Volume 1 / Part 3 — *104. ASCENDING CARDINALS.
Reading edition
*104. ASCENDING CARDINALS.
Summary of *104.
In this number we have to consider cardinals derived from a relation of similarity which goes from the type of to that of , or to that of . The propositions to be proved can be extended, by a mere repetition of the proofs, to , , etc. This extension must, however, be made afresh in each instance; we cannot prove that it can be made generally, because mathematical induction cannot be applied to the series
Ascending cardinals, though less important than homogeneous cardinals, yet have considerable importance in arithmetic, because and are defined as the cardinals of classes of higher types than those of and , and the same applies to the product of the cardinals of members of a class of classes. In these cases, however, we also need cardinals of relational types, which will be dealt with in *106.
We have to deal, in this number, with three different sets of notions, namely
*104·01.
*104·02.
*104·03.
with similar definitions of , etc. Thus consists of all classes similar to but of the next higher type, i.e. it is the cardinal number of in the type next above that of ; is the class of all such cardinals as , and is a typically ambiguous symbol, though is typically definite when is given; (if is a cardinal which is not null) is the "same" cardinal in the next higher type, so that, e.g., if is 1 determined as consisting of unit classes of individuals, will be 1 determined as consisting of unit classes of classes of individuals. (When is not an existent cardinal, is unimportant.)
The following are the most useful propositions in the present number:
*104·12.
*104·2.
*104·21.
*104·24.
*104·25.
*104·26.
*104·265.
*104·27.
*104·35.
*104·43.
*104·01.
This defines the cardinal number of in the next type above that of ; thus consists of all classes similar to and of the next type above that of .
*104·011.
Similar definitions are to be assumed for , etc.
*104·02.
, like , is typically ambiguous; but is typically definite.
*104·021.
Similar definitions are to be assumed for , etc.
*104·03.
Here, if is a cardinal, is the same cardinal in the next higher type. For example, if is couples of individuals, is couples of classes of individuals.
*104·031.
Similar definitions are to be assumed for , etc.
*104·1.
*104·101.
*104·102.
*104·11.
*104·111.
*104·112.
*104·12.
Dem.
*104·121.
Dem.
*104·122.
*104·123.
*104·13.
*104·14.
*104·141.
When the hypothesis "" is omitted, this proposition is still true, but with a difference. E.g. let us put Then . Thus . But we still have Thus , but is not the same cardinal as in a higher type, i.e. there are classes whose cardinal in one type is , but whose cardinal in the next higher type is not .
*104·142.
*104·15.
*104·2.
Dem.
*104·201.
Dem.
*104·21.
It follows from this proposition that ascending cardinals are never null. The proof has to be made separately for each kind of ascending cardinal, i.e. , , etc.
*104·211.
*104·23.
Dem.
*104·231.
Dem.
*104·232.
*104·24.
*104·25.
This proposition holds for each possible determination of the typical ambiguities, i.e. for every we have
*104·251.
*104·252.
*104·26.
Dem.
*104·261.
Dem.
*104·262.
Dem.
*104·263.
Dem.
*104·264.
Dem.
*104·265.
*104·27.
*104·28.
*104·29.
Dem.
*104·3.
Dem.
*104·31.
*104·311.
*104·32.
Dem.
*104·33.
Dem.
*104·34.
Dem.
*104·35.
*104·36.
Dem.
*104·37.
Dem.
The following propositions are concerned with the proof that, given any two cardinals and , of the same type, we can find two mutually exclusive classes one of which has terms while the other has terms. The proof requires that we should raise the types of both and one degree above that in which they were originally given, i.e. that we should turn and into and . Thus, for example, suppose the total number of individuals in the universe were finite (a supposition which is consistent with our primitive propositions), and suppose were this number. Then unless , a class of individuals will be an existent sub-class of the only class which consists of individuals, and therefore we shall have
But if we consider classes of classes and classes, we shall always be able to find a and a such that
The existence of such a and is important in connection with the arithmetical operations, and is therefore proved here.
*104·4.
Dem.
*104·41
Dem.
This proposition proves the desired conclusions provided , and consists of at least three terms. The following propositions deal with the cases in which this hypothesis is not verified.
*104·411.
Dem.
*104·412.
Dem.
*104·413.
Dem.
*104·42.
*104·43.
Dem.
The above proposition gives the desired result. The following propositions re-state this result in other forms.
*104·44.
*104·45.
*104·46.