Volume 1 / Part 5 — *213. SECTIONAL RELATIONS.
Reading edition
*213. SECTIONAL RELATIONS.
Summary of *213.
If is a section of , is called a sectional relation of ; and if is a segment of , is called a segmental relation of . If is serial, sectional relations may be arranged in a series by the relation of inclusion (*213·153). That is, if we call the series of sectional relations , we shall so define as to secure that if is serial, The natural definition to take would be But this has the disadvantage that if , Thus does not imply ; and when is serial, is not serial, because . In order to obviate this inconvenience, we confine ourselves to sections which are not null, putting With the above definition, we have (*213·151·152), if , and .
The relation is very useful in dealing with well-ordered series; in this case, we have (as will be shown later) It will be seen that, if , whenever exists, (*213·158); and whenever exists, (*213·155).
We have, if , Hence (*213·246·242).
If is serial, the sectional relations of are all relations such that by adding something to them they become , i.e. they are Hence their relation-numbers are those that can be made equal to that of by being added to. This fact is important in connection with the theory of greater and less among relation-numbers.
The propositions of this number are rendered complicated by the necessity of taking account of the possibility of a section being a unit class. This necessitates a good many propositions which are merely lemmas; but in the end the complications mostly disappear.
We begin with propositions on the field, etc., of . We have
*213·141.
*213·142.
*213·16.
*213·161.
*213·162.
We then prove:
*213·17.
If is finite, it follows from the above that is not similar to ; but if is infinite and has a beginning and is well-ordered, we find
*213·172.
We then have a set of propositions (*213·2—·251) chiefly concerned with the sections of , where . Besides those already mentioned, the following are important:
*213·24.
*213·243.
*213·25.
Our next set (*213·3—·32) is concerned with and . We have
*213·3.
*213·32.
We then have three propositions (*213·4·41·42) showing that a sectional relation of is one which becomes by being added to. We proceed to a set of propositions (*213·5—·58) on , and , leading to
*213·57.
*213·58.
*213·01.
*213·1.
*213·11.
*213·12.
Dem.
*213·121.
Dem.
*213·122.
*213·123.
*213·124.
*213·125.
Dem.
The hypothesis , in the above proposition, restricts more than is necessary for the truth of the conclusion. What we really require is , i.e. . This holds if either (1) the field of does not consist of a single family, or (2) there is a member of which does not have the relation to itself. Thus the only case excluded is that of a single cyclic family. The hypothesis may be substituted for in most of the subsequent propositions of this number in which occurs in the hypothesis. We have, however, preferred the hypothesis , as it gives a more immediate application to the case of , which is the case in which the propositions of the present number are important.
*213·126.
Dem.
*213·13.
Dem.
*213·131.
*213·132.
*213·133.
*213·134.
*213·14.
*213·141.
*213·142.
*213·143.
*213·144.
*213·145.
*213·146.
Dem.
*213·15.
Dem.
*213·151.
Dem.
*213·152.
*213·153.
*213·154.
*213·155.
Dem.
*213·156.
*213·157.
*213·158.
Dem.
*213·16.
Dem.
*213·161.
Dem.
*213·162.
Dem.
*213·163.
Dem.
*213·164.
*213·17.
Dem.
*213·171.
Dem.
*213·172.
*213·18.
Dem.
*213·2.
Dem.
*213·21.
Dem.
*213·22.
*213·23.
*213·24.
Dem.
*213·241.
Dem.
*213·242.
Dem.
*213·243.
Dem.
*213·244.
Dem.
*213·245.
Dem.
*213·246.
*213·247.
*213·25.
Dem.
*213·251.
Dem.
*213·3.
Dem.
*213·301.
*213·302.
Dem.
*213·31.
Dem.
*213·32.
*213·4.
Dem.
*213·41.
*213·42.
*213·5.
Dem.
*213·51.
Dem.
*213·52.
Dem.
*213·53.
Dem.
*213·531.
Dem.
*213·54.
Dem.
*213·541.
Dem.
*213·55.
Dem.
As in *213·54,
*213·56.
*213·561.
Dem.
*213·57.
Dem.
*213·58.
Dem.