Volume 1 / Part 4 — *177. PROPOSITIONS CONNECTING WITH PRODUCTS AND POWERS.
Reading edition
*177. PROPOSITIONS CONNECTING WITH PRODUCTS AND POWERS.
Summary of *177.
The principal proposition on this subject is
*177*13.
which is the analogue of *116·72, or rather leads to the analogue of *116·72 as soon as powers of relation-numbers have been defined; for then it becomes Another proposition is an extension of *171·69, namely
*177·22.
where we put .
The remaining propositions of this number are lemmas for the above two.
*177·13 shows, for example, that all classes of finite integers can be arranged in a series of which the relation-number is , where is the relation-number of the series of finite integers. is not the relation-number of the continuum, but is closely allied to it.
*177·1.
In the propositions of *116 referred to, and appear in place of and , but no property of and is used in the proof except .
*177·11.
Dem.
*177·12.
*177·13.
*177·2.
*177·21.
The proof proceeds as the proof of *174·24 proceeds. If , we shall have, if ,
Hence we easily obtain whence whence the result follows easily.
*177·22.