Volume 1 / Part 5 — *200. RELATIONS CONTAINED IN DIVERSITY.
Reading edition
*200. RELATIONS CONTAINED IN DIVERSITY.
Summary of *200.
Some of the propositions of this number are repetitions or immediate consequences of previous propositions, especially those of the propositions of *50 which deal with diversity. But we are chiefly concerned here with propositions which will be useful in the theory of series; this leads us to introduce propositions on and on matters connected with relation-arithmetic and other topics. It will be seen that "" (i.e. " is asymmetrical") is an important hypothesis, as is also , of the use of which we have already had examples in *96 and *121.
The following are among the most useful propositions in this number:
*200·12.
This is the proposition which makes it impossible to define an ordinal number 1 which shall take its place among relation-numbers applicable to series.
*200·35.
This is a consequence of *200·12.
*200·36.
*200·361.
I.e. if , no term precedes itself or any of its predecessors, and no term succeeds itself or any of its successors.
*200·38.
*200·39.
We then have a collection of propositions concerned with relation-arithmetic.
*200·211.
I.e. the property of being contained in diversity is invariant for likeness-transformations;
*200·4.
*200·41.
and other such propositions.
We then have a set of propositions concerned with and . The most important are
*200·5.
*200·52.
*200·53.
I.e. if is asymmetrical, the terms which precede part of do not succeed the whole of , and vice versa.
*200·11.
*200·12.
Dem.
*200·2.
Dem.
*200·21.
Dem.
*200·211.
The properties of relations are very frequently common to all relations which are like a given relation, and this applies specially to the kinds of properties with which we are most concerned. The above proposition is an illustration of this fact: it shows that the property of being contained in diversity is invariant for likeness-transformations.
*200·22.
Dem.
We have, without the need of typical definiteness, both of which are immediate consequences of *200·211. The converse implications, however, fail if is taken in a type in which .
*200·3.
*200·31.
*200·32.
*200·33.
*200·34.
*200·35.
Dem.
*200·36.
*200·361.
Dem.
*200·37.
Dem.
*200·38.
*200·381.
Dem.
*200·39.
Dem.
*200·391.
Dem.
The above proposition is useful in the theory of segments.
The following propositions are concerned with the ideas of relation-arithmetic. Analogous propositions will be proved for transitiveness and connection in *201 and *202, whence analogous propositions concerning series will be deduced in *204.
*200·4.
Dem.
This proposition is part of the proof that the sum of two mutually exclusive series is a series.
*200·41.
*200·42.
Dem.
The following propositions (*200·421·422·423) are lemmas for *204·53.
*200·421.
Dem.
*200·422.
Dem.
*200·423.
Dem.
*200·43.
Dem.
The following propositions, with the exception of *200·52, are concerned with and , i.e. the class of terms preceding (or succeeding) the whole of .
*200·5.
Dem.
*200·51.
Dem.
*200·52.
Dem.
This proposition is often used in the theory of well-ordered series.
*200·53.
Dem.
The above proposition is frequently used. If is an existent class contained in , and are the two parts of the Dedekind "cut" determined by (excluding the maximum of , if any). The above proposition shows that these two parts are mutually exclusive.
*200·54.
Dem.
This proposition is a lemma whose purpose is to avoid the necessity of introducing the hypothesis or in proofs in which it is not really necessary. The first use of this proposition occurs in *206·551.