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Bertrand Russell

Principles of Mathematics

Part 5: Infinity and Continuity

Ch.32: The Correlation of Series

Brief Summary:

Members of one class can correspond in an order-preserving way to those of another class, as for example is the case of mathematical functions.

Summary

§249

The problem of continuity in mathematics was once thought of as an issue relating to the infinitesimal, but not any more. [261] The theory of infinity takes two forms: (1) cardinal, from which the theory of number derives, and (2) ordinal, under which the theory of continuity belongs. In the ordinal/continuity theory of infinity, each member of an infinite progression can be given a name. [261-262] As a result of these investigations, we will see that only this kind of continuity applies to time and space, and also that by means of the concept of limits we will not need to resort to the problematic concept of the infinitesimal. [262]

§250

From Cantor we have learned two important properties of infinite numbers: 1) they do not obtain from mathematical induction, namely, the successor function (+1) applied to finite numbers and thus “they do not form part of a series of numbers beginning with 1 or 0, proceeding in order of magnitude, containing all numbers intermediate in magnitude between any two of its terms” [262]. This gives us the true definition of an ordinal infinite or of the infinite term in a series. 2) Regarding cardinal infinity, “an infinite number of terms always contains a part consisting of the same number of terms” [262]. This defines an infinite collection, and many philosophers would say it is self-contradictory. In the following chapters, Russell will prove that there is really no contradiction involved in this concept. [262]

§250

We will need to correlate series in order to determine whether they are denumerable. [263] Series are correlated if there is a 1-to-1 order-preserving isomorphism holding between them:

Two

seriess, s'are said to be correlated when there is a one-one relationRcoupling every term ofswith a term ofs', andvice versâ, and when, ifx, ybe terms ofs, andxprecedesy, then their correlatesx' , y'ins'are such thatx'precedesy'.[263]

Series correlations maintain ordinal relations. Two classes or collections are correlated when their terms are related one-to-one without one class having any terms left over:

Two

classesor collections are correlated whenever there is a one-one relation between the terms of the one and the terms of the other, none being left over.

[263]

Class correlation does not require that the order be preserved, and thus there is only a cardinal correspondence between them.

Thus two series may be correlated as classes without being correlated as series; for correlation as classes involves only the same cardinal number, whereas correlation as series involves also the same ordinal type—a distinction whose importance will be explained hereafter. In order to distinguish these cases, it will be well to speak of the correlation of classes as correlation simply, and of the correlation of series as ordinal correlation. Thus whenever correlation is mentioned without an adjective, it is to be understood as being not necessarily ordinal. Correlated classes will be called

similar; correlated series will be calledordinally similar; and their generating relations will be said to have the relation oflikeness.

[263]

[[In the following, Russell will make use of a logic and notation of relations. Let’s for a moment jump back to where he explains these ideas, in the 9th Chapter “Relations.” We are dealing with propositions that state a relation between terms. If we exchange the order, and the terms are not identical, then we have a new proposition. (*a *is the same size as *b*: aRb; and thus b is the same size as *a*: bRa. However, if *a* is taller than b (aRb) then the reordering bRa is not equivalent (b is not taller than *a*).

A relation between two terms is a concept which occurs in a proposition in which there are two terms not occurring as concepts, and in which the interchange of the two terms gives a different proposition. This last mark is required to distinguish a relational proposition from one of the type “a and b are two”, which is identical with “b and a are two”. A relational proposition may be symbolized by aRb, where R is the relation and a and b are the terms; and aRb will then always, provided a and b are not identical, denote a different proposition from bRa. That is to say, it is characteristic of a relation of two terms that it proceeds, so to speak, from one to the other. This is what may be called the sense of the relation, and is, as we shall find, the source of order and | series. It must be held as an axiom that aRb implies and is implied by a relational proposition bR'a, in which the relation R' proceeds from b to a, and may or may not be the same relation as R. But even when aRb implies and is implied by bRa, it must be strictly maintained that these are different propositions. [95|96]

The first term [for example a of aRb] is the referent, and the second term is the relatum. If aRb, then bRa is the converse of R, denoted R˘

We may distinguish the term

fromwhich the relation proceeds as thereferent, and the termtowhich it proceeds as therelatum. The sense of a relation is a fundamental notion, which is not capable of definition. The relation which holds between b andawhenever R holds betweenaand b will be called the converse of R, and will be denoted (following Schröder) by R˘. The relation of R to R˘ is the relation of oppositeness, or difference of sense; and this must not be defined (as would seem at first sight legitimate) by the above mutual implication in any single case, but only by the fact of its holding for all cases in which the given relation occurs.

[97]

Let’s skip again now to chapter 25. Consider if R means ‘is succeeded by’. This means there is a domain R of terms which are succeeded by other terms.

Let us examine the relation of a relation R to its terms x and y. In the first place, there certainly is such a relation. To be a term which has the relation R to some other term is certainly to have a relation to R, a relation which we may express as “belonging to the domain of R”. Thus if xRy, x will belong to the domain of R, and y to that of R˘ .

[212]

Now we might note another relation. We have x and we have R, which is the relation of x to its successor. We also have the domain R, which is the set of all terms that have a successor. Now we also have another relation, “belonging to domain R”. x belongs to R. Or if by R we mean just the relation, we can say that x is related to R because there is another relation x can have, namely, “having the relation R”. We would then call this relation of belonging to R (or to having R) as E.

