Volume 1 / Part 4 — *170. ON THE RELATION OF FIRST DIFFERENCES AMONG THE SUB-CLASSES OF A GIVEN CLASS.
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*170. ON THE RELATION OF FIRST DIFFERENCES AMONG THE SUB-CLASSES OF A GIVEN CLASS.
Summary of *170.
The definition to be given in this number of the relation of first differences among the sub-classes of a given class is by no means the only one possible, in fact a different definition will be considered in *171. In the present number, the definition we choose is this: is said to precede according to this definition when has at least one member which neither belongs to nor follows any term belonging to and not to ( and being both sub-classes of ). In other words, if we consider the two classes and , there are members of which are not preceded by any members of . Pictorially, we may conceive the relation as follows ( being supposed serial): and each pick out terms from , and these terms have an order conferred by ; we suppose that the earlier terms selected by and are perhaps the same, but sooner or later, if , we must come to terms which belong to one but not to the other. We assume that the earliest terms of this sort belong to , not to ; in this case, has to the relation . That is, where and begin to differ, it is terms of that we come to, not terms of . We do not assume that there is a first term which belongs to and not to , since this would introduce undesirable restrictions in case is not well-ordered.
A few of the propositions of the present number will be used in the next number, which deals with a slightly different form of the relation of first differences, but with this exception the propositions of this number will not be referred to again until we come to series. Their chief use occurs in the section on compact series, rational series, and continuous series (Part V, Section F), especially in *274 and *276, which respectively establish the existence of rational series (assuming the axiom of infinity) and the fact that the cardinal number of terms in a continuous series is the same as the number of classes contained in the field of a progression, i.e. . The definitions and a few of the simpler propositions are also used in connection with the series of segments of a series, since, as explained above, the segments of a series are arranged in the series generated by .
The propositions of this number which will be used in dealing with series are the following:
*170·1.
*170·101.
*170·102.
(These propositions merely embody the definitions.)
*170·11.
This form is often more convenient than *170·1.
*170·16.
I.e. every sub-class of has the relation to every proper part of itself.
*170·17.
*170·2.
This proposition deals with the case where there is a definite first term y which belongs to and not to , and whose predecessors all belong to both or neither.
*170·23.
This proposition is useful in case is well-ordered, since then must have a minimum if it exists ( and being supposed sub-classes of ).
*170·31.
This follows from *170·16, as does the following proposition:
*170·32.
*170·35.
*170·38.
*170·6.
Besides the above, the following propositions should be noted:
*170·36.
*170·37.
*170.44.
*170·64.
This proposition shows that every term added to doubles the number of terms in ; hence it is not surprising that (when is well-ordered) has a power of for its relation-number (cf. *177).
*170.67.
whence
*170·69.
*170·01.
*170·02.
*170·1.
*170·101.
*170·102.
Thus means, roughly speaking, that goes on longer than , just as means that begins sooner. Thus if is the relation of earlier and later in time, and and are the times when and respectively are out of bed, "" will mean that gets up earlier than , and "" will mean that goes to bed later than .
*170·103.
Dem.
*170·11.
*170·12.
*170*121.
*170*13.
Dem.
*170·14.
*170·141.
*170·15.
Dem.
*170·16.
Dem.
*170·161.
Dem.
*170·17.
Dem.
In order that should be serial, we need further that it should be transitive and connected. is transitive if is transitive and connected. But may still not be connected: there may be many distinct families in its field, though all of them must begin with and end with . For example, if is a regression, the class which takes every odd member does not have either of the relations , to the class which takes every even member. In order that should be serial, we require that should be not only serial, but well-ordered, i.e. that every existent sub-class of should have a first term. When is serial but not well-ordered, will, however, generate various series contained in it by imposing suitable limitations on the field.
*170·2.
*170·21.
Dem.
*170·22.
Dem.
*170·23.
Dem.
*170·3.
*170·31.
*170·32.
*170·33.
Dem.
*170·34.
Dem.
*170·35.
*170·36.
Dem.
*170·37.
*170·371.
*170·38.
The following propositions lead up to *170·44.
*170·4.
Dem.
*170·41.
*170·42.
Dem.
*170·43.
Dem.
*170·44.
*170·5.
Dem.
*170·51.
Dem.
*170·52.
Dem.
*170·6.
*170·601.
*170·61.
This and the following propositions are lemmas for
Dem.
*170·62.
Dem.
*170·63.
Dem.
*170·64.
Dem.
The following propositions are lemmas for *170·67, i.e. which itself leads to *170·69, i.e.
*170·65.
Dem.
*170·651.
Dem.
*170·652.
Dem.
*170·653.
Dem.
*170·66.
Dem.
*170·67.
Dem.
*170·68.
Dem.
*170·69.