Volume 1 / Part 3 — *106. CARDINALS OF RELATIONAL TYPES.
Reading edition
*106. CARDINALS OF RELATIONAL TYPES.
Summary of *106.
In this number we have to consider the cardinals whose members are classes of relations which have a given relation of type to some given class. For example, we have , and has a given relation of type to when is given. Thus we want a notation for and all the associated ideas. In this number, we shall deal only with relations in which the referent and relatum have a relation, as to type, which can be expressed by the notations of *63, i.e. roughly speaking, when, for suitable values of , , , our relations are contained in
Thus if has been defined, we shall put with analogous definitions for , and .
Much the most important case is that of . For this case we have
*106·1.
Thus will be the number of a class of relations whose fields are of the same type as , provided this class of relations is similar to . E.g. the number of terms such as , where , will be .
We have
*106·21.
*106·22.
*106·23.
*106·32.
*106·4·41·411.
*106·53.
whence it follows that
*106·54.
The propositions of this number, except *106·21, are never referred to again (except in *154·25·251·262, which are themselves never used again), but they have a somewhat greater importance than the propositions of *105, owing to the fact that the arithmetical operations are defined by means of classes of relations, i.e. the sum of two cardinals (for instance) is defined as the cardinal number of a certain class of relations (cf. *110).
*106·01
*106·011.
*106·012.
*106·02.
*106·021.
*106·03.
*106·04.
*106·041.
*106·1.
*106·101.
Similar propositions hold for any other double index for which has been defined.
*106·11.
Similar propositions hold for any other double suffix for which has been defined.
*106·12.
*106·121.
Similar propositions hold for any other index and suffix for which or has been defined.
*106·13.
Similar propositions hold for etc.
*106·14.
*106·141.
Similar propositions hold for , , etc.
*106·2
Dem.
*106·201.
*106·202.
*106·203.
*106·204.
*106·21.
*106·211.
*106·212.
*106·213.
*106·22.
Dem.
The proof requires, in addition to *106·12, its analogue for . Such analogues will be assumed as required.
*106·221.
*106·222.
*106·223.
Other propositions of the same kind as the above may be proved by observing that, if and are indices for which and have been defined, we have of which the proof is direct and simple. Hence, since we always have , we also always have We have in like manner But we do not always have
*106·23.
Dem.
*106·231.
*106·24.
*106·241.
The analogues of the above propositions for other indices or suffixes are similarly proved.
*106·25.
*106·251.
*106·31.
*106·311.
*106·312.
Dem.
*106·32.
Dem.
*106·4.
Dem.
*106·401.
Dem.
*106·402.
Dem.
*106·41.
Dem.
*106·411.
*106·43.
Dem.
*106·44.
The following propositions are analogous to *102·71 ff., and similar remarks apply to them.
*106·5.
Dem.
*106·51.
Dem.
*106·52.
Dem.
*106·53.
*106·54.
Dem.
*106·55.