Volume 1 / Part 3 — *105. DESCENDING CARDINALS.
Reading edition
*105. DESCENDING CARDINALS.
Summary of *105.
In this number, we consider cardinals generated by a relation of similarity which goes from a higher to a lower type, i.e. given any class of classes , we consider in the type of members of (which we shall call ) or in some lower type. Thus e.g. we shall have where "" means "classes similar to but of the next lower type." Similarly and so on. We shall have generally and so on. The chief difference between ascending and descending cardinals is that is one of the latter, but not one of the former. Otherwise the propositions of the present number are mostly analogous to corresponding propositions of *104.
On the analogy of the definitions in *104, we put with similar definitions for and .
No proposition of the present number is ever referred to in the sequel, and the reader who is not interested in the subject may therefore omit it without detriment to what follows. The principal propositions proved are the following:
*105·25.
*105·251.
*105·26.
Thus or , in any given type, only differs from in that type by the addition of .
*105·3.
*105·322.
*105·34.
*105·35.
*105·38.
*105·01.
We might write which would be equivalent to the above. But we choose the above form for the sake of uniformity. If is any suffix, we put, provided has been defined, and if is any index for which has been defined, we put
Thus for the sake of uniformity it is better, in the above definition *105·01, to write "" rather than "."
*105·011.
*105·02.
*105·021.
*105·03.
*105·031.
*105·1.
*105·101.
*105·11.
*105·111.
*105·12.
*105·121.
*105·13.
*105·131.
*105·14.
Dem.
*105·141.
*105·142.
*105·143.
*105·15.
*105·151.
*105·16.
*105·161.
In what follows, propositions concerning or have proofs exactly analogous to those of the corresponding propositions concerning or .
*105·2.
Dem.
*105·201.
*105·21.
*105·211.
*105·22.
*105·221.
*105·23.
*105·231.
*105·24.
*105·241.
*105·25.
*105·251.
*105·252.
Dem.
*105·26.
Dem.
*105·261.
*105·27.
*105·271.
*105·28.
*105·281.
*105·29.
*105·3.
Dem.
*105·301.
*105·31.
*105·311.
*105·312.
*105·313.
*105·314.
*105·315.
*105·316.
Dem.
*105·317.
*105·32.
Dem.
*105·321.
*105·322.
*105·323.
*105·324.
*105·325.
*105·326.
Dem.
*105·327.
*105·33.
Dem.
*105·331.
*105·34.
*105·341.
*105·342.
Dem.
*105·343.
*105·344.
Dem.
*105·345.
*105·35.
Dem.
*105·351.
*105·352.
*105·353.
*105·354.
*105·355.
*105·356.
*105·357.
*105·36.
Dem.
*105·361.
Dem.
*105·362.
*105·37.
*105·371.
Dem.
*105·372.
*105·38.
Dem.
*105·4.
Dem.
*105·41.
*105·42.
*105·43.
Dem.
*105·44.
Dem.