Contributions to the Founding of the Theory of Transfinite Numbers by Georg Cantor (PDF)

Description

2010 Reprint of 1915 Edition. Georg Ferdinand Ludwig Philipp Cantor was a German mathematician, best known as the inventor of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between sets, defined infinite and well-ordered sets, and proved that the real numbers are “more numerous” than the natural numbers. In fact, Cantor’s theorem implies the existence of an “infinity of infinities”. He defined the cardinal and ordinal numbers and their arithmetic. Cantor’s work is of great philosophical interest, a fact of which he was well aware. In 1895-97 Cantor fully propounded his view of continuity and the infinite, including infinite ordinals and cardinals, in his best known work, Contributions to the Founding of the Theory of Transfinite Numbers . This work contains his conception of transfinite numbers, to which he was led by his demonstration that an infinite set may be placed in a one-to-one correspondence with one of its subsets.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Cantor probably contributed more to our modern theory than any other mathematician. In my opinion, a true genius who dedicated most of his life to getting a clearer and more in-depth understanding of the topic than almost any other human. However, reading this book is for those who might enjoy reading “Euclid’s Elements.” A wonderful co-joining of a truly powerful mind and allowing insights into how they approached the topic, but definitely not a modern textbook. For a historian, it is where modern foundations of numbers were first treated. I hope you enjoy it as much as I did.

⭐I love number theory and my first argument with a mathematics teacher was when I was eight years old on the topic of inequalities, integers, and real numbers. Enjoy reading Mr. Cantor’s foundations.

⭐Dover has up ’till now done a flawless job in its Kindle transcriptions, but I had to return this particular book. The scans of Cantor’s equations and theorems are low quality blow-ups from the print edition and, what’s worse, when some are transcribed into actual text, they often get garbled into question marks. Perhaps it’s a technological problem and the kindle software just cannot parse some of the more esoteric mathematical symbols used in this publication. In any case, I advise everyone to stick with their hardcopies until Dover issues an improved edition.

⭐Bought this over the Dover edition because I wanted a hardcover, and had great experience with Fermat’s Oeuvres in reproduction. Nope. This one is a repro of a Harvard library copy that is badly marked up. Underlines in the text, check marks and scribbling in the margins. Get the Dover paperback instead.

⭐Good book.

⭐Georg Ferdinand Ludwig Philipp Cantor (1845-1918) was a German mathematician, best known as the inventor of set theory.The translator wrote in the Preface to this book, “This volume contains a translation of the two very important memoirs of Georg Cantor on transfinite numbers which appeared in the Mathematishe Annalen for 1895 and 1897 under the title, `Beiträge zur Begründung der transfiniten Mengen-lehre.’… These memoirs are the final and logically purified statement of many of the most important results of the long series of memoirs begun by Cantor in 1870… It was in these researches that the need for the transfinite numbers first showed itself, and it is only by the study of these researches that the majority of us can annihilate the feeling of arbitrariness and even insecurity about the introduction of these numbers… The philosophical revolution brought about by Cantor’s work was even greater, perhaps, than the mathematical one… mathematicians joyfully accepted, built upon, scrutinized, and perfected the foundations of Cantor’s undying theory; but very many philosophers combated it. This seems to have been because very few understood it. I hope that this book may help to make the subject better known to both philosophers and mathematicians.”He adds in the Introduction, “It is of the utmost importance to realize that, whereas until [Karl[ Weierstrass’s time such subjects as the theory of points of condensation of an infinite aggregate and the theory of irrational numbers, on which the founding of the theory of functions depends, were hardly ever investigated… Weierstrass carried research into the principles of arithmetic farther than it had been carried before. But we must also realize that there were questions, such as the nature of whole number itself, to which he made no valuable contributions. These questions… were… historically the last to be dealt with. Before this could happen, arithmetic had to receive a development, by means of Cantor’s discovery of transfinite numbers, into a theory of cardinal and ordinal numbers, both finite and transfinite, and logic had to be sharpened, as it was by Dedekind, Frege, Peano and Russell—to a great extent owing to the needs which this theory made evident.” (Pg. 22-23)”All so-called proofs of the impossibility of actually infinite numbers,” said Cantor, “are, as may be shown in every particular case and also on general grounds, false in that they begin by attributing to the numbers in question all the properties of finite numbers, whereas the infinite numbers, if they are to be thinkable in any form, must constitute quite a new kind of number as opposed to the finite numbers, and the nature of this new kind of number is dependent on the nature of things and is an object of investigation, but not of our arbitrariness or our prejudice.” (Pg. 74)The translator adds, “When Cantor said that he had solved the chief part of the problem of determining the various powers in nature, he meant that he had almost proved that the power of the arithmetical continuum is the same as the power of the ordinal numbers of the second class. In spite of the fact that Cantor firmly believed this, possibly on account of the fact that all known aggregates in the continuum had been found to be either of the first power or of the power of the continuum, the proof or disproof of this theorem has not even now been carried out, and there is come ground for believing that it cannot be carried out.” (Pg. 76)He adds in the Notes at the end of the book, “Although Frege worked out, in the first volume of his

