Contributions to the Founding of the Theory of Transfinite Numbers

by Georg Cantor

One of the greatest mathematical classics of all time, this work established a new field of mathematics which was to be of incalculable importance in topology, number theory, analysis, theory of functions, etc., as well as in the entire field of modern logic. It is rare that a theory of such fundamental mathematical importance is expressed so simply and the reader with a good grasp of college mathematics will be able to understand most of the basic ideas and many of the proofs.
Cantor first develops the elementary definitions and operations of cardinal and ordinal numbers and analyzes the concepts of "canlinality" and "ordinality." He covers such topics as the addition, multiplication, and exponentiation of cardinal numbers, the smallest transfinite cardinal number, the ordinal types of simply ordered aggregates, operations on ordinal types, the ordinal type of the linear continuum, and others. He then develops a theory of well-ordered aggregates, and investigates the ordinal numbers of well-ordered aggregates and the properties and extent of the transfinite ordinal numbers.
An 82-page introduction by the eminent mathematical historian Philip E. B. Jourdain first sketches the background of Cantor's theory, discussing the contributions of such predecessors as Veicrstrass, Cauchy, Dedekind, Dirichlet, Riemann, Fourier, and Hankel; it then traces the development of the theory by summarizing and analyzing Cantor's earlier work. A bibliographical note provides information on further investigations in the theory of transfinite numbers by Frege, Peano, Whitehead, Russell, etc.
"Would serve as well as any modern text to initiate a student in this exciting branch of mathematics." — Mathematical Gazette.

Genres: Mathematics, Science, Nonfiction, Philosophy, Classics, History, Logic

224 pages, Paperback

First published January 1, 1915

Reviews

[M · Rating 5 out of 5]

“Consider for a moment what is called the 'history' of the sciences, for example, that of mathematics: a continual dissolution of the definitions of mathe­matical objects through new imaginations which not only extend the totality constituted by these objects to new beings, but completely modify the nature of mathematics... This progress is in time what, in the space of imperialism, the extension of frontiers is to the empire: displacement of a border (of an anteriority), beyond which, it is agreed, there is nothing to hear… This moment can be described as Caesarism and the exploita­tion of barbarous frontiersmen; this would be to forget the moment of madness, when Lobatchevsky says, ‘I do geometry without recourse to the Euclidean postulate of the parallel', when Cantor says, 'I include infinity amongst the operational numbers'. These moments are not ones of permanence, but of discontinuity, not ones of inhibition, but of delirium assumed and carried to its end. They do not reduce the unknown to the known, they make everything one thought one knew unstable in proportion to what one used to know, for a moment one means to speak like a barbarian on the agora; these moments are to science what Beethoven's late quartets are to harmony.”
—Lyotard, Libidinal Economy

You and your friend and their friend and his friend and her friend, etc., walk into a bar, one at a time, pausing briefly to scratch a tally mark on an infinitely large wall as you pass through. Your friend remarks on the unclear meaning of “infinitely large” in this instance: “So is this a progressively growing wall? Is it a potential infinite? An actual infinite? Is it a completed infinite? I sure hope not, or we’ll never actually make it past this thing and into the bar itself. But if it’s a progressive infinite, what’s the rate of its growth? What can such a thing mean?”

A Hegelian pipes up: “This bad infinite is in itself the same as the perpetual ought; it is indeed the negation of the finite, but in truth it is unable to free itself from it; the finite constantly resurfaces in it as its other, since this infinite only is with reference to the finite, which is its other. The progress to infinity is therefore only repetitious monotony, the one and the same tedious alternation of this finite and infinite.”

“Uh, okay,” your friend replies with a snort. “Okay, obscurantist. Okay then. Sure, yeah. Okay then.”

—

Have you ever heard that the repeating decimal 0.999... = 1 before? A lot of people find that fact surprising, suspicious even, and are often a little sceptical of the proofs offered in support of the claim, which usually look something like this:

Master: The decimal representation of 1/3 is 0.333...
Scholar: Why?
Master: Imagine you’ve got a cake and you wish to slice it into perfect thirds. First you might cut it into ten pieces, take three for yourself and give two sets of three away. You’ve got one piece left, so you cut that into ten pieces and repeat the procedure. If you do this over and over again, you’ll find you’ve got 0.3 slices for yourself, then 0.33, then 0.333, and you approach closer and closer to a perfect third as you add more and more threes to the end of your decimal representation. I could make this rigorous, but epsilon-delta definitions are beyond you, I suspect.
Scholar: So what’s this I hear about 0.999... being equal to 1?
Master: Well, let 0.999... = x. Then 10x = 9.999... and 10x - x = 9x = 9. Manifestly, then, x = 1, which was what was to be proved.
Scholar: I’m beginning to think this is a trivial and uninteresting consequence of the properties of limits.
Master: You want something nontrivial and interesting, do you?

