Volume 1 / Part 5 — *206. SEQUENT POINTS.
Reading edition
*206. SEQUENT POINTS.
Summary of *206.
A "sequent" of a class is a minimum of the terms that come after the whole of ; that is, we put Thus the sequents of are its immediate successors. If has a maximum, the sequents are the immediate successors of the maximum; but if has no maximum, there will be no one term of which is immediately succeeded by a sequent of ; in this case, if has a single sequent, the sequent is the "upper limit" of . Whenever is connected, and therefore whenever is serial, every class has one sequent or none with respect to , by *205·3.
It will be seen that the sequents of are the same as the sequents of , and therefore that depends only upon : if has terms not belonging to , they are irrelevant.
For the immediate predecessors of a class , we put We have , so that propositions about result from those about by merely writing in place of ; they will therefore not be given in what follows.
Among the elementary properties of with which this number begins, the following are the most important:
*206·13.
This merely embodies the definition.
*206·131.
*206·134.
*206·14.
Thus if has a first term, this is the sequent of the null class, or of any other class which has no members in common with .
*206·16.
This follows at once from *205·3. It leads to
*206·161.
Thus if is a connected relation, no class has more than one sequent. This is not in general the case with relations which are not connected, even where the idea of sequents is quite naturally applicable. Take, e.g., the relation of descendent to ancestor, and let be the class of monarchs of England. Then will be such parents of monarchs as were not themselves monarchs.
*206·171.
This proposition states that is a sequent of if the whole of precedes , but every term that precedes either belongs to or precedes some term of . When is a series and has no maximum, we have i.e. the sequent of , if any, is a term whose predecessors are identical with the predecessors of members of . This is the case of a limit (cf. *207).
We have next a set of propositions (*206·211·28) concerned with and . When is transitive and connected, and is an existent class contained in and having a sequent, we shall have That is, the predecessors of the sequent are the members of and the predecessors of members, while the sequent and its successors are the successors of the whole of . The various parts of this statement require various parts of the hypothesis. Thus we have
*206·211.
*206·213.
*206·22.
*206·23.
If is transitive, the value of is unchanged if we add to any set of terms contained in (*206·24); thus in particular, (*206·25). Thus we can fill up any gaps in , and take the whole series up to the end of , without altering the sequent.
We have next a set of propositions (*206·3—*206·38) on the sequent of , i.e. of the segment defined by . If is a series, is the maximum of if has a maximum, the sequent of if has a sequent but no maximum, and non-existent if has neither a maximum nor a sequent (*206·35·331·36).
Our next set of propositions (*206·4—·52) concerns the sequents of unit classes, especially of , and of classes of the form . We have
*206·4.
*206·42.
whence the three following propositions:
*206·43.
*206·45.
*206·46.
From the above propositions it results that, when is a series, any member of is the sequent of the class of its predecessors, is the sequent of if either exists, and the sequent of a class which has a maximum is the immediate successor (if any) of the maximum, i.e.
*206·5.
We then have a set of propositions (*206·53—·57) on the sequent of , i.e. the sequent of the predecessors of the whole of . These propositions are specially useful in connection with "Dedekindian" series, i.e. series in which every class has either a maximum or a sequent (*214). These propositions all require the full hypothesis that is a series. In this case, , i.e. the sequent (if any) of the predecessors of the whole of is the minimum (if any) of . Moreover by definition the maximum of , if any, is the precedent of . Hence has either a minimum or a precedent if has either a sequent or a maximum (*206·54). Moreover the sequent and maximum of are respectively (if they exist) the sequent and maximum of the predecessors of all the successors of the whole of (*206·551). Hence we arrive at the conclusion that the assumption that every class of the form has either a maximum or a sequent is equivalent both to the assumption that every class has either a maximum or a sequent (*206·56) and to the assumption that every class has either a minimum or a precedent (*206·55). It follows that these two latter assumptions are equivalent (*206·57), i.e. that a series is Dedekindian when, and only when, its converse is Dedekindian (*214·14).
We deal next (*206·6—*206·63) with correlations, showing that if two relations are correlated, the sequents of the correlates of any class are the correlates of the sequents, i.e.
*206·61.
