Volume 1 / Part 4 — *164. DOUBLE LIKENESS.
Reading edition
*164. DOUBLE LIKENESS.
Summary of *164.
The subject of this number is of great importance throughout relation-arithmetic and its applications. Double likeness, or double ordinal similarity, is a relation which is to hold between and when (1) and are like, (2) correlated members of the fields of and are like, with a specific given correlator in each case. (It is necessary, in general, to have a given correlator in each case, to avoid the necessity of the multiplicative axiom for selecting among correlators.) This definition can be somewhat simplified by starting from a relation correlating and . If is such a correlator, so that we want to be such that it not only correlates the whole of with the whole of , but also correlates each member of with the corresponding member of , i.e. such that, if is any member of , is the corresponding member of . This requires i.e. writing , in place of , it requires When and , we have by *162·35. Hence double likeness will subsist if there is a relation such that
A relation fulfilling this condition will be called a double correlator of and . Thus two relations and have double likeness when there exists a double correlator of and , i.e. when A double correlator of and is a relation which is a correlator of and and is such that is a correlator of and .
It will be seen that this definition has the usual analogy to the corresponding definition in cardinals (*111·01). The two inverted commas of the cardinal definition are replaced by the semi-colon, and is replaced by , and is replaced by or . The propositions of the present number consist largely of analogues of the propositions of *111, in accordance with the above substitutions.
If it were not for the difficulty of choice among correlators, we could define two relations as having double likeness when they are like relations of like relations, i.e. when, if and are the two relations, they have a correlator such that, if , then . In this case, . Thus we have to consider the relations of the class to the class of double correlators, and we have to consider the relation of the relation "" to the relation of double likeness. The propositions to be proved on this subject in the present number are analogous to the propositions of *111. But at a later stage (*251·61) we shall show that if the field of consists entirely of relations which generate well-ordered series, then the use of the multiplicative axiom ceases to be necessary in identifying double likeness with the relation , the reason being that two well-ordered series can never be correlated in more than one way.
Our definitions are
*164·01.
*164·02.
The principal propositions of this number are
*164·15.
whence
*164*151.
*164*18.
This is usually the most convenient proposition when a double correlation has to be proved.
*164·201·211·221. Double likeness is reflexive, symmetrical and transitive.
*164·31.
(Cf. note to *164·31, below.)
We then have a set of propositions (*164·4 to the end) on the identification of with double likeness by means of the multiplicative axiom. We have
*164·43.
That is to say, given that and are like relations of like mutually exclusive relations, if we can pick out one correlator for each pair of correlated members of and , then the sum of such selected correlators is a double correlator of and . Hence, observing that if is a double correlator of and , (*164·15·16), we arrive at
*164·45.
From *164·43 we deduce also
*164·46.
*164·48.
I.e. in effect, assuming the multiplicative axiom, if two series and ) can each be divided into sets of terms (, being relation-numbers), then the two series are ordinally similar, and the sets in the one case have double similarity with the sets in the other. (Here we have written , in place of the and of the enunciation.)
It is by means of the above propositions that ordinal addition and multiplication are connected, as will appear in *166.
*164·01.
*164·02.
*164·1.
*164·11.
*164·12.
*164·13.
*164·131.
Dem.
*164·14.
The two following propositions are required for proving *164·18.
*164·141.
*164·142.
*164·143.
Dem.
*164·15.
Dem.
*164·151.
*164·16.
Dem.
*164·17.
This proposition states that when and have double likeness, there is a correlator of and which couples like with like relations; i.e. if is the correlator, then, if , and are ordinally similar. The converse of this proposition, namely, that if and have a correlator which couples ordinally similar relations, then and have double likeness, can be proved if the multiplicative axiom is assumed, but not otherwise, except in special cases, such as that of well-ordered series.
The following proposition is used frequently, owing to the fact that, in the cases we are concerned with, double correlators generally have the form , where is some relation for which we have .
*164·18.
Dem.
*164·181.
Dem.
The following propositions are concerned in proving that double likeness is reflexive, symmetrical, and transitive.
*164·2.
Dem.
*164·201.
*164·21.
Dem.
*164·211.
*164·22.
Dem.
*164·221.
*164·23.
Dem.
*164·3.
Dem.
*164·301.
*164·31.
Dem.
This proposition has the merit of reducing the ordinal element in double likeness to a minimum. The proof of is a cardinal problem, and what has to be added for ordinal purposes is merely .
*164·32.
In this proposition, the various 's need not be of the same type. Hence "" is not an immediate consequence of *164·201.
Dem.
*164·33.
Dem.
*164·34.
The following propositions are concerned in showing that, if and are like relations, and the correlator of and is contained in likeness (i.e. correlates relations which have the relation of likeness), a correlator being given for each pair of relations coupled by the correlator of and , then the logical sum of such correlators is a double correlator of and , provided and are relations of mutually exclusive relations. That is, assuming to be the correlator of and , and assuming that whenever , let it be possible to choose one correlator out of the class of correlators , for every which belongs to . That is, assume that it is possible to make a selection from the class of classes of correlators. If is such a selection, then will be a double correlator of and , if , .
The following propositions, down to *164·421, are lemmas for *164·43.
*164·4.
Dem.
*164·41.
Dem.
*164·411.
Dem.
*164·412.
*164·413.
Dem.
*164·414.
*164·42.
*164·421.
The following proposition, besides being used in proving all subsequent propositions of this number (except *164·432·433, which are mere lemmas for *164·44), is used in *251·6, in the theory of ordinal numbers.
*164·43.
Dem.
*164·431.
*164·432.
Dem.
*164·433.
All the remaining propositions of the number are important.
*164·44.
*164·45.
*164·46.
*164·47.
Dem.
*164·48.