—

Take the nonnegative integers, starting from 0, and count up by 1 per second for some number of seconds = n. Neglecting senescence, boredom, laryngeal stress, the heat death of the universe, etc., it is manifestly always possible to spend another second counting up, rather than stopping after n seconds. This tempting little n+1 is like an ever-receding horizon, and no matter how long you count, you’ll only ever have counted for a finite number of seconds—a finite n.

Still, we might ask—if we measure the number of elements (Cantor calls this the “power” or “cardinal number” of a set) in our set of integers as we count up, where a set with n elements has a cardinality of n—what does the cardinal number of our set of integers approach as we add more and more integers? Not a finite integer, no, for you could always add another 1—the size of this set is the first (and least) transfinite cardinal number: aleph-null.

I know. Yeah. The smallest infinity. Implication of larger infinities. I know. It’s absurd. It’s brilliant. Just thinking about it makes me ill. It’s genius.

Imagine, if you can (you can’t), a collection of all the sets of aleph-null size. What is the size of this set? Well, it’s not aleph-null. It can’t be. It’s a bigger infinity than that, which Cantor calls aleph-one. The explanation of this is involved enough that I can’t come up with a clever analogy, and don’t even get me started on transfinite ordinals.

If you know about this already from more modern educational resources, there might be something a bit uninteresting about this idea of differently sized infinities, but there’s something about going back and reading Cantor that brings the ridiculousness of the whole thing to life. Combine with revisiting some of Kant’s antinomies for additional, agonising angst.

[Nick Black · Rating 3 out of 5]

2008-11-18. Picked up at full price at the ogreish GT Barnes & Noble (which is already playing accursed christmas music, aieeee) to celebrate wafflestomping a cs6262 exam this morning, hurrah! I'm gonna go sit on the porch until it's time for cs6290 and read this straightaway....good stuff, but nothing you wouldn't already know from undergraduate mathematics education. It's nice to read source material that's modern enough to be understood (contrast with Gauss or Newton's works, which I've tried and horribly failed to work through).

[Alexander Robert · Rating 5 out of 5]

A great source text, and a refreshing mathematical text in general. A must read for any interested in concepts of infinity.

[William Schram · Rating 5 out of 5]

Infinity is a concept. It isn't a set number; it merely implies a number greater than any other.

Georg Cantor was ahead of his time. I checked Wikipedia and found that mathematicians now call transfinite numbers infinite numbers. Due to the names in the book, I will call them transfinite, though.

The first part of the book discusses Cantor's predecessors and contemporaries. A mathematical historian named Philip E B Jourdain wrote that part. The idea is rigor. Fourier discovered the Fourier series, trigonometric functions with value to physics and other fields of knowledge. Eventually, there was a split between Dirichlet and Cauchy about what was more significant to equations.

The second part starts with a paper by Georg Cantor on explaining and manipulating sets. He writes lucidly. I understand what he is discussing, and I never took any mathematics with set theory. Either I bombarded my brain with enough information for the idea to click, or Cantor is a skilled writer. He discusses Aleph-zero and other concepts related to transfinite cardinal and ordinal numbers. Cantor offers proof for all of his statements.

The third and final part of the book is another paper on transfinite numbers by Georg Cantor. Cantor states that there are different types of infinity. Among sets of infinity, some are larger than others.

The book is fascinating. It is short enough to read in one sitting, but the content might confound you at first. Thanks for reading my review, and see you next time.

[Evan Ford · Rating 5 out of 5]

In the era of German Philosophy from Kant all the way, sadly to Hitler and Heidegger, this is just as pivotal a discovery as Pythagoras or Xeno in Ancient Greece. The majority of the work is the tedium of getting to exactly what Cantor is implying in the discovery or transfinite numbers which is altogether a difficult read throughout it, but that is where the brilliant underscores of how they shine at the beginning of modern and onto post-modern mathematics which, Cantor would be the father of.

[Poet for fun · Rating 3 out of 5]

No entendí nada, ja, ja. Hacía tiempo que no me ocurría (quizá la única situación análoga que recuerdo sea Ética de Spinoza)... Hablando desde la ignorancia, que siempre lo deja a uno en ridículo, eso de inventarse variables aleatoriamente, acompañadas de condiciones propicias autoimpuestas y de signos para representarlas con el único fin de forzar el argumento hacia el resultado esperado pues... Así hasta yo demuestro que rojo es azul y viceversa, je, je.