Volume 1 / Part 5 — *214. DEDEKINDIAN RELATIONS.
Reading edition
*214. DEDEKINDIAN RELATIONS.
Summary of *214.
We call a relation "Dedekindian" when it is such that every class has either a maximum or a sequent with respect to it. As a rule, the hypothesis that a relation is Dedekindian is only important in the case of serial relations. Dedekindian series have considerable importance, especially in connection with limits.
When is transitive, the hypothesis that is Dedekindian is equivalent to the hypothesis that every section of has a maximum or a sequent (*214·13); it is also equivalent to the assumption that every segment of has a maximum or a sequent (*214·131), i.e. to the assumption that every segment of which has no maximum has a limit, i.e. to When is a series, the hypothesis that it is Dedekindian is equivalent to the hypothesis that every segment has a sequent (*214·15), i.e. to the hypothesis that the class of segments is the class (*214·151). If is a Dedekindian series, so is , and vice versa (*214·14). Whenever is connected and not null, is a Dedekindian series (*214·32), and so is if it exists (*214·34); whenever is transitive and connected and not null, is a Dedekindian series (*214·33). All these propositions have been virtually proved already: almost the only thing new in the present number is the definition, which is
*214·4—·43 give properties of series which have Dedekindian continuity. We have
*214·4.
*214·41.
I.e. in a series, Dedekindian continuity is equivalent to the assumption that the classes which have a maximum are the same as the classes which have no sequent.
*214·42.
This proposition is important in dealing with Dedekind "cuts."
*214·43.
*214·5 shows that a Dedekindian relation has a beginning and an end; the following propositions deal with when is Dedekindian.
*214·6 shows that a relation which is similar to a Dedekindian relation is Dedekindian.
We call a relation "semi-Dedekindian" if it becomes Dedekindian by the addition of one term at the end; the definition is
*214·02.
*214·01.
*214·02.
*214·1.
*214·101.
*21·411.
*214·12.
*214·13.
*214·131.
*214·132.
*214·14.
*214·141.
*214·15.
*214·151.
*214·2.
*214·21.
*214·22.
*214·23.
Dem.
*214·24.
*214·241.
*214·3.
*214·31.
*214·32.
*214·33.
*214·34.
*214·4.
*214·41.
*214·42.
Dem.
*214·43.
Dem.
The following propositions are no longer mere restatements of previous results.
*214·5.
Dem.
*214·51.
Dem.
*214·52.
*214·53.
Dem.
*214·531.
Dem.
*214·532.
Dem.
*214·54.
Dem.
*214·6.
Dem.
*214·7.
*214·71.
*214·72.
*214·73.
The proof of the following proposition is given in a somewhat compressed form, since, if given with the usual fullness, it would require various lemmas not required elsewhere.
*214·74.
Dem.
*214·75.