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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.37: Transfinite Cardinals

Brief Summary

Sets of terms can have cardinal values [being equivalent to the number of items in it.] In transfinite classes, a part of its set will have the same cardinal value as that of the whole.

Summary

§283

Our recent [ca. 1900] mathematical understanding of infinity comes from Cantor. Cantor describes transfinite ordinals and transfinite cardinals. Russell begins with cardinals.

§284

Another term for transfinite cardinals is ‘powers.’ We may first define them in such a way that we include finite cardinals. This then allows us secondly to distinguish the two.

[Russell gives Cantor’s definition which seems to indicate that the cardinal transfinite is like the amount of numbers regardless of their order].

“We call the power or cardinal number of M that general idea which, by means of our active faculty of thought, is deduced from the collection M, by | abstracting from the nature of its diverse elements and from the order in which they are given.” [307-308]

This is not sufficiently a definition however [it is more like a description].

Russell then proceeds to define cardinality without reference to number [It seems he is saying that by matching one set to another in a one-one fashion, the placing of a term in one set can serve to indicate the cardinality of the other. Or please read the following:

By means, however, of the principle of abstraction, we can give, as we saw in Part II, a formal definition of cardinal numbers. This method, in essentials, is given by Cantor immediately after the above informal definition. We have already seen that, if two classes be called similar when there is a one-one relation which couples every term of either with one and only one term of the other, then similarity is symmetrical and transitive, and is reflexive for all classes. A one-one relation, it should be observed, can be defined without any reference to number, as follows: A relation is one-one when, if x has the relation to y, and x' differs from x, y' from y, then it follows that x' does not have the relation to y, nor x to y' . In this there is no reference to number; and the definition of similarity also is therefore free from such reference. Since similarity is reflexive. transitive and symmetrical, it can be analysed into the product of a many-one relation and its converse, and indicates at least one common property of similar classes. This property, or, if there be several, a certain one of these properties, we may call the cardinal number of similar classes, and the many-one relation is that of a class to the number of its terms. In order to fix upon one definite entity as the cardinal number of a given class, we decide to identify the number of a class with the whole class of classes similar to the given class. This class, taken as a single entity, has, as the proof of the principle of abstraction shows, all the properties required of a cardinal number.

(308)

]

So similarity between sets enables us to avoid enumeration, and so this holds for finite and non-finite sets. [308-309] [Note, in his demonstration, Russell evokes an example that is similar to one Leibniz uses to illustrate the law of continuity. Russell says “the points on a given line are similar to the lines through a given point and meeting the given line”. But are there not points very far down the line which would not meet the other point with any possible line going through them, because at infinity the lines become parallel and not convergent?]

§285

Russell then examines the chief properties of cardinal numbers.

(1) If two sets of classes are similar in pairings, then their logical sums are similar.

(2) [For infinite classes:] “Again, the cardinal number of a class u is said to be greater than that of a class v, when no part of v is similar to u, but there is a part of u which is similar to v. In this case, also, the number of v is said to be less than that of u.” (309) [I do not know how a smaller class has no part similar to a larger one while the reverse is the case. Consider a set with 5 terms and a set with 4 (putting aside these enumerations). Would not 4 of the terms be similar in both cases? Perhaps he is saying this only holds for infinite classes. In that case, perhaps so long as one class has part that is similar to another, then that is enough to make them equal. As we come to see, in transfinite classes, a part of the class is similar to the whole, as the parts can be put into a one-one relation with the whole. Since there is one part that is similar, that means according to this above definition that the whole is not greater.]

(3) A transfinite class has a part which is equal to the whole. [“It is to be observed that the definition of *greater* contains a condition not required in the case of finite cardinals. If the number of v be finite, it is sufficient that a proper part of u should be similar to v. But among transfinite cardinals this is not sufficient. For the general definition of greater, therefore, both parts are necessary. This difference between finite and transfinite cardinals results from the defining difference of finite and infinite, namely that when the number of a class is not finite, it always has a proper part which is similar | to the whole; that is, every infinite class contains a part (and therefore an infinite number of parts) having the same number as itself. Certain particular cases of this proposition have long been known, and have been regarded as constituting a contradiction in the notion of infinite number. ... The proposition itself may be taken as the definition of the transfinite among cardinal numbers, for it is a property belonging to all of them, and to none of the finite cardinals.” p.309-310]

§286

Russell proceeds to show the arithmetical properties of cardinals. Addition: “The addition of numbers is defined, when they are transfinite, exactly as it was defined in the case of finite numbers, namely by means of logical addition.” [310] Multiplication: “defined by Cantor: If M and N be two classes, we can combine any element of M with any element of N to form a couple (m, n); the number of all such couples is the product of the numbers of M and N.” [310] Powers: “ The definition of powers of a number (a_{b}) is also effected logically (ib. § 4).” [p.311 see for details.]

§287

“Transfinite integers differ from finite ones, however, both in the properties of their relation to the classes of which they are the numbers, and also in regard to the properties of classes of the integers themselves.”

(1) “The number of finite numbers, it is plain, is not itself a finite number; for the class finite number is similar to the class even finite number, which is a part of itself.” [311] “The number of finite numbers, then, is transfinite. This number Cantor denotes by the Hebrew Aleph with the suffix 0; for us it will be more convenient to denote it by α_{0}. Cantor proves that this is the least of all the transfinite cardinals. This results from the following theorems (loc. cit. § 6): (A) Every transfinite collection contains others as parts whose number is α0. (B) Every transfinite collection which is part of one whose number is α0, also has the number α_{0}. (C) No finite collection is similar to any proper part of itself. | (D) Every transfinite collection is similar to some proper part of itself.”. [311-312]

(2) “From these theorems it follows that no transfinite number is less than the number of finite numbers. Collections which have this number are said to be denumerable, because it is always possible to count such collections, in the sense that, given any term of such a collection, there is some finite number n such that the given term is the nth. This is merely another way of saying that all the terms of a denumerable collection have a one-one correlation with the finite numbers, which again is equivalent to saying that the number of the collection is the same as that of the finite numbers. It is easy to see that the even numbers, the primes, the perfect squares, or any other class of finite numbers having no maximum, will form a denumerable series. For, arranging any such class in order of magnitude, there will be a finite number of terms, say n, before any given term, which will thus be the (n + 1)th term. What is more remarkable is, that all the rationals, and even all real roots of equations of a finite degree and with rational coefficients (i.e. all algebraic numbers), form a denumerable series. And even an n-dimensional series of such terms, where n is a finite number, or the smallest transfinite ordinal, is still denumerable.” [312]

(3) “All denumerable series have the same cardinal number α0, however different they may appear. But it must not be supposed that there is no number greater than α_{0}. On the contrary, there is an infinite series of such numbers.” [312]

§288

“Of the transfinite numbers other than α0, the most important is the number of the continuum. Cantor has proved that this number is not α0,* and hopes to prove that it is α1†—a hope which, though he has long cherished it, remains unfulfilled.” [312]

§289

“In what respect do the finite and transfinite cardinals together form a single series? ... There are an infinite number of infinite classes in which any given finite class is contained; and thus, by correlation with these, the number of the given finite class precedes that of any one of the infinite classes. Whether there is any other sense in which all integers, finite and transfinite, form a single series, I leave undecided” [313]

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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