Volume 1 / Part 2 — *96. ON THE POSTERITY OF A TERM.
Reading edition
*96. ON THE POSTERITY OF A TERM.
Summary of *96.
By the "posterity" of a term with respect to a relation we mean the class . In the present number, we shall be chiefly concerned with the relation (, i.e. the relation confined to the posterity of . We shall also be concerned with ( and (, which, as is proved in *96·13, are respectively
The most interesting case is when . In this case, is in general shaped like a , with at the tip of the tail; that is, may be divided into two parts, the first an open series, the second a closed series. If is the junction of the two, we shall have
We have also, when ,
It thus appears that is divided into two parts, the first consisting of those terms for which , the second of those for which . The first wholly precedes the second; the first exists if , the second if . Every term in has one and only one immediate predecessor, except the term (if it exists) at the junction of the tail and circle of the ; this term has just two immediate predecessors, one in the tail and one in the circle. But if either the tail or the circle is null, then every term in has only one immediate predecessor, and therefore (these definitions being only to apply within *96). Then is the open part of the series , and is the circular part. The open part wholly precedes the circular part, provided ; i.e.
If and both exist, has a last term, say . The successor of this term, , is the only term in which has two immediate predecessors in , namely and .
The most important applications of the propositions of the present number are in the theory of finite and infinite, both cardinal and ordinal. When is many-one, then if exists, or, more generally, if has a last term, is a finite class, i.e. what we shall call a "" (cf. *120). That is, we have
If exists, but has no last term, is a progression (cf. *122) when its terms are arranged in the order generated by . That is, giving to and the meanings given by Cantor (cf. *123 and *263), and using "" for the class of one-one relations which generate progressions, we have
Another very important proposition in the proof of which the present number is useful is *121·47, which proves that if is either one-many or many-one, and and are any two terms whatever, then (which we call the "interval" from to ) is always a finite class. The proof that progressions are well-ordered series depends upon the propositions of this number, since it uses *122·23, which depends upon *96·52.
The present number begins with a series of propositions (ending with *96·16) on and , both in general and when . We then proceed to a few propositions (*96·2-·25) on ( when ; with the exception of *96·24, these propositions are all used in the cardinal theory of finite and infinite. They are, however, less important than the subsequent propositions, which are concerned with when .
If is a many-one relation, and x is a member of , the relation in general arranges (i.e. the posterity of ) in a figure such as is here given. The relation holds between each dot and the next, starting from , and travelling round the circle in the sense indicated by the arrow. The dots from to constitute , and the dots in the circle constitute . is the last term of , i.e. ; is , and is , or, what comes to the same thing, . is the only term which has more than one immediate predecessor in ; always exists if neither nor is null, and conversely, if exists, neither nor is null. The proof of these propositions is long; the following are useful stages in the proof.
If , the whole posterity of is itself (*96·33); if and , and constitute the whole posterity of (*96·331), and so on. The successors of members of belong to (*96·341), and the predecessors of members of , if they belong to , belong to (*96·351). (It should be observed that, since is only assumed to be many-one, not one-one, every member of may have any number of predecessors which do not belong to . We have a series of propositions, beginning with *96·4, which deal with the hypothesis . We prove (*96·42) that if and , then , i.e. belongs to . We prove (*96·431) that wholly precedes ; that ( and ( are both one-one (*96·45), so that if , one of and must belong to and the other to (*96·441). Hence it follows (*96·453) that if either (in which case or ( (in which case , then ( is a one-one relation. (This proposition is used twice in the cardinal theory of finite and infinite, namely in *121·43 and *122·17.) Hence we arrive at the proposition (*96·47) that if two different members and of both immediately precede a term , then one of and (say ) is the last term of , is its immediate successor and is the immediate predecessor of in , i.e. we have Thus , , are unique if they exist. We prove next (*96·475) that , , exist when, and only when, neither nor is null.
It follows from the above propositions that if is one-one, either or must be null (*96·491), i.e. the posterity of a term is either an open series or a cycle, and cannot have the -shape.
*96·01.
*96·02.
*96·1.
*96·101.
*96·102.
*96·103.
Dem.
*96·104.
Dem.
*96·11.
Dem.
*96·111.
Dem.
*96·112.
Dem.
*96·121.
*96·122.
*96·13.
*96·131.
*96·14.
*96·141.
Dem.
*96·142.
*96·143.
Dem.
*96·144.
Dem.
*96·15.
Dem.
*96·151.
Dem.
*96·152.
*96·153.
*96·154.
*96·155.
Dem.
*96·156.
Dem.
*96·157.
*96·158.
Dem.
*96·159.
*96·16.
Dem.
*96·2.
Dem.
*96·21.
Dem.
*96·22.
Dem.
*96·23.
Dem.
*96·24.
Dem.
*96·25.
Dem.
The following propositions lead up to *96·32, i.e. which is a proposition used in the following number (*97).
*96·3·301·302·303 are also frequently used elsewhere.
*96·3.
*96·301.
*96·302.
*96·303.
*95·31.
*96·311.
*96·32.
Dem.
*96·33.
Dem.
*96·331.
Dem.
This process of proof can obviously be extended to any finite cycle of terms.
*96·34.
Dem.
*96·341.
Dem.
*96·342.
*96·35.
*96·351.
Dem.
*96·352.
The following propositions are lemmas for *96·45·47.
*96·4.
Dem.
*96·401.
Dem.
*96·402.
Dem.
*96·403.
Dem.
*96·41.
Dem.
*96·42.
Dem.
*96·421.
Dem.
*96·431.
Dem.
*96·432.
Dem.
*96·44.
*96·441.
Dem.
*96·442.
The following proposition (*96·45) is important.
*96·45.
*96·451.
*96·452.
Dem.
*96·453.
Dem.
*96·46.
Dem.
*96·461.
Dem.
*96·462.
Dem.
The above proposition, since it exhibits , , as functions of and , shows that there is at most one in having more than one immediate predecessor, and that this one has exactly one immediate predecessor in and one in . (These results require *96·441, in addition to *96·462.) Thus we arrive at the following proposition:
*96·47.
We still have to prove or, what comes to the same thing because of *96·441, This is effected in the following propositions.
*96·472.
Dem.
*96·473.
*96·474.
Dem.
*96·475.
This proposition and *96·45·47 embody the main results of this number.
*96·48.
Dem.
In the above proposition we write "" rather than "," because the latter implies the existence of .
*96·49.
Dem.
*96·491.
Dem.
*96·492.
Dem.
The above proposition is used in *122·52.
The following propositions, not being needed in the sequel, are merely stated:
*96·5.
Dem.
*96·501.
*96·502.
Dem.
*96·51.
Dem.
*96·52.
Dem.
The proposition is used in *122·23.