Volume 1 / Part 2 — *91. ON POWERS OF A RELATION.
Reading edition
*91. ON POWERS OF A RELATION.
Summary of *91.
In the present number, we consider the class of relations
Each of these has to its predecessor the relation ; we have Thus every term of the series has to the relation (; hence the powers of may be defined as those relations which have to the relation (. The series of powers starting with instead of with is similarly composed of those relations which have to the relation . (This class consists of the previous class together with .) To say that the relation holds between and turns out to be equivalent to saying that one of the relations holds between and ; and to say that the relation holds between and turns out to be equivalent to saying that one of the relations holds between and . Thus we might have begun by defining powers of , and proceeded to define as their sum.
For notational convenience we put Then the definition of powers of excluding is and the definition of powers of including is (Here the letters "id" are added to suggest that identity is to be added to .)
We put also
Many of the propositions in this number are very often used. Among the more important propositions are the following:
*91·17.
*91·171.
*91·373.
These are formulae of induction. The first two state that if the property is hereditary with respect to , then if belongs to it belongs to any member of , while if belongs to it belongs to any member of . The third gives a form of induction which is sometimes more powerful than the second. It states that if is hereditary provided its argument is a power of , and if , then every power of satisfies , and vice versa.
*91·23.
*91·24.
These two propositions are very useful as giving relations of and .
*91·27.
*91·271.
We do not have in general . If is the sort of relation which generates a series (i.e. is either itself serial, or such that is serial), the above would characterize a series without a first or last term. To illustrate the matter, consider a series of four terms, , , , , and let be the relation of immediately preceding in this series. Thus holds between and , and , and . Then holds between and , and ; thus , which belongs to , does not belong to . holds only between and ; thus neither nor belongs to . All powers of beyond the third are null. On the other hand, if we take a cyclic relation, such as that of left-hand neighbour at a dinner-table, we shall always have , whatever power of may be.
*91·282.
This proposition shows that is a hereditary class with respect to .
*91·34.
This proposition states that the relative product is commutative when each factor is or a power of .
We come next to propositions concerning . We have
*91·502.
*91·504.
*91·511.
*91·52.
*91·54.
*91·52·54 are fundamental in the theory of inductive relations.
*91·542.
This proposition is particularly useful when (as often happens) we have . In that case, it gives .
*91·55.
*91·56.
Thus is always transitive, which is one of the three characteristics of serial relations (cf. *204). We shall find that is often serial when is not so.
*91·574.
*91·602.
*91·01.
*91·02.
*91·03.
*91·04.
*91·05.
The first two of the above definitions are introduced merely for notational convenience. The other three represent ideas of great importance. The last is especially useful when a series is given as the field of a one-one relation between consecutive terms—as, e.g., when the series of natural numbers is given as the field of the relation of to . Then is the relation of any earlier term to any later term—e.g., in the above case of the natural numbers, the relation of a less integer to a greater.
*91·1.
Dem.
*91·11.
*91·12.
*91·13.
*91·14.
*91·15.
*91·16.
*91·17.
*91·171.
These propositions are of great importance, because they enable us to prove that a property belongs to every power of if it belongs to (or ) and also belongs to whenever it belongs to .
*91·2.
Dem.
*91·201.
*91·204.
Dem.
*91·205.
*91·21.
Dem.
*91·211
*91·212
Dem.
*91·213
*91·22
*91·221
*91·23
Dem.
*91·231
*91·24
Dem.
*91·241
Dem.
The last line of the above proof is obtained as follows: writing for (1) becomes
But by *91·11, writing for the of *91·11, and for the , we have
Hence, by (1) and (2), , i.e. which is the proposition to be proved.
*91·242.
Dem.
*91·25.
Dem.
*91·251.
*91·26.
*91·261.
*91·262.
*91·263.
*91·264.
*91·27.
Dem.
*91·271.
Dem.
*91·28.
*91·281.
*91·282.
*91·283.
The following propositions show that the relative product of two powers of is commutative, i.e. (cf. *91·34)
We also have (cf. *91·341)
It is these propositions (as will appear in the sequel) which are the source of the commutative law for the addition of finite ordinals. Ordinals in general are not commutative, just as relative products in general are not commutative; but owing to the fact that relative products whose factors are powers of a given relation are commutative, finite ordinals are commutative.
*91·3.
Dem.
*91·301.
*91·302.
Dem.
*91·303.
*91·304.
*91·31.
*91·33.
Dem.
*91·331.
Dem.
*91·34.
Dem.
This is the commutative law for the relative product of two powers of .
*91·341.
Dem.
*91·342.
Dem.
*91·343.
*91·35.
*91·351.
*91·352.
*91·36.
*91·37.
Dem.
*91·371.
*91·372.
*91·373.
*91·41.
*91·411.
*91·42.
*91·421.
*91·43.
Dem.
*91·431.
*91·44.
Dem.
*91·45.
Dem.
*91·46.
The remainder of this number is concerned with and its relations to .
*91·502.
*91·503.
*91·504.
Dem.
The following propositions are concerned mainly with the relations of and . These relations are embodied in the propositions
*91·51.
Dem.
*91·511.
*91·512.
Dem.
*91·513.
Dem.
*91·514.
Dem.
*91·52.
*91·521.
Dem.
*91·522.
*91·53.
Dem.
*91·54.
*91·541.
*91·542.
*91·543.
Dem.
*91·544.
*91·545.
*91·546.
*91·55.
Dem.
*91·56.
Dem.
*91·561.
*91·562.
*91·57.
*91·571.
*91·572.
*91·573.
*91·574.
Dem.
*91·575.
Dem.
*91·58.
*91·581.
*91·59.
Dem.
*91·6.
Dem.
*91·601.
Dem.
*91·602.
Dem.
*91·603.
Dem.
*91·62.
This formula should be compared with *90·11, in which an analogous formula is given for . It will be observed that here we do not require to add , for if , the above formula leads to , i.e. to . Hence , i.e. . It will be observed that holds whenever belongs to every hereditary class which contains the immediate successors of , whereas holds whenever belongs to every hereditary class to which itself belongs.
*91·7.
*91·71.
Dem.
*91·711.
Dem.
The above proposition is used in the theory of minimum points in a series (*205·68).
*91·72.
Dem.
*91·721.
*91·73.
Dem.
*91·731.
By means of *91·73 or *91·731, the powers of can often be arranged in a series, the rule of arrangement being that comes earlier than if , and later in the converse case. But we shall only get an open series from this arrangement if ; otherwise the powers from a certain point onwards form a cyclic series.
*91·732.
Dem.
*91·74.
*91·75.
Dem.