Volume 1 / Part 2 — *55. ORDINAL COUPLES.
Reading edition
*55. ORDINAL COUPLES.
Summary of *55.
Ordinal couples, which are now to be considered, are much more important, even in cardinal arithmetic, than cardinal couples. Their properties are in part analogous to those of cardinal couples, but in part also to those of unit classes; for they are the smallest existent relations, just as unit classes are the smallest existent classes. The properties which are analogous to those of unit classes do not demand that the two terms of the couple should be distinct, i.e. they hold for as well as for (where ); on the other hand, the properties which are analogous to those of cardinal couples do in general demand that the two terms of the ordinal couple should be distinct.
The notation is cumbrous, and does not readily enable us to exhibit the couple as a descriptive function of for the argument , or vice versa. We therefore introduce a new symbol, "," for the couple. In a couple , we shall call the referent of the couple, and the relatum. In virtue of the definitions in *38, this gives rise to two relations, and ; hence we obtain the notations , , , and so on, which will be much used in the sequel. It should be observed that means the class of ordinal couples in which is referent and a member of is relatum, while or denotes the class of couples having as relatum and a member of as referent; denotes all such classes of couples as , where y is any member of ; and in virtue of *40·7, denotes all ordinal couples of which the referent is a member of , while the relatum is a member of . This is a very important class, which will be used to define the product of two cardinal numbers; for it is evident that the number of members of is the product of the number of members of and the number of members of .
The first few propositions of the present number are immediate consequences of the definition of and the notations introduced in *38. We then proceed to various elementary properties of the relation , of which the most used are the following:
*55·13.
*55·15.
*55·16.
*55·202.
This proposition should be contrasted with *54·22, as giving one reason why ordinal couples are more useful in arithmetic than cardinal couples. In virtue of the above proposition, when two ordinal couples are identical, their referents are identical, and their relata are identical.
We proceed next to various properties of the relations and . These relations play a great part in arithmetic. It will be observed that if two terms have the relation , the referent is a couple whose relatum is the relatum in the relation , i.e. when we have , we have (cf. *55·122). Similar remarks apply to the relation . The class , consisting of all couples whose referent is a member of , while the relatum is , is important. We have
*55·232.
This proposition is frequently useful.
We proceed next (*55·3—·51) to give various properties of which are analogous to the properties of unit classes. Among the more important of these properties are the following:
*55·3.
This is the analogue of *51·31.
*55·34.
This is the analogue of *51·4.
*55·5.
This is the analogue of *54·4.
We then proceed to such properties of ordinal couples as are not analogous to those of unit classes. For connecting the cardinal number 2 with the ordinal number , we have the proposition
*55·54.
This proposition shows that the only asymmetrical relations which have a given cardinal couple for their field are the two corresponding ordinal couples and . We have next a set of propositions on the relative products of couples and other relations, i.e. on , , and . These propositions are very useful in arithmetic. The chief of them is
*55·61.
Finally we have four propositions which belong, by their subject, to *43, but could not be given there, because the proofs make use of ordinal couples.
*55·01.
*55·02.
This definition serves merely for the avoidance of brackets.
*55·1.
*55·11.
*55·12.
*55·121.
*55·122.
*55·123.
*55·13.
Dem.
*55·132.
*55·134.
*55·14.
*55·15.
*55·16.
Dem.
The above proposition is important, and will be frequently used.
*55·161.
Dem.
*55·17.
*55·2.
Dem.
*55·201.
*55·202.
Dem.
The above proposition is important.
*55·21.
*55·22.
*55·221.
*55·222.
Dem.
*55·223.
*55·224.
Dem.
*55·23.
*55·231.
*55·232.
Dem.
*55·233.
The above two propositions are frequently useful in arithmetic.
*55·24.
Dem.
*55·241.
*55·25.
Dem.
*55·251.
This proposition is used in the theory of cardinal multiplication (*113·142).
*55·26.
*55·261.
*55·262.
*55·27.
*55·28.
*55·281.
*55·282.
*55·283.
*55·29.
*55·291.
*55·292.
The following propositions, down to *55·51 inclusive, give properties of ordinal couples which are analogous to the properties of unit classes.
*55·3.
The first half of this proposition is the analogue of *51·2; like that proposition, it gives a means of reducing propositions to the form of inclusions. For the second half, compare *51·31.
*55·31.
This proposition is the analogue of *51·23.
Dem.
*55·32.
Dem.
*55·33.
*55·34.
Dem.
*55·341.
Dem.
*55·35.
Dem.
*55·36.
Dem.
*55·37.
Dem.
The following proposition is the analogue of *51·232.
*55·4.
*55·41.
Dem.
The above proposition is the analogue of *51·234. The following proposition (*55·42) is the analogue of *51·235.
*55·42.
Dem.
*55·43.
This proposition is the analogue of *51·41.
Dem.
*55·431.
Dem.
*55·44.
Dem.
The above proposition is the analogue of *51·43.
*55·5.
Dem.
The above proposition is the analogue of *54·4.
*55·51.
Dem.
In the remainder of the present number, we are concerned with properties of ordinal couples which have no analogues for unit classes.
*55·52.
*55·521.
*55·53.
Dem.
*55·54.
Dem.
*55·57.
*55·571.
*55·572.
*55·573.
*55·58.
*55·581.
*55·582.
*55·583.
The above propositions are frequently useful in arithmetic. Their use arises as follows. Let , , , be classes of which is correlated with by the relation , and with by the relation . Then if , the couple consisting of the correlate of and the correlate of is (, i.e., by the above, , i.e. (. Thus the relation correlates the couples, in and , composed of the correlates of terms in and . The most useful form, in practice, of *55·583, is that given below in *55·61.
*55·6.
*55·61.
*55·62.
Dem.
*55·621.
The four following propositions belong to *43, but are inserted here because the proof uses *55·13.
*55·63.
Dem.
*55·631.
*55·632.
Dem.
*55·64.