Volume 1 / Part 5 — *208. THE CORRELATION OF SERIES.

*208. THE CORRELATION OF SERIES.

Summary of *208.

The propositions of this number are chiefly important on account of their consequences in the theory of well-ordered series (*250 ff.) and in the theory of vector-families (*330 ff.). When two well-ordered series are ordinally similar, they have only one correlator; and a well-ordered series is not ordinally similar to any of its segments. Of these two propositions, the first is an immediate consequence of *208·41, and the second is an immediate consequence of *208·47.

Propositions concerning correlators of two relations and are obtained from propositions concerning correlators of with itself, by means of the fact that, if , are two correlators of and , is a correlator of with itself. Again, correlators of with itself are considered, in this number, as a special case of correlators of with parts of itself. This latter is a notion which will prove important for other reasons than those for which it is used in our present context. If is connected, and correlates with part of itself (so that ), will contain terms of three kinds, (1) those for which , (2) those for which , (3) those for which . Our propositions result from the non-existence (under certain circumstances) of maxima or minima of classes (2) and (3).

The following definition defines "correlations of P with parts (or the whole) of itself." The letters "" stand for "ordinal correlation." For a cardinal correlation, should occasion arise, we should use "," i.e. we should put so that . For the present, we are concerned with the corresponding ordinal notion; thus we require This is secured by putting

It will be observed that if is what we called a "non-reflexive" class (cf. *124), , and . When is non-reflexive, the same is true of ; and when is reflexive, is also reflexive, in the sense that it contains proper parts similar to itself, though if is well-ordered, such proper parts cannot be segments of , but must extend to the end of .

The class of correlators of with the whole of itself, i.e. , is a sub-class of , and is specially important. This class differs widely in its properties from the corresponding cardinal class. If has more than one member, the class (which is the "permutations" of in the usual elementary sense) always has more than one member. But the class (which consists of such permutations of as keep the order unchanged) will consist of the single term , unless contains classes which have neither a minimum nor a maximum, in which case there will be many correlators of with itself. As a simple illustration, take the series of negative and positive integers in their natural order. Then if is any one of these integers, is a correlator of the whole series with itself. If we take only the positive integers, is no longer a correlator of the whole series with itself, since all integers less than are omitted from the correlate.

The first important use of the propositions of this number is in the beginning of the theory of well-ordered series (*250). The propositions there used are

*208·41.

I.e. if is connected and asymmetrical, and every existent sub-class of has either a minimum or a maximum, and cannot have more than one correlator.

*208·42.

*208·43.

I.e. if every existent sub-class of has a minimum, a correlator of with part of itself can never move terms backwards. Thus for example, to take a simple instance, an infinite series consisting of some of the natural numbers in order of magnitude cannot have its th term less than .

*208·45.

I.e. if is connected and every existent sub-class of has both a maximum and a minimum, no proper part of is similar to . This proposition is important in the theory of finite series and finite ordinals.

*208·46.

I.e. if every existent sub-class of has a minimum, a part of which is similar to must go up to the end of , i.e. must not wholly precede any member of .

*208·47.

This is an immediate consequence of *208·46.

The proof of the above propositions proceeds simply by showing that if and , then , so that is not the earliest term for which , since is an earlier term for which the same thing holds. Hence can have no minimum; and similarly can have no maximum (*208·14). So far we require no hypothesis as to . Assuming now , we show similarly that if correlates the whole of with itself, can have no maximum and can have no minimum.

Propositions about correlators of with follow from the above by taking two correlators and , and applying the above propositions to , which is a correlator of with the whole of itself.

*208·01.

*208·1.

Dem.

*208·11.

Dem.

*208·111.

*208·12.

*208·13.

Dem.

*208·131.

*208·14.

Dem.

Thus the proof that has no minimum, and no maximum, requires no hypothesis as to . The proof that has no maximum, and no minimum, requires the hypothesis . This proof results from the following propositions.

*208·2.

Dem.

*208·21.

Dem.

*208·211.

*208·22.

Dem.

Observe that, in virtue of *208·111, the above hypothesis gives , so that . Hence we are led to *208·3.

*208·3.

Dem.

*208·31.

*208·32.

*208·4.

Dem.

*208·41.

The above proposition is of great importance in the theory of well-ordered series.

*208·42.

*208·43.

*208·431.

*208·44.

Dem.

In virtue of this proposition, if is a finite series, no proper part of is ordinally similar to . (It will be shown later that a finite series is one in which every existent contained class has both a maximum and a minimum.) The following proposition gives a more explicit form of the above result.

*208·45.

Dem.

The following propositions are useful in the theory of segments of well-ordered series, since they show that a well-ordered series is never ordinally similar to any of its segments.

*208·46.

Dem.

*208·461.

*208·47.

Dem.