Volume 1 / Part 2 — *80. ELEMENTARY PROPERTIES OF SELECTIONS.
Reading edition
*80. ELEMENTARY PROPERTIES OF SELECTIONS.
Summary of *80.
In this number, we shall give such properties of as follow most directly from the definition, without any restrictive hypothesis as to .
If , selects one member of , whenever , as the selected referent of . For, since , we have ; and since , we have , i.e. . Calling the selected referent of , it is evident that we may replace by any other member of , and still have a member of . (This is proved in *80·4.) Thus if has any members at all, we can get as many members as there are members of by merely altering the selected referent of , leaving the other selected referents unchanged.
In the present section, we first prove various simple properties of . Most of these are almost immediate consequences of
*80·14.
The most useful of them are
*80·2.
*80·291.
*80·3.
*80·33.
We then have various propositions (*80·4—·46) concerned with when . Of these the most important are the following:
*80·41.
I.e. given a selective relation , the selected referent of (where ) may be replaced by any other term having the relation to , and we shall still have a selective relation.
*80·45.
We then have a set of propositions (*80·5—·54) connecting ( with and . These are chiefly useful as leading to the next set
(*80·6—·69), connecting with and . The most useful of these are the following:
*80·6.
*80·65.
*80·66.
We have next a set of propositions (*80·7—·78) dealing with the relations of and when (e.g.) and . These propositions are seldom used, but they would be useful in considering division.
We next have a set of propositions (*80·8—·84) dealing with the relations of and . The most useful are
*80·81.
*80·82.
Finally, we have four propositions (*80·9—·93) on and one on . The most useful of these is
*80·9.
*80·01.
*80·1.
*80·11.
*80·12.
*80·13.
*80·14.
*80·15.
*80·16.
Dem.
*80·17.
Dem.
This proposition is used in the theory of ordinal multiplication (*172·162).
*80·2.
Dem.
*80·21.
*80·22.
Dem.
*80·23.
Dem.
*80·24.
*80·25.
*80·26.
Dem.
Note that is a unit class, not the null-class. It is owing to this fact (as will appear later) that, if is any cardinal, . See the note to *83·15.
*80·27.
Dem.
*80·28.
Dem.
*80·29.
Dem.
*80·291.
Dem.
*80·3.
Dem.
*80·31.
Dem.
*80·32.
Dem.
*80·33.
Dem.
*80·34.
Dem.
*80·35.
*80·36.
Dem.
This proposition is used in dealing with greater and less among cardinals (*117·68).
*80·4.
This proposition is important. It shows that, if and is the selected referent of (i.e. is , then may be replaced by any other member of without our ceasing to have a member of .
Dem.
*80·41.
Dem.
*80·42.
Dem.
*80·43.
Dem.
*80·44.
Dem.
*80·45.
Dem.
*80·46.
*80·5.
Dem.
*80·51.
Dem.
*80·511.
Dem.
*80·52.
Dem.
*80·53.
Dem.
*80·54.
Dem.
*80·6.
Dem.
*80·61.
Dem.
*80·62.
*80·621.
Dem.
*80·63.
*80·64.
Dem.
*80·65.
*80·651.
Dem.
*80·66.
Dem.
*80·661.
Dem.
*80·67.
Dem.
*80·68.
Dem.
*80·69.
Dem.
*80·7.
Dem.
*80·71.
Dem.
*80·72.
*80·73.
Dem.
*80·731.
Dem.
*80·732.
Dem.
*80·74.
Dem.
*80·75.
*80·76.
Dem.
*80·761.
Dem.
*80·77.
Dem.
*80·771.
Dem.
*80·78.
Dem.
*80·8.
Dem.
*80·81.
Dem.
*80·82.
Dem.
The following proposition is used in *80·84 and in the theory of double similarity (*111·3).
*80·83.
Dem.
*80·84.
Dem.
The three following propositions are useful both in cardinal and in ordinal multiplication (*113 and *172).
*80·9.
Dem.
*80·91.
Dem.
80·9·91 can be extended, by precisely similar proofs, to any finite number of variables , , .... They will, on occasion, be assumed for three or four variables, without fresh proofs.
*80·92.
Dem.
*80·93.
*80·94.
From this proposition, together with *80·26 (which gives , we shall obtain an inductive proof that exists whenever is a finite class contained in (cf. *120·611).