by Corry Shores

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[The following is summary and quotation. All boldface, underlining, and bracketed commentary are mine. Please see the original text, as I did not follow it closely. Proofreading is incomplete, so mistakes are still present]

Bertrand Russell

Principles of Mathematics

Part 5: Infinity and Continuity

Ch.38: Transfinite Ordinals

Brief Summary

The cardinal number of a set is like its tally or total. The ordinal number might be more like the nth value of its last term. But infinite sets do not have a last term. We call their cardinal and ordinal values ‘transfinite’ in this Cantorian set theory. The natural numbers have no largest term, for example. However, they tend toward an upper limit cardinal value, which we call ** ,** or in Russell’s notation α_{0}. While this number is not in the set of natural numbers, it is the lowest term in transfinite sets, and it is the limit which still defines the natural numbers’ cardinality. Since no natural number can designate the ordinality of a set with α_{0} terms, (that is, since there is no designatable natural number which can be the ‘n’ of the nth term to which the ordering of progressive succession is tending toward), we designate that ordinal value as ω. All the natural numbers are of what is called the first class of ordinals. But a larger set, the real numbers, has the cardinal value α_{1 }and is considered the second class of ordinals. So α_{1 }is both the upper limit beyond the infinity of all natural numbers, but it is the lowest possible transfinite number. The natural numbers are infinite but countable. The real numbers are infinite but uncountable.

Summary

§290

[For transfinite cardinals, the order does not matter. But for transfinite ordinals, it does. So] “For every transfinite cardinal, or at any rate for any one of a certain class, there is an infinite collection of transfinite ordinals, although the cardinal number of all ordinals is the same as or less than that of all cardinals.” [315] We call “second class of ordinals” those ordinals which belong to the series whose cardinal is a_{0}, and “third class” to a_{1}, and so on. [The order of a set can be determined by a generating principle like the successor function, so] “The ordinal numbers are essentially classes of series, or better still, classes of generating relations of series; they are defined, for the most part, by some relation to mathematical induction.” [315] [An ordinal number indicates that there is a series of numbers of a particular order with that many terms.]

The finite ordinals, also, may be conceived as types of series: for example, the ordinal number n may be taken to mean “a serial relation of n terms;” or, in popular language, n terms in a row. This is an ordinal notion, distinct from “nth”, and logically prior to it.* In this sense, n is the name of a class of serial relations. It is this sense, not that expressed by “nth”, which is generalized by Cantor so as to apply to infinite series.

[316]

§291

[**Recall from §287****: consider a set with an infinity of terms. The ‘number’ or quantity of terms it has is called ‘transfinite’ rather than infinite.**** The smallest of all transfinite cardinal numbers is denoted aleph (subscript) 0, , which** **Russell renders as α _{0}. Then α_{1} would be the next highest infinite cardinal (however I am not sure if it is considered to be 1 greater in quantity or greater in another sense). The cardinal number is like the tally or total of the terms in the set, which in these cases is infinite. But the terms could be in a different order and still have the same cardinality. It would have a different ordinality, meaning for example that the nth term in both orderings can be a different term. Since the ordinality are the terms’ places in the series, 1st, 2nd, … nth, then we use the natural numbers as the set to which we correlate the terms’ order. All the natural numbers together have a infinite but countable cardinal value, and they are considered the first class of ordinals. But a larger class, like the real numbers, is infinite and uncountable. It is thought to have the cardinal number α_{1}, and it is considered the ‘second-class of ordinals’.]** Russell now quotes Cantor’s definition of the second class of ordinal numbers. The natural numbers does not have a largest number whose ordinal value we can assign numerically. However, on the basis of the set’s law of succession, we can still think of a new number that expresses the set’s ordinality on a while in the sense of the limit to which it this law makes the ordinality tend toward. Cantor calls this upper limit of the natural numbers’ ordinal value omega, ω. We think of ω as the first number which is larger than all those in the natural numbers. Now, if we continue the ordering law of successive unitary addition, adding one and one and one etc. to ω, we again have an infinite series which will tend toward a new higher limit, 2ω. We obtained all the numbers up to ω using the first principle of formation, but to get to 2ω we used the second principle of formation [I am not sure how it is different, since it again is the successor function. What is different is that we start not with the value of one but with the value of infinity.]

Let us begin with Cantor’s definition of the second class of ordinal numbers.

“It is now to be shown”, he says, “how we are led to the definitions of the new numbers, and in what way are obtained the natural sections, which I call classes of numbers, in the absolutely endless series of real integers. . . . The series (1) of positive real whole numbers 1, 2, 3, . . . ν, . . . arises from repeated | positing and combination of units which are presupposed and regarded as equal; the number ν is the expression both for a certain finite amount (Anzahl) of such successive positings, and for the combination of the units posited into a whole. Thus the formation of finite real whole numbers rests on the addition of a unit to a number which has already been formed; I call this moment, which, as we shall see immediately, also plays an essential part in the formation of the higher integers, the first principle of formation. The amount (Anzahl) of possible numbers ν of the class (1) is infinite, and there is no greatest among them. Thus however contradictory it would be to speak of a greatest number of the class (1), there is yet nothing objectionable in imagining a new number, which we will call ω, which is to express that the whole collection (1) is given by its law in its natural order of succession. (In the same way as ν expresses the combination of a certain finite amount of units into a whole.) It is even permissible to think of the newly created number ω as a limit, towards which the numbers ν tend, if by this nothing else is understood but that ω is the first integer which follows all the numbers ν, i.e. is to be called greater than each of the numbers ν. By allowing further additions of units to follow the positing of the number ω we obtain, by the help of the first principle of formation, the further numbers

ω + 1, ω + 2, . . . . . . . . . ω + ν, . . . . . . . . .;

Since here again we come to no greatest number, we imagine a new one, which we may call 2ω, and which is to be the first after all previous numbers ν and ω + ν.

