Volume 1 / Part 4 — *182 ON SEPARATED RELATIONS.
Reading edition
*182 ON SEPARATED RELATIONS.
Summary of *182.
In this number, we have to consider, as a preliminary to the addition of the relation-numbers of a field, the properties of the relation , which is defined as follows. If is any function of two arguments in the sense of *38, we put . Thus , i.e. . Hence is the relation of to when . Thus the symbol is only significant when is a relation of relations; when this is the case, is the relation which results when, for every which is a member of , every member of is replaced by . The result is a , whose arithmetical properties serve to define the arithmetical properties of the sum of the relation-numbers of members of the next number, we shall put We shall put later and we shall find Thus we might have dispensed with as a fundamental notion, using instead, and putting But this course is on the whole less convenient than that adopted in *172 and *173.
The notation is thus required in connection with ordinal addition, where it is almost indispensable. It has besides certain minor uses. The object of the notation is to enable us to exhibit as a function of an expression of the form , where is any descriptive double function which exists for all possible pairs of arguments. Thus for example is a function of , but the notations hitherto introduced do not enable us to exhibit it in the form . Hence if we wish (say) to deal with the class we cannot write it in the form unless we introduce a new notation. We put whence . We introduce the notation generally for all descriptive double functions which exist for all possible pairs of arguments. Thus "" in this number corresponds to "" in *38.
In the present number, we shall begin by a few propositions illustrating possible uses of the notation . Thus for example if is a class of relations, we have hitherto had no simple notation for expressing the class of their squares. But since , the class of the squares of 's is . The notation is, however, introduced chiefly in order to be applied to and . We therefore proceed almost at once to propositions on , and especially on . We have
*182·16·162.
*182·2.
*182·21.
We next prove (*182·27) that if , then has double likeness to , the double correlator being with its converse domain limited to (*182·26). We then prove (*182·33) that if is a double correlator of with , then (with its converse domain limited) is a double correlator of and , whence we deduce
*182·34.
We next proceed to prove
*182·42.
The proof of this is as follows: In virtue of *182·21 and the associative law for , we have Now (*182·413), and (*172·51).
Hence our proposition results. Hence we arrive at
*182·44.
Finally we have some propositions showing how the notation can be applied in cardinals. It is then applied to , instead of, as above, to . We have (*182·5·51·52) . Thus the notation of the present number might have been employed in dealing with cardinal addition (*112) instead of the notation . The general notation was, however, required for other purposes (cf. *85) and could not have been dispensed with.
In *183 we shall put and by *182·52 we have It will be seen that these formulae have the usual kind of analogy.
*182·01.
*182·02.
*182·021.
*182·022.
*182·023.
*182·03.
Thus if is a class of relations, the class of their squares is .
*182·031.
*182·032.
*182·033.
*182·04.
Observe that in , we first take , and then put a circumflex over it. If we first took , we could not then place two commas under it, because is a relation, not a double descriptive function, and two commas can only significantly be placed under a double descriptive function.
*182·05.
The relation for the sake of which the above notation is chiefly introduced is , where is a relation of relations. If relates and , then relates and . This is stated in the following proposition:
*182·1.
*182·11.
*182·12.
*182·13.
*182·14.
Dem.
*182·15.
Dem.
*182·16.
*182·161.
Dem.
*182·162.
*182·17.
Dem.
*182·18.
Dem.
*182·19.
Dem.
*182·2.
*182·21.
The following propositions lead up to *182·26·27.
*182·22.
*182·23.
*182·24.
*182·25.
Dem.
*182·26.
Dem.
*182·27.
The following propositions lead up to *182·33·34.
*182·3.
Dem.
*182·31.
Dem.
*182·32.
Dem.
*182·33.
Dem.
*182·34.
The converse of the above proposition is false. For example, if , we shall have , by *182·16·27, but we shall not have unless , as appears from *182·16 and *164·23.
*182·411·412 are lemmas for *182·413. All the following propositions lead up to *182·42, which leads to *182·44.
*182·411.
Dem.
*182·412.
Dem.
*182·413.
*182·414.
Dem.
*182·415.
Dem.
The purpose of the above proposition is to enable us to apply *174·221·231 to , as is done in *182·42·43·431 below.
*182·42.
Dem.
*182·43.
Dem.
*182·431.
*182·44.
*182·45.
The following propositions are concerned with cardinals. They show how to express the propositions and definitions of *112 in the notation of this number, and they thereby illustrate the analogy of cardinal and ordinal addition.
*182·5.
*182·51.
*182·52.
*182·53.
Dem.
*182·54.