Volume 1 / Part 2 — *60. THE SUB-CLASSES OF A GIVEN CLASS.
Reading edition
*60. THE SUB-CLASSES OF A GIVEN CLASS.
Summary of *60.
Our definitions in this number are as follows:
*60·01.
This defines the relation to a class of the class of all its sub-classes.
*60·02.
This defines the relation to a class of the class of all its existent sub-classes, i.e. of all its sub-classes except . This is often required, as, for example, in the statement of Zermelo's axiom: "Given any class , there is a relation such that, if is any existent sub-class of , is a member of ," i.e. This axiom, or its equivalent the multiplicative axiom, plays (as will appear hereafter) an important part as the hypothesis to many propositions in cardinal arithmetic.
*60·03.
A is a class whose members are classes.
*60·04.
A is a class whose members are classes whose members are classes, i.e. a is a class of classes of classes.
Apart from propositions which merely embody the definitions, the most useful propositions in this number are the following:
*60·3.
*60·32.
*60·34.
*60·362.
I.e. and are the only sub-classes of a unit class .
*60·5.
*60·57.
*60·6.
The propositions of this number are chiefly useful in cardinal and ordinal arithmetic, but uses also occur in the theory of series; hardly any uses occur before cardinal arithmetic.
*60·01.
*60·02.
*60·03.
*60·04.
*60·1.
*60·11.
*60·12.
*60·13.
*60·14.
*60·15.
*60·2.
*60·21.
*60·22.
*60·23.
*60·24.
*60·3.
*60·31.
*60·32.
Dem.
*60·321.
Dem.
*60·33.
We write "" on the right, to indicate that the concerned is of higher type than the on the left.
Dem.
*60·34.
*60·35.
*60·36.
*60·361.
*60·362.
*60·37.
Dem.
*60·371.
Dem.
*60·38.
Dem.
*60·39.
*60·391.
This proposition is used in the theory of the continuity of functions (*234·202).
*60·4.
*60·41.
The following proposition is used in the theory of well-ordered series (*250·14).
*60·42.
*60·43.
*60·44.
The following proposition is required in the theory of "first differences" (*170·65).
*60·45.
Dem.
*60·5.
Dem.
*60·501.
Dem.
The above proposition is used in the theory of cardinal multiplication (*115·17).
*60·51.
The following proposition is used in the cardinal theory of finite and infinite (*124·541).
*60·52.
*60·53.
Dem.
*60·54.
*60·55.
Dem.
*60·56.
The following proposition is used frequently.
*60·57.
Dem.
*60·6.
The following proposition is used in connection with cardinal multiplication and with greater and less (*115·17 and *117·66).
*60·61.
*60·62.
*60·7.
Dem.
*60·71.
*60·72.