Volume 1 / Part 2 — *64. RELATIVE TYPES OF RELATIONS.
Reading edition
*64. RELATIVE TYPES OF RELATIONS.
Summary of *64.
In the present number, we introduce notations defining the type of a relation relatively to the types of its domain and converse domain, when these types are given relatively to some fixed class . If is any relation, it is of the same type as . If and are both of the same type as , is of the same type as , which is of the same type as . The type of we call , and the type of we call , and the type of we call , and the type of we call , and the type of we call . We thus have a means of expressing the type of any relation in terms of the type of , provided the types of the domain and converse domain of are given relatively to .
The most useful propositions of the present number are the following:
*64·16.
*64·201.
*64·231.
Here "" will only be significant if and are homogeneous relations, which is not required by the rest of the proposition. When and are homogeneous relations we have
*64·24.
This proposition is useful in connecting ordinal and cardinal existence-theorems.
*64·312.
*64·5.
This proposition is frequently used. It states that the class of relations whose referents are of the type of members of while its relata are of the type of members of (i.e. the class of all relations contained in ) is the type of and is also the type of .
*64·55.
*64·57.
The propositions of the present number are mostly obvious, though formal proofs are sometimes not very easily found. The use of the propositions of this number occurs chiefly in the first section on relation-arithmetic and in the proofs of existence-theorems in ordinal arithmetic and the theory of ratio.
*64.01.
*64.011.
*64.012.
*64.013.
*64.014.
*64.02.
*64.021.
*64.022.
*64.03.
*64.031.
*64.04.
*64.041.
*64.1.
Dem.
*64.11.
*64.12. )
Dem.
*64·13.
*64·14.
*64·15.
*64·16. )
Dem.
By putting (where and are some index and suffix which have been defined) for and for , the above propositions give results applicable to any of the types defined at the beginning of this number, because of .
*64·2.
*64·201.
Dem.
*64·21. )
Dem.
*64·22.
*64·23.
Dem.
*64·231.
Dem.
*64·24.
This proposition is only significant when and are homogeneous relations.
Dem.
*64·3.
Dem.
*64·31.
*64·311.
*64·312.
*64·313.
*64·32.
Dem.
Similarly the other equivalences are proved.
*64·33.
Dem.
Similarly the other equivalences are proved.
*64·34.
*64·5.
*64·51.
*64·52.
*64·53.
Dem.
This proposition is used in connection with cardinal addition (*110·18).
*64·54.
*64·55.
Dem.
*64·56.
Dem.
*64·57.
*64·6.
Dem.
*64·61.
Dem.
*64·62.
Dem.
*64·63.
Dem.