I've got a reprint version of Georg Cantor's Contributions to the Foundings of the Theory of Transfinite Numbers (see http://www.maths.ed.ac.uk/~aar/papers/cantor1.pdf), but the print quality is quite poor (lots of missing dots and commas). Additionally, there's a lot of outdated terminology and notation, such as using the words "aggregate" and "part" instead of set and subset, and using $A = (M, N)$ instead of $A = M \cup N$.

Does anyone know if there exists a modern $\mathrm{\LaTeX}$ or Word rewrite of any of Cantor's Set Theory works, whether they be a single paper, several works, or a complete bibliography?

If not, I was considering doing it myself but I'm stuck as to whether I should update the terminology and notation or keep it. Furthermore, say I do make it myself, where should I publish it free of charge?

EDIT: Just to be clear, I'm not just looking for an entire collection of all of Cantor's works, just what currently exists in modern formats (e.g. $\mathrm{\LaTeX}$) and what there is left to rewrite.

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    $\begingroup$ the print quality is quite poor --- Strange to read this, because the print quality looks quite good to me. However, in case it helps for a hard-to-read part, I've included a link to a digital copy of the original 1915 printing (not the much later Dover reprint) in my answer. The Dover edition was first published in 1952. Incidentally, I got a copy of the Dover edition in the 1970s, and even then the copy I got was rather old, as the price printed on the upper right of its front cover is $1.25. $\endgroup$ Commented Dec 25, 2019 at 15:08
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    $\begingroup$ If you do end up doing this, I'd suggest that you set up semantic macros for the points where you update terminology, so that you can build both "original" and "updated" versions in parallel. $\endgroup$
    – RLH
    Commented Dec 25, 2019 at 20:10
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    $\begingroup$ Yes it is hard to read, and some of the mathematical functions can be ambiguous (parentheses left out, etc). $\endgroup$
    – john
    Commented Dec 25, 2019 at 22:22
  • $\begingroup$ @DaveL.Renfro The scan quality in the link provided is very good but the print quality of the hardcover book I got is sometimes only just barely readable. $\endgroup$
    – HarrisonO
    Commented Dec 26, 2019 at 0:03

2 Answers 2


Besides Philip E. B. Jourdain's 1915 translation of Cantor's two long 1890s papers and the translation of one of Cantor's papers in Edgar's Classics on Factrals, the following are the only English translations of Cantor's papers that I know about. I also have a personally procured translation of Cantor's review of Hermann Hankel's 1870 memoir Untersuchungen über die unendlich oft oszillierenden und unstetigen funktionen (see reference [3] here for publication details about Hankel's memoir), but this translation is not deposited anywhere on the internet.

There are several French translations of Cantor's work that were published in the 1880s, but these are fairly well known. For example, see the bibliography of Dauben's biography of Cantor. In fact, I cited a few of these French translations about 3 weeks ago in a Mathematics Stack Exchange answer.

Since the translation by Bingley [3] seems to not be very well known, I've included all of Bingley's introductory comments.

[1] William Bragg Ewald, From Kant to Hilbert: A Source Book in the Foundations of Mathematics, two volumes, Clarendon Press, 1996, xviii + 1340 pages (both volumes). reprinted in 2005

Volume II contains English translations of the following items: Cantor’s 1874 paper Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen (pp. 840-843); 1872-1882 correspondence between Cantor and Dedekind (pp. 843-878); Cantor’s 1883 booklet Grundlagen einer allgemeinen Mannigfaltigkeitslehre: Ein mathematisch-philosophischer Versuch in der Lehre des Unendlichen (pp. 881-920); Cantor’s 1892 paper Ueber eine elementare Frage der Mannigfaltigkeitslehre (pp. 920-922); 1897-1899 correspondence between Cantor and Dedekind & Hilbert (pp. 926-940).

[2] Shaughan M. Lavine, Understanding the Infinite, Harvard University Press, 1994, xii + 372 pages.

An English translation of Cantor’s 1892 paper Ueber eine elementare Frage der Mannigfaltigkeitslehre is given in Chapter IV, Appendix B, pp. 99-102.

