Volume 1 / Part 2 — *56. THE ORDINAL NUMBER .
Reading edition
*56. THE ORDINAL NUMBER .
Summary of *56.
In this number, we have to consider the class of those relations which are each constituted by a single couple. In case the two members of this couple are not identical, the class of such relations is (as will be shown later) the ordinal number 2, which, to distinguish it from the cardinal number 2, we denote by "." (Here the suffix is intended to suggest "relational.") The class of all relations consisting of a single couple, without the restriction that the two members of the couple are to be distinct, will be denoted by "." This is not an ordinal number. It will be observed that there is no ordinal number 1, because ordinal numbers apply to series, and series must have more than one member if they have any members. This will appear more fully when we come to deal with series.
The properties of are largely analogous to those of 1, while the properties of are more analogous to those of 2.
Most of the propositions of the present number are seldom referred to in the sequel, but such references as occur are important. The most useful propositions in the present number are the following.
*56·111.
*56·112.
*56·113.
Observe that "" means "relations whose fields have two terms."
*56·13.
*56·37.
I.e. is the class of asymmetrical relations whose fields have two terms.
*56·381.
*56·39.
I.e. the relations which are couples whose referent and relatum are identical are the relations whose fields consist of a single term.
*56·01.
*56·02.
*56·03.
*56·1.
*56·101.
Dem.
*56·102.
Dem.
*56·103.
Dem.
*56·104.
*56·11.
*56·111.
Dem.
*56·112.
Dem.
*56·113.
Dem.
*56·114.
*56·12.
Dem.
*56·121.
*56·122.
*56·13.
Dem.
might be defined as the ordinal number 1, since it is what we shall call a relation number (cf. *153). But we wish our ordinal numbers to be classes of serial relations, and such relations have the property of being contained in diversity. Hence if we were to define as the ordinal number 1, we should introduce a tiresome exception, from which trivial complications would be introduced into ordinal arithmetic. We have, therefore, not adopted this course.
*56·14.
Dem.
*56·141.
*56·15.
Dem.
*56·151.
*56·16.
Dem.
*56·17.
Dem.
*56·18.
Dem.
*56·19.
Dem.
*56·191.
*56·2.
*56·21.
*56·22.
*56·24.
*56·25.
*56·26.
This proposition is the analogue of *52·4.
Dem.
*56·261.
Dem.
*56·262.
Dem.
*56·27.
Dem.
*56·28.
Dem.
*56·281.
Dem.
*56·29.
Dem.
*56·3.
Dem.
The steps from (2) to the conclusion are analogous to those from (2) of *56·29 to the conclusion of *56·29. Analogous steps in succeeding proofs will be merely indicated as above.
*56·31.
*56·32.
Dem.
*56·33.
Dem.
*56·34.
Dem.
*56·35.
Dem.
*56·36.
Dem.
The following proposition, in addition to being used in *56·38, is used in the elementary theory of series (*204·463).
*56·37.
*56·38.
Dem.
This proposition is important as establishing the connection between the cardinal and ordinal 2. It shows that the ordinal 2 consists of those asymmetrical relations whose fields have (cardinal) 2 terms. It is used in the theory of well-ordered series (*250·44).
The following proposition, in addition to being used in *56·39, is used in relation-arithmetic (*165·38) and in the theory of series (*205·4).
*56·381.
Dem.
*56·39.
Dem.
This proposition establishes the connection between and 1, showing that is the class of those relations whose fields consist of a single term. It is used in the discussion of and and as relation-numbers (*153·301).
*56·4.
Dem.
This proposition is the analogue of *53·23. It is used in the number on exponentiation in relation-arithmetic (*176·19).