We end with a set of propositions (*206·7—·732) showing that the sequent of a class is unchanged if we remove from the class any term other than its maximum (*206·72); that if a class has terms in , and has both a precedent and a sequent, the precedent has the relation to the sequent (*206·73), and that the precedent is not identical with the sequent (*206·732). These propositions are in the nature of lemmas, whose use is chiefly in the theory of stretches (*215).
*206·01.
*206·02.
*206·1.
*206·101.
We shall not enunciate any other propositions on (unless for some special reason), since the above proposition enables them to be immediately deduced from the corresponding propositions on .
*206·11.
Observe that when is not null, , so that the factor on the right is unnecessary; but when , we have , so that the factor becomes relevant. Owing to this factor, the sequents of are , so that if exists, is the sequent of .
*206·12.
*206·13.
*206·131.
*206·132.
*206·133.
*206·134.
Dem.
This formula for is usually more convenient than *206·13·132.
*206·14.
Dem.
*206·141.
Dem.
*206·142.
*206·143.
*206·144.
*206·15.
*206·16.
*206·161.
Thus in a series, or in any connected relation, no class has more than one sequent.
*206·17.
Dem.
The following propositions give simplified formulae for in various special cases.
*206·171.
Dem.
*206·172.
*206·173.
*206·174.
Dem.
The propositions *206·173·174 deal with limits. When a class has no maximum, i.e. when , its sequent (if any) is called its limit. By the above propositions, the limit is a term such that precedes , but every predecessor of precedes some member of (*206·173); it is also a term whose predecessors are identical with the predecessors of (*206·174). The subject of limits will be explicitly treated in *207.
*206·18.
*206·181.
*206·2.
Dem.
*206·21.
*206·211.
Dem.
*206·212.
Dem.
*206·213.
Dem.
*206·22.
*206·23.
Dem.
*206·24.
Dem.
*206·25.
*206·26.
Dem.
*206·27.
Dem.
*206·28.
Dem.
*206·3.
*206·31.
*206·32.
Dem.
In the hypothesis of *206·32, we have both and . So long as is not contained in diversity, these are both necessary. For example, suppose we take Then is transitive and connected, but not contained in diversity. We have Also Thus in this case exists, but does not exist. When is serial, i.e. when is contained in diversity, in addition to being transitive and connected, the existence of involves that of , and therefore the hypothesis , which appears in *206·32, becomes unnecessary.
*206·33.
Dem.
*206·331.
*206·34.
Dem.
*206·35.
Dem.
*206·36.
Dem.
The condition is the definition of what may be called "Dedekindian" series, i.e. series in which, when any division of the field into two parts is made in such a way that the first part wholly precedes the second, then either the first part has a last term or the second part has a first term. (When these alternatives are also mutually exclusive, the series has "Dedekindian continuity.") If is any class, is the segment of defined by . In virtue of the above proposition, every segment of a Dedekindian series has a sequent. The sequent of a class having no maximum is what is commonly called a limit. Thus in a series having Dedekindian continuity (in which segments never have maxima), every segment has a limit.
*206·37.
Dem.
*206·38.
Dem.
*206·4.
Dem.
*206·401.
*206·41.
*206·42.
Dem.
*206·43.
*206·44.
*206·45.
*206·451.
Dem.
*206·46.
Dem.
*206·47.
Dem.
*206·48.
Dem.
*206·5.
Dem.
*206·51.
Dem.
*206·52.
Dem.
*206·53.
Dem.
*206·531.
Dem.
*206·54.
Dem.
*206·55.
*206·551.
Dem.
*206·56.
Dem.
*206·57.
This proposition is important, since it shows that when a serial relation satisfies Dedekind's axiom, so does its converse. Thus if all classes which have no maximum have an upper limit, then all classes which have no minimum have a lower limit, and vice versa.
*206·6.
Dem.
*206·61.
Dem.
*206·62.
*206·63.
*206·7.
Dem.
*206·71.
Dem.
*206·72.
Dem.
*206·73.
Dem.
*206·731.
Dem.
Note that "" is not the same proposition as . The former involves , while the latter does not, in virtue of the conventions as to descriptive symbols explained in *14.
*206·732.
Dem.