Volume 1 / Part 1 — *43. THE RELATIONS OF A RELATIVE PRODUCT TO ITS FACTORS.
Reading edition
*43. THE RELATIONS OF A RELATIVE PRODUCT TO ITS FACTORS.
Summary of *43.
The purpose of the present number is to give certain propositions on the relation which holds between and whenever , or whenever , or whenever , where and are fixed. In virtue of the general definitions of *38, these relations are respectively , , and (. Such relations are of great utility both in cardinal and in ordinal arithmetic; they are also much used in the theory of induction (Part II, Section E). In place of the notation (, which is cumbrous, we adopt the more compact notation . If is a class of relations, will be the class of relations where , will be the class of relations where , and ( will be the class of relations where . These classes of relations are often required in subsequent work.
In virtue of our definitions, we have
*43·112.
The propositions most used in the present number (except such as merely embody definitions) are the following:
*43·302.
*43·411.
*43·421.
The remaining propositions are used seldom, but their uses, when they are used, are important.
*43·01.
At a later stage (in *150) we shall introduce a simpler notation for the special case of . The following propositions are for the most part immediate consequences of the definitions, and proofs are therefore usually omitted.
*43·1.
*43·101.
*43·102.
*43·11.
*43·111.
*43·112.
*43·12.
*43·121.
*43·122.
*43·2.
Dem.
*43·201.
*43·202.
*43·21.
*43·211.
*43·212.
*43·213.
*43·22.
*43·3.
*43·301.
*43·302.
*43·31.
Dem.
*43·311.
*43·312.
*43·34.
*43·4.
*43·401.
*43·41.
*43·411.
*43·42.
Dem.
*43·421.
*43·43.
Dem.
*43·48.
*43·481.
*43·49.
Dem.
*43·491.
*43·5.
*43·51.
Dem.
The above proposition is used in the proof of *74·773.