Volume 1 / Part 3 — *123. .
Reading edition
*123. .
Summary of *123.
In this number we are concerned with the arithmetical properties of , the smallest of Cantor's transfinite cardinals. Cantor defines as the cardinal number of any class which can be put into one-one relation with the inductive cardinals. This definition assumes that , when is an inductive cardinal; in other words, it assumes the axiom of infinity; for without this, the inductive cardinals would form a finite series, with a last term, namely . For this reason among others, we do not make similarity with the inductive cardinals our definition. We define as the class of those classes which can be arranged in progressions, i.e. as . We then have to prove that so defined is a cardinal, and that if it is not null, it is the number of the inductive numbers.
For convenience we put for the moment for the relation of to when is an inductive cardinal. We then easily prove
*123·21·23.
The only thing further required to prove is , i.e.
By *120·311, this holds if , which holds if holds. Hence
*123·25·26.
whence, by *123·36,
*123·27.
Again it is obvious from *122·34·341 that if is a progression, can always be put into a relation to the inductive cardinals (*123·3) since consists of the terms , , ... , ..., and all the inductive cardinals are used in putting into this form. Hence
*123·31.
whence also
*123·311.
It remains to prove that any class similar to the inductive cardinals is an ; this can only be proved by assuming the axiom of infinity. We prove first (*120·32) that if is a progression, and is a one-one whose converse domain is , then is a progression whose domain is . Hence
*123·321.
From this and , , we obtain
*123·322.
Hence by our previous results
*123·34.
Also we have, by *120·322 above, whence, since , we obtain at last
*123·36.
As to the existence of in various types, if holds, i.e. if, given any inductive cardinal , there are classes having terms and composed of terms of the same type as , then . Thus
*123·37.
The arithmetical properties of in regard to addition, multiplication and exponentiation by an inductive cardinal are easily proved. We have
*123·41.
*123·421.
*123·422.
*123·52.
*123·53.
All these propositions are well known.
The early propositions of the present number are for the most part immediate consequences of propositions proved in *122.
*123·01.
*123·02.
*123·1.
*123·101.
*123·11.
*123·12.
*123·13.
Dem.
*123·14.
*123·15.
*123·16.
*123·17.
Dem.
*123·18.
*123·19.
*123·191.
*123·192.
Dem.
*123·2.
*123·21.
Dem.
*123·22.
*123·23.
Dem.
*123·24.
Dem.
*123·25.
*123·26.
*123·27.
*123·3.
Dem.
*123·31.
*123·311.
It is not assumed here that and are of the same type.
*123·312.
Dem.
*123·313.
Dem.
*123·32.
*123·321.
*123·322.
Dem.
*123·323.
*123·33.
*123·34.
*123·35.
*123·36.
*123·361.
*123·37.
Dem.
*123·39.
Dem.
*123·4.
*123·401.
Dem.
*123·41.
*123·411.
*123·42.
Note that is the odd terms and the even terms of .
Dem.
*123·421.
Dem.
*123·422.
Dem.
*123·43.
Dem.
*123·44.
Dem.
*123·45.
*123·46.
Dem.
*123·47.
Dem.
The following propositions are concerned in proving . The proof given is roughly Cantor's. It consists in showing that the relation defined in the hypothesis of *123·5 is a progression.
*123·5.
Dem.
*123·501.
Dem.
*123·502.
Dem.
*123·503.
Dem.
*123·504.
*123·51.
*123·52.
*123·53.
*123·7.
Dem.