Volume 1 / Part 5 — *204. ELEMENTARY PROPERTIES OF SERIES.
Reading edition
*204. ELEMENTARY PROPERTIES OF SERIES.
Summary of *204.
In this number we give the definition and a few of the simpler properties of series. Most of the propositions of this number result immediately from those of *200, *201, and *202. Our definition is
We have
*204·16.
either of which might have been taken as the definition.
After a few propositions giving other possible forms of the definition of series, we proceed to a set of propositions which follow immediately from those of *200, *201, and *202. Such are
*204·2.
*204·21.
*204·24.
*204·25.
Another important proposition on couples is
*204·272.
so that couples are the only series having unit classes for their domains or converse domains.
We then proceed to a set of propositions on . We have
*204·33.
Also, if is a one-one and (*204·34·35).
We then have some propositions (*204·4—·44) on relations with limited fields. The most important of these are
*204·4.
*204·41.
This proposition is important, since it shows that any series contained in a given series is wholly determined when its field is given.
We have next a number of propositions (*204·45—·59) applying relation-arithmetic to series. The first set of these (*204·45—·483) are concerned with the proof that if a "cut" is made in a series, the series is the sum of the two parts into which the cut divides it, where the sum is taken in the sense of *160 or *161, according as one part of the cut does not or does consist of a single term. Most of these propositions do not require the full hypothesis that is a series, but only some part of it. Thus we have for instance
*204·46.
with a similar proposition for and (*204·461).
We next prove that if , are mutually exclusive series, their sum is a series, and vice versa (*204·5); that if is a series to which does not belong, and are series, and vice versa (*204·51); that if is a series of mutually exclusive series, its sum is a series (*204·52); that if , are series, so is (*204·55); that if is a series of series, is contained in diversity and is transitive (*204·561), while if is also well-ordered, i.e. such that every existent sub-class of has a first term, then is a series (*204·57); and that if and are series, and is well-ordered, then and are series (*204·59). These propositions are essential to ordinal arithmetic, but they will not be referred to again until we reach that stage (Sections D and E of this Part).
We have next a collection of propositions (*204·6—·65) on for various values of , and finally three propositions on . Two of these are much used, namely
*204·7.
*204·71.
*204·01.
*204·1.
*204·11.
*204·12.
*204·121.
*204·13.
Dem.
*204·14.
*204·15.
Dem.
*204·151.
*204·16.
We have also For, by *200·37, since , it follows that
A relation such as , where , satisfies , but not . On the other hand, satisfies , but not .
*204·2.
*204·21.
*204·22.
*204·23.
*204·24.
*204·25.
*204·26.
The three following propositions deal with couples. Couples often require special treatment, owing to the fact that, if is a couple, , so that , whereas in any other case, if is a series, . Hence the following propositions are often required.
*204·27.
Dem.
*204·271.
Dem.
*204·272.
*204·3.
*204·32.
Dem.
*204·33.
Dem.
The three following propositions only require , but are required for application to series, and are therefore convenient in the form here given.
*204·331.
*204·34.
*204·35.
This proposition shows that the series of segments which have upper limits is like the original series, for a segment whose upper limit is is , and the series of such segments is .
The following propositions (*204·4—·44) are concerned with relations with limited fields.
*204·4.
*204·41.
In virtue of the above two propositions, the series contained in a given series are the relations resulting from limitations of the field; the process of limiting the field is merely the process of selecting a part of the original series without changing the order.
*204·42.
*204·421.
*204·43.
Dem.
*204·44.
The following propositions (*204·45—*204·483) are concerned with the division of a series into two parts, one of which wholly precedes the other. The case where one of the parts consists of a single term requires special treatment, and so does the case where both parts consist of single terms, i.e. where the series is a couple.
*204·45.
Dem.
*204·46.
Dem.
*204·461.
*204·462.
*204·463.
Dem.
*204·47.
Dem.
*204·48.
Dem.
*204·481.
*204·482.
Dem.
*204·483.
The following propositions are concerned with the application of relation-arithmetic to series.
*204·5.
*204·51.
*204·52.
Dem.
*204·53.
Dem.
*204·54.
Dem.
*204·55.
Dem.
*204·551.
Dem.
*204·56.
Dem.
*204·561.
Dem.
In order to prove that is connected, we require a further hypothesis, namely that is well-ordered, i.e. that every class contained in and not null has a first term.
*204·562.
Dem.
*204·57.
*204·58.
Dem.
*204·581.
*204·59.
Dem.
The two following propositions are lemmas for *204·62.
*204·6.
Dem.
*204·61.
Dem.
*204·62.
Dem.
*204·63.
Dem.
*204·64.
Dem.
The following proposition is used in *234·101.
*204·65.
Dem.
*204·7.
On this proposition, compare the remarks preceding *201·6.
*204·71.
*204·72.
Dem.
The above proposition is used in *274·23.