members of the first which have no maximum (*211·41). If every member of the third class has a limit, i.e. if then every class has either a sequent or a maximum, i.e. the series is Dedekindian; and the converse also holds (*211·47).
When is connected, of any two sections one must be contained in the other (*211·6). Moreover, if is contained in any one of the three classes , , , then is a member of that class (*211·63·64·65). Hence the propositions of *210 become available. It is thus that the existence of limits in series of segments or sections is proved: the maximum or upper limit of any class consisting of segments or sections is , and the minimum or lower limit is the sum of the segments that are contained in every .
We begin, in this number, with elementary properties of . The sections of are the segments of (*211·13) and the sections of (*211·17). We have
*211·26.
We then proceed to the elementary properties of segments, i.e. of (*211·3—·38). We have
*211·3.
*211·301.
*211·302.
*211·351.
We then proceed to elementary properties of segments having no maximum, i.e. of (*211·4—·47). We have
*211·42.
*211·44.
*211·451.
Our next set of propositions (*211·5—·553) is concerned with compact series, i.e. with the hypothesis . We have
*211·51.
*211·551.
I.e. a series is compact when, and only when, no class has both a maximum and a sequent.
We come next to the application of the propositions of *210 (*211·56—·692). These propositions proceed from
*211·56.
(Here "" may be substituted in the hypothesis: cf. *211·561.) The propositions of this set, which are very important, have been already mentioned.
Our next set of propositions (*211·7—·762) are concerned with the complements of sections and segments. Some of these propositions have been already mentioned; others of importance are:
*211·7.
*211·703.
*211·726.
*211·727.
*211·728.
The remaining propositions are mainly occupied with relation-arithmetic. The most important of them is
*211·82.
That is, given any series contained in , if something can be added to make it into , its field is a section of , and vice versa.
*211·01.
*211·1.
*211·11.
*211·12.
Dem.
*211·13.
Dem.
In virtue of the above proposition, the properties of can be deduced from those of or by substituting for .
*211·131.
Dem.
*211·132.
Dem.
*211·133.
Dem.
*211·14.
Dem.
*211·15.
Dem.
*211·16.
Dem.
*211·17.
The following propositions are useful in dealing with sectional relations, i.e. relations of the form , where . Unit sections often need special treatment, owing to the fact that for them we do not have .
*211·18.
Dem.
*211·181.
Dem.
*211·182.
*211·2.
Dem.
*211·21.
Dem.
*211·22.
Dem.
*211·23.
Dem.
*211·24.
Dem.
*211·26.
Dem.
*211·27.
Dem.
*211·271.
Dem.
*211·272.
Dem.
*211·28.
Dem.
*211·281.
Dem.
*211·282.
*211·283.
Dem.
The following propositions are concerned with . This is to be compared with two other classes, namely and . The members of which do not belong to are those which have a maximum but no sequent, i.e. (if is a series), those classes which consist of a term together with all its predecessors, where x has no immediate successor. In series in which every term except the last has an immediate successor, will be the only member of , if the series has a last term; if the series has no last term, .
The members of which are not members of are those that have no sequent, i.e. those that have no upper limit (for a member of which has no sequent has also no maximum). These are the members of corresponding to a "gap," i.e. to a Dedekind section in which neither the earlier terms have a maximum nor the later terms a minimum. Hence in a Dedekindian series, ; and conversely, if , the series is Dedekindian. These properties of are proved in the following propositions.
*211·3.
*211·301.
*211·302.
Dem.
*211·31.
*211·311.
*211·312.
Dem.
*211·313.
Dem.
*211·314.
Dem.
The above proposition and the two following propositions enable us in certain cases to prove propositions concerning the relations of and without assuming that is transitive. An example of the use of these propositions occurs in *211·754, where the hypothesis assumes . If we used *211·31 and its consequences instead of *211·314 and its consequences, the hypothesis of *211·754 would have to assume .
*211·315.
Dem.
*211·316.
*211·317.
Dem.
*211·32.
*211·321.
*211·33.
Dem.
*211·34.
Dem.
*211·35.
Dem.
*211·351.
Dem.
*211·36.
Dem.
*211·361.
Dem.
*211·371.
*211·372.
Dem.
*211·38.
Dem.
