Volume 1 / Part 2 — *74. ON ONE-MANY AND MANY-ONE RELATIONS WITH LIMITED FIELDS.
Reading edition
*74. ON ONE-MANY AND MANY-ONE RELATIONS WITH LIMITED FIELDS.
Summary of *74.
The purpose of the present number is to collect together various propositions in which we have such hypotheses as or in which such hypotheses are shown to be deducible from others. Hypotheses of this kind occur very frequently, and it is important to be able to deal with them easily. For the sake of completeness, we shall here repeat propositions previously proved on this subject.
The propositions of this number are mostly of the nature of lemmas, to be used in the theory of selections (Part II, Section D), and in cardinal and ordinal arithmetic. The most useful of them are *74·772·773·774·775. These propositions are concerned with circumstances under which or , with or without some limitation of the converse domain, is a one-one relation. The reason they are important is that the correlators by means of which many of the fundamental theorems of cardinal and ordinal arithmetic are proved are such relations as (with the converse domain limited) for suitable values of and . The above-mentioned propositions are as follows:
*74·772.
The hypothesis of this proposition will be verified if we put, for example, . Thus . This proposition is used in *116·531, which is used in proving one of the formal laws of exponentiation, namely .
*74·773.
This proposition is used in connection with both cardinal and ordinal multiplication and exponentiation. If and correlate with and with , then if we take for the class of all ordinal couples that can be formed of an and a , ( will be the class of all couples that can be formed of a and a . Thus in virtue of the above proposition, if is similar to and is similar to , the class of ordinal couples formed of a and a is similar to the class of ordinal couples formed of an and a . This result is useful because we define the product of the number of members of and the number of members of as the number of ordinal couples formed of an and a .
*74·774.
This proposition is useful when, for example, is .
*74·775.
This is a particular case of *74·773, and has similar uses.
*74·1.
Dem.
*74·11.
*74·12.
*74·13.
*74·131.
*74·14.
*74·141.
*74.15.
*74·151.
*74·16.
*74·161.
*74·17.
*74·171.
*74·2.
Dem.
*74·201.
*74·21.
*74·211.
*74·22.
*74·221.
*74·23.
*74·231.
*74·24.
*74·25.
*74·251.
*74·26.
Dem.
*74·27.
Dem.
*74·271.
*74·3.
Dem.
*74·301.
*74·31.
Dem.
*74·311.
*74·32.
Dem.
*74·4.
Dem.
*74·41.
Dem.
*74·42.
*74·43.
*74·44.
*74·5.
Dem.
*74·51.
Dem.
*74·511.
*74·52.
Dem.
*74·521.
*74·53.
Dem.
*74·531.
*74·6.
Dem.
*74·61.
Dem.
*74·62.
Dem.
*74·63.
*74·631.
*74·632.
*74·7.
Dem.
*74·701.
*74·71.
*74·711.
*74·72.
Dem.
*74·721.
*74·73.
*74·731.
*74·74.
*74·741.
*74·75.
Dem.
*74·751.
*74·76.
*74·761.
*74·77.
Dem.
*74·771.
*74·772 and its immediate successors are of very great use in cardinal and ordinal arithmetic.
*74·772.
*74·773.
Dem.
*74·774.
Dem.
*74·775.
*74·8.
Dem.
*74·801.
*74·81.
Dem.
*74·811.
*74·82.
Dem.
*74·821.
*74·822.
*74·823.
*74·83.
*74·831.
*74·832.
*74·833.
*74·84.
Dem.
*74·841.
*74·842.
*74·843.