[3] Transfinite Numbers. Three Papers on Transfinite Numbers from the Mathematische Annalen, translated by George Althoff Bingley (1888-1966), The Classics of the St. Johns Program, 1941, ii + 150 pages.

On pp. 92-150 Bingley has translated Cantor’s paper Ueber unendliche, lineare punktmannichfaltigkeiten 5 [On infinite, linear point sets 5], Mathematische Annalen 21 #4 (1883), 545-591. In Cantor’s original paper, pages 587-591 are Cantor’s “Endnotes”, which are longer footnotes put at the end of the paper. These five pages are sometimes omitted in the paging information for this paper in bibliographies. French translation (omitting the philosophical comments that make up the first half of the Mathematische Annalen version): Fondements d'une théorie générale des ensembles, Acta Mathematica 2 (December 1883), 381-408. Reprinted in Cantor’s 1932 “Collected Works” [4] (pp. 165-204 + Zermelo’s notes on pp. 204-209), and published separately as Grundlagen einer allgemeinen Mannigfaltigkeitslehre: Ein mathematisch-philosophischer Versuch in der Lehre des Unendlichen [Foundations of a General Theory of Manifolds: A Mathematical-Philosophical Essay in the Theory of the Infinite], B. G. Teubner (Leipzig), 1883, 47 pages. This 47-page separately published booklet (what Ewald translates on pp. 881-920) includes a half-page preface and 4 newly added footnotes, in addition to all the original endnotes of the Mathematische Annalen version.

Foreward [sic] (pp. i-ii): The translator offers in a single volume three papers of Georg Cantor on transfinite numbers all of which appeared in issues of the Mathematische Annalen. The first two, of volumes 46 and 49, form together a kind of text-book on transfinite numbers and were written after Cantor’s ideas had reached full maturity and had been generally accepted by mathematicians and logicians, and were no longer on the defensive. The earlier paper, of volume 21, usually called the “Grundlagen”, is an a̲p̲o̲l̲o̲g̲i̲a̲ and is indispensable if one is to know something of the genesis and development of Cantorian theory. Some of Cantor’s arguments in his own defence may now seem superfluous in a world which generally accepts them. The recent attack on Cantor by Brouwer and the intuitionalist school, however, calls for a new and thorough reassessment of the theory of point-sets. The beautiful superstructure of Cantor’s edifice cannot give us genuine pleasure if there linger doubts as to the soundness of the foundation. Cantor has been frequently defended by emiment [sic] mathematicians and logicians but in accordance with the St. John’s plan, we prefer that Cantor should speak for himself. This the “Grundlagen” in its labored and involved style, does. Also some will find in Cantor a modern phase of the age-old Platonic-Aristotelian controversy which runs through the entire list of the Great Books. The first two papers have already appeared in an English translation by Philip E. B. Jourdain. So far as the translator is aware no English translation of the “Grundlagen” has been published, although a French translation appeared some time ago in Acta Mathematica. These papers do not cover the entire range of Cantor’s ideas but are limited to that important field which best exemplifies the Cantorian point of view, that of transfinite numbers. In translating a work in a technical field it is quite impossible to please everyone in the matter of terminology. The translator in his own defence can merely say that he adopted the present terminology only after considerable thought and discussion. “Set” rather than “aggregate” seemed obviously preferable for “Menge”, for example. The difference between German rhetorical expression and English − a difference which superficially at least seems greater than that between ancient Greek and English − has made the translation of the “Grundlagen” no light task. The translation may be found to be too literal for comfortable reading but any attempt to improve Cantor’s style is attended with the danger of altering the meaning. The translator is greatly indebted to Mrs. Edward Flint Lathrop who has patiently and cheerfully cooperated in an earnest effort to keep the text reasonably free from typographical errors.

(ADDED NEXT DAY) Because many of those who study Cantor's original works rely almost entirely on the versions that appear in his 1932 collected works [4] (although in the last couple of decades this reliance is probably a lot less, since it's easy to find digitized versions of the original published versions), I thought it would be of interest to point out that there are many slight variations and even omissions between the original versions of Cantor's papers and those that appear in [4].