The following propositions are concerned with , i.e. with those sections of which have no maximum. If is compact (i.e. if ), . If is also a Dedekindian series, . This is the mark of Dedekindian continuity, since it states that, if has no maximum, there is an for which , and this is the upper limit of has no maximum, so that the series is compact.
*211·4.
Dem.
*211·41.
Dem.
*211·411.
Dem.
*211·42.
Dem.
*211·43.
Dem.
*211·431.
*211·44.
*211·45.
Dem.
*211·451.
Dem.
*211·452.
*211·46.
Dem.
*211·47.
Dem.
The following propositions are concerned with certain consequences of the hypothesis . This hypothesis is important because it is the defining characteristic of compact series.
*211·5.
Dem.
*211·51.
Thus in compact series there is no distinction between the two sorts of segments.
*211·52.
Dem.
*211·53.
Dem.
The condition is the Dedekindian definition of continuity. In virtue of the above proposition, this is equivalent, in a series, to compactness combined with Dedekind's axiom, namely
*211*54.
Dem.
*211·541.
Dem.
*211·55.
*211·551.
*211·552.
*211·553.
The following propositions are concerned in showing that , , and all verify the hypotheses of *210, if taken as the of that number.
*211·56.
Dem.
*211·561.
*211·562.
*211·6.
*211·61.
*211·62.
In the hypothesis of *211·61, it is necessary that should be transitive as well as connected. Take, for example, Then is connected, but not transitive; also we have Hence . Thus connection is not sufficient in the hypothesis of *211·61.
*211·63.
Dem.
This proposition shows that verifies the hypothesis of *210·251, with the exception of , which requires .
*211·631.
Dem.
*211·632.
Dem.
*211·633.
This proposition shows that verifies the hypothesis of *210·252, with the exception of , which requires .
*211·64.
Dem.
*211·65.
Dem.
*211·66.
Dem.
*211·661.
Dem.
The following propositions sum up the above results in relation to the hypotheses of *210. The relation with its field limited to sections or segments, which occurs in the following propositions, is important, and will be considered at length in the following number.
*211·67.
*211·671.
*211·68.
*211·681.
*211·69.
*211·691.
*211·692.
The following propositions are concerned with the relations of sections and segments of to sections and segments of . When , , and vice versa. Also, if is connected, the maximum of (if any) is the precedent with respect to (i.e. the sequent with respect to of , and the sequent of (if any) is the minimum with respect to (i.e. the maximum with respect to ) of . Hence the relations to be proved follow easily.
*211·7.
Dem.
*211·701.
Dem.
*211·702.
*211·703.
*211·71.
Dem.
If is a section of , we shall call the complement of . By the above proposition, if is a section of having a maximum, its complement is a section of which is a member of .
*211·711.
*211·712.
Dem.
*211·713.
Dem.
*211·714.
Dem.
The above hypothesis is not sufficient to secure , as may be seen by putting We then have . Thus . It will be seen that , so that it is useless to add to the hypothesis of *211·714. A sufficient addition is , as is proved in the following proposition.
*211·715.
Dem.
*211·72.
*211·721.
Dem.
*211·722.
Dem.
We have always, if , The converse inclusion does not always hold, as appears (on writing in place of ) from the note to *211·714. To secure the converse implication, it is sufficient to assume or or .
*211·723.
Dem.
*211·724.
Dem.
*211·725.
*211·726.
Dem.
*211·727.
*211·728.
Dem.
*211·729.
*211·73.
Dem.
*211·74.
Dem.
The following propositions sum up our previous results.
*211·75.
*211·751.
Dem.
In the above proposition, "" is necessary in order that may be contained in , and "" is necessary in order that "" may imply "." Hence the full hypothesis "" becomes necessary.
*211·752.
Dem.
*211·753.
*211·754.
Dem.
*211·755.
*211·756.
*211·757.
*211·76.
Dem.
*211·761.
*211·762.
Dem.
*211·8.
Dem.
The above proposition is used in *232·352 and *234·242.
The following propositions lead up to *211·82, which is used in *213·4. *211·83·841·9 are also used in *213.
*211·81.
Dem.
*211·811.
Dem.
*211·812.
Dem.
*211·82.
*211·83.
Dem.
*211·84.
Dem.
*211·841.
*211·9.
Dem.