‘The logical function which has given us the two numbers ω and 2ω is evidently different from the first principle of formation; I call it the second principle of formation of real integers, and define it more exactly as follows: If we have any determinate succession of defined real integers, among which there is no greatest number, by means of this second principle of formation a new number is created, which is regarded as the limit of those numbers, i.e. is defined as the next number greater than all of them.”

[316-317]

§292

Series made through the successor function like the natural numbers are progressions, and since no limit is implied in that generating relation, it has no upper limit. All such infinite series point to the entity or class ω.

Mathematical induction, starting from any finite ordinal, can never reach ω, since ω is not a member of the class of finite ordinals. Indeed, we may define the finite ordinals or cardinals—and where series are concerned, this seems the best definition—as those which, starting from 0 or 1, can be reached by mathematical induction.

[319]

Russell then notes that the notion of infinity in philosophy has never had a precise definition, but now it has. Finite values are reachable by means of progression, and there is no part of the series with the same cardinality as the whole. However, infinite series have parts which do share the same cardinality as the whole.

At this point, a word to the philosophers may be in season. Most of them seem to suppose that the distinction between the finite and the infinite is one whose meaning is immediately evident, and they reason on the subject as though no precise definitions were needed. But the fact is, that the distinction of the finite from the infinite is by no means easy, and has only been brought to light by modern mathematicians. The numbers 0 and 1 are capable of logical definition, and it can be shown logically that every number has a successor. We can now define finite numbers either by the fact that mathematical induction can reach them, starting from 0 or 1—in Dedekind’s language, that they form the chain of 0 or 1—or by the fact that they are the numbers of collections such that no proper part of them has the same number as the whole. These two conditions may be easily shown to be equivalent. But they alone precisely distinguish the finite from the infinite, and any discussion of infinity which neglects them must be more or less frivolous.

[319]

§293

[Now Russell says that for finite sets, different orderings still produce a set with the same ordinality, which might contradict our accounts above. Perhaps he is saying here that two finite sets with the same cardinality still have a largest (or final) term that is countable by mapping the series to the natural numbers. Russell will also say that we can have an infinite set of rational numbers with a limiting upper bound an as well an infinite set of rational numbers without one, that is, as a progression. We saw this in ch.33. Note that a finite segment, let’s say all the rational values between 1 and 2, is infinite on account of infinite divisibility. But all the rationals from 0 to positive and to negative infinity is also infinite, but has no limiting bounds contained within it. Although both are infinite, they have a different order and thus different ordinal values. I am not certain about this, but it seems the limited case has the order of divisibility rather than succession (between each value is another middle value), where the progressive case has the order of succession (to each value is a positive and negative successor for all possible increments). But please check the original to get a better interpretation. Russell further says that changing the order of any or all terms does not change the ordinality. Please see pages 319-320]

§294 & §295

Russell now explains additional and multiplication of transfinite ordinals. They obey associative but not communative law. [So grouping does not matter but order does. For details see pages 321-323]

§296

“The term ordinal number is reserved for well-ordered series” with certain properties [see pages 323 to 324 for more details on these specifications].

§297

In this section Russell discusses series that are not well ordered. He says they are important but they have less affinity to arithmetic. [See pages 324-326]

§298

“The consideration of ordinals not expressible as functions of ω shows clearly that ordinals in general are to be considered—as I suggested at the beginning of this chapter—as classes or types of serial relations, and to this view Cantor himself now apparently adheres”. [326]

§299

In this section, Russell repeats “the definitions of general notions involved in terms of what may be called relation-arithmetic” [see pages 325-326, and refer as well to Ch.29, §221]

§300

It is to be observed that the merit of the above method is that it allows no doubt as to existence-theorems—a point in which Cantor’s work leaves something to be desired. As this is an important matter, and one in which philosophers are apt to be sceptical, I shall here repeat the argument in outline. [see pages 326-327 for that argument]

§301

M. Burali-Forti “infers that of two different ordinals, as of two different cardinals, it is not necessary that one should be greater and the other less.” This contradicts one of Cantor’s theorems that Russell has discussed already, and Russell cannot find anything wrong with its proof. [327]

§302

In this section Russell discusses successive derivatives of a series.

Popularly speaking, the first derivative consists of all points in whose neighbourhood an infinite number of terms of the collection are heaped up; and subsequent derivatives give, as it were, different degrees of concentration in any neighbourhood. Thus it is easy to see why derivatives are relevant to continuity: to be continuous, a collection must be as concentrated as possible in every neighbourhood containing any terms of the collection. But such popular modes of expression are incapable of the precision which belongs to Cantor’s terminology. [329]

Sources [unless otherwise notes, all bracket page citations are from]:

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

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