[4] Georg Ferdinand Ludwig Philip Cantor, Gesammelte Abhandlungen Mathematischen und Philosophischen Inhalts [Collected Papers of Mathematical and Philosophical Content], edited by Ernst Friedrich Ferdinand Zermelo, Springer, 1932, viii + 486 pages.

Some of Cantor’s papers in this 1932 collection are not reproduced in their exact original form, as noted by: Dauben (1979) [see: p. 44 (after equation 2.18) and p. 335 (note 1)]; Ferreirós (1999) [see: p. 160 (footnote 2) and p. 203 (footnote 2)]; Grattan-Guinness (1980) [see: p. 65, footnote 12]; Hallett (1984) [see: p. 5 (near bottom)]; Purkert (1989) [see: p. 52 (middle, beginning with Unfortunately, this footnote $\ldots)].$

  • $\begingroup$ I'm afraid I'm not familiar with all of these books. Can you please clarify which of Cantor's works are yet to be rewritten in LaTeX or Word for modern readers. $\endgroup$
    – HarrisonO
    Commented Dec 26, 2019 at 0:30
  • $\begingroup$ You need to get a bibliography of Cantor's works, which can be assembled by looking at several of the better known biographies and studies of his works (Dauben, Hallett, Ferreirós, Meschkowski, etc.). If doing this is too much, then I'm afraid the several years long project you have in mind is out of your reach. For what it's worth, it took about 8-10 hours just to translate and type what I think is a reasonably accurate translation of the 2 page review by Cantor I mention above, which I did with someone who could read German and who knows some basic mathematics (has a Ph.D. in philosophy). $\endgroup$ Commented Dec 26, 2019 at 7:51
  • $\begingroup$ the 2 page review by Cantor I mention above --- Actually, it's about $1/4$ this length, being roughly the right-side column of one page. $\endgroup$ Commented Dec 26, 2019 at 8:54
  • $\begingroup$ @DavidL.Renfro I'm afraid there's been a bit of a misunderstanding: I'm not looking to compile, translate, and modernise all of Cantor's works in one book. First things first, I want to rewrite Cantor's articles regarding set theory one at a time in LaTeX, if not already done so. For now, this means the two articles translated by Philip E. B. Jourdain and the third one found in [3]. Speaking of which, is there a digital or otherwise easily accessible version of [3]? Your link only shows a handful of libraries I do not have access to. $\endgroup$
    – HarrisonO
    Commented Dec 27, 2019 at 5:32
  • $\begingroup$ I have a photocopy of pp. i-ii & 92-150 of [3] (found in a university library many years ago, not sure when, but probably before the late 1990s), but I do not have a scanned version or an easy way of making one at this time. I couldn't find [3] on the internet, but then again, it seems every year or two since the late 1990s I'll find things on the internet that had not been previously available anywhere on the internet just 2 or 3 years before, so it will likely show up at some point I suppose. If it's of interest, I have personal LaTeX translations of Cantor's 1874 and 1892 papers. $\endgroup$ Commented Dec 27, 2019 at 9:24

All of Cantor's works? I doubt it.

There in one Cantor paper (translated into English) here:

Edgar, Gerald A. (ed.), Classics on fractals, Reading, MA: Addison-Wesley Publishing Company. x, 366 p. (1993). ZBL0795.28007.

It is Cantor's paper “On the power of perfect sets of points” where the "Cantor set" can be found. It does have explanations about outdated notations and terminologies.

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    $\begingroup$ Maybe of others know of other works of Cantor done in modern language, they could be added as additional answers. $\endgroup$
    – GEdgar
    Commented Dec 25, 2019 at 12:05
  • $\begingroup$ what is your take about updating the terminology and notation? Seems an interesting question per se. $\endgroup$
    – lcv
    Commented Dec 25, 2019 at 12:47
  • $\begingroup$ My approach was: keep the original terminology and notation, but explain it in footnotes and commentary for a modern reader. That continued through the whole volume. For example: I note at a certain point in history the switch from $AB$ to $A \cap B$ for intersection of sets. $\endgroup$
    – GEdgar
    Commented Dec 25, 2019 at 15:46

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