If we express this relation between x and R, or between y and R˘ , by E, we shall have xER, yER˘.

[212]

Now, R is ‘being succeeded by’ and R˘ is ‘being the successor of’ (or the domains with this relation). We now have another relation then, ‘being the converse relation of,’ which is a relation we can call *I*. (*I *does not seem to have a converse *I˘*, since it is a relation which is equivalent both ways).

If further we express the relation of R to R˘ by I, we shall have R˘IR and RIR˘.

[212]

Thus we have xER means ‘x belongs to the domain of R’ (or x bears relation R). Consider if we wrote yEIR. This would mean ‘y belongs to the domain of R’s converse relation’. So yEIR is equivalent to yER˘.

Thus we have xER, yEIR.

[212]

Returning now to Ch.35:]]

Russell then explains how correlation is a method that allows other series to be generated from one series. He has us consider a series whose generating relation is P. [For suppose for example P is ‘succeeded by’. Then we have xPy] We also suppose that there is a one-one relation between this series and another, and we will call this one-one mapping relation R. It gets us from x to x_{R}. So we also have xRx_{R}. Thus its converse is x_{R}R˘x. So R means ‘maps from the first series to the sub-R series,’ and R˘ means ‘maps from the sub-R series to the first series.’

Correlation is a method by which, when one series is given, others may be generated. If there be any series whose generating relation is P, and any one-one relation which holds between any term x of the series and some term which we may call xR, then the class of terms xR will form a series of the same type as the class of terms x. For suppose y to be any other term of our original series, and assume xPy. Then we have x

_{R}R˘x, xPy and yRy_{R}.

[263]

Russell now seems to perform an algebreic-like operation to combine these three to get: x_{R}R˘PRy_{R}. This might mean ‘x_{R} of the sub-R series maps onto x of the first series which is succeeded by y, which maps onto y_{R} of the sub-R series.’ But since we have eliminated the middle terms, it would maybe mean something like ‘x_{R} of the sub-R series is succeeded by y_{R}, which is the correspondent to the x_{R}’s correspondent’s successor.’ So in this case, we have not just relation P [in our example the successor relation], we also have R˘PR, which I think is to be thought of as one whole relation which means [in our example] ‘is succeeded by the correspondent to the successor of the first term’s correspondent’. This means that the correspondence is order preserving.

Then we have x

_{R}R˘x, xPy and yRy_{R}.Hence x

_{R}R˘PRy_{R}.Now it may be shown that if P be transitive and asymmetrical, so is R˘ PR; hence the correlates of terms of the P-series form a series whose generating relation is R˘PR. Between these two series there is ordinal correlation, and the series have complete ordinal similarity. In this way a new series, similar to the original one, is generated by any one-one relation whose field includes the original series. It can also be shown that, conversely, if P, P' be the generating relations of two similar series, there is a one-one relation R, whose domain is the field of P, which is such that P' =R˘PR.

[263]

So there is a mathematical symmetry between the two series, and each can be thought of as derivative of the other. But not all have this symmetry. It is possible for members of one set to map one-to-one on the other, but not the other way around. For example, each letter on a text page occupies a location. So for each location in the domain of locations corresponds a letter in the domain of the alphabet. However, many letters repeat. So each of the letters maps onto multiple points in space.

§253

So classes have likeness when they correspond one-to-one in this way.

§254

A mathematical function is a relation of this sort. It relates a term from one class to another. The member in the first class is an independent variable, and it is called the referent. The member of the referred-to class is called the relatum.

But functions in math would only be useful if the referents have unique relatum and if the independent variables are in series (and thus there is an ordering principle that is preserved by the function).

For most purposes, we can identify function and relation, so “if y = f x) is equivalent to xRy, where R is a relation, it is convenient to speak of R as the function, and this will be done in what follows” [266].

§255

The dependent variables can also be independent on their own right. An object in motion passes through a number of points throughout a number of instants. To each instant corresponds a point. But those points are a series all their own as well. [268]

§256

The formulation for the function can either be true or false, if its terms are defined, or it can serve to define those terms.

§257

A series can be complete or incomplete. It is complete if all its terms involved in its generation are contained within it, but it is incomplete when the terms within it make reference to a term outside it. The cardinal integers are complete, because they all relate by means of the successor function either directly or transitively to the number 1, which is included in that set.

If R be the defining relation of a series, the series is complete when there is a term x belonging to the series, such that every other term which has to x either the relation R or the relation R˘ belongs to the series. It is connected (as was explained in Part IV) when no other terms belong to the series. Thus a complete series consists of those terms, and only those terms, which have the generating relation or its converse to some one term, together with that one term. Since the generating relation is transitive, a series which fulfils this condition for one of its terms fulfils it for all of them. A series which is connected but not complete will be called incomplete or partial. Instances of complete series are the cardinal integers, the positive and negative integers and zero, the rational numbers, the moments of time or the points on a straight line. Any selection from such a series is incomplete with respect to the generating relations of the above complete series. Thus the positive numbers are an incomplete series, and so are the rationals between 0 and 1. When a series is complete, no term can come before or after any term of the series without belonging to the series; when the series is incomplete, this is no longer the case.

[271]

Bertrand Russell. Principles of Mathematics. London/New York: Routledge, 2010 [1st published 1903].

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