⭐, an important part of arithmetic, with a logical accuracy previously unknown and for years afterward almost unknown, his ideas did not become at all widely known until Bertrand Russell… gave prominence to them in his

⭐of 1903. The two chief reasons in favour of this definition are that it avoids, by a construction of `numbers’ out of the fundamental entities of logic, the assumption that there are certain new and undefined entities called `numbers’; and that it allows us to deduce at once that the class defined is not empty, so that the cardinal number of u `exists’ in the sense defined in logic: in fact, since u is equivalent to itself, the cardinal number of u has u at least as a member. Russell also gave an analogous definition for ordinal types of the more general `relation numbers.'” (Pg. 202-203)If set theory is “your thing” (particularly the historical development of it), you will appreciate (and perhaps even treasure!) this book. More “casual” readers should probably avoid it, however.

⭐Georg Ferdinand Ludwig Philipp Cantor (1845-1918) was a German mathematician, best known as the inventor of set theory.The translator wrote in the Preface to this book, “This volume contains a translation of the two very important memoirs of Georg Cantor on transfinite numbers which appeared in the Mathematishe Annalen for 1895 and 1897 under the title, `Beiträge zur Begründung der transfiniten Mengen-lehre.’… These memoirs are the final and logically purified statement of many of the most important results of the long series of memoirs begun by Cantor in 1870… It was in these researches that the need for the transfinite numbers first showed itself, and it is only by the study of these researches that the majority of us can annihilate the feeling of arbitrariness and even insecurity about the introduction of these numbers… The philosophical revolution brought about by Cantor’s work was even greater, perhaps, than the mathematical one… mathematicians joyfully accepted, built upon, scrutinized, and perfected the foundations of Cantor’s undying theory; but very many philosophers combated it. This seems to have been because very few understood it. I hope that this book may help to make the subject better known to both philosophers and mathematicians.”He adds in the Introduction, “It is of the utmost importance to realize that, whereas until [Karl[ Weierstrass’s time such subjects as the theory of points of condensation of an infinite aggregate and the theory of irrational numbers, on which the founding of the theory of functions depends, were hardly ever investigated… Weierstrass carried research into the principles of arithmetic farther than it had been carried before. But we must also realize that there were questions, such as the nature of whole number itself, to which he made no valuable contributions. These questions… were… historically the last to be dealt with. Before this could happen, arithmetic had to receive a development, by means of Cantor’s discovery of transfinite numbers, into a theory of cardinal and ordinal numbers, both finite and transfinite, and logic had to be sharpened, as it was by Dedekind, Frege, Peano and Russell—to a great extent owing to the needs which this theory made evident.” (Pg. 22-23)”All so-called proofs of the impossibility of actually infinite numbers,” said Cantor, “are, as may be shown in every particular case and also on general grounds, false in that they begin by attributing to the numbers in question all the properties of finite numbers, whereas the infinite numbers, if they are to be thinkable in any form, must constitute quite a new kind of number as opposed to the finite numbers, and the nature of this new kind of number is dependent on the nature of things and is an object of investigation, but not of our arbitrariness or our prejudice.” (Pg. 74)The translator adds, “When Cantor said that he had solved the chief part of the problem of determining the various powers in nature, he meant that he had almost proved that the power of the arithmetical continuum is the same as the power of the ordinal numbers of the second class. In spite of the fact that Cantor firmly believed this, possibly on account of the fact that all known aggregates in the continuum had been found to be either of the first power or of the power of the continuum, the proof or disproof of this theorem has not even now been carried out, and there is come ground for believing that it cannot be carried out.” (Pg. 76)He adds in the Notes at the end of the book, “Although Frege worked out, in the first volume of his

⭐, an important part of arithmetic, with a logical accuracy previously unknown and for years afterward almost unknown, his ideas did not become at all widely known until Bertrand Russell… gave prominence to them in his

⭐of 1903. The two chief reasons in favour of this definition are that it avoids, by a construction of `numbers’ out of the fundamental entities of logic, the assumption that there are certain new and undefined entities called `numbers’; and that it allows us to deduce at once that the class defined is not empty, so that the cardinal number of u `exists’ in the sense defined in logic: in fact, since u is equivalent to itself, the cardinal number of u has u at least as a member. Russell also gave an analogous definition for ordinal types of the more general `relation numbers.'” (Pg. 202-203)If set theory is “your thing” (particularly the historical development of it), you will appreciate (and perhaps even treasure!) this book. More “casual” readers should probably avoid it, however.

⭐This small volume provides the original text of Cantor’s revolutionary ideas and an excellent introduction to the context into which these ideas were introduced. Personally, I believe that we are in a period where the fundamental paradigm shift that “Cantor’s paradise” has provided for Mathematics and Philosophy is not discussed, or even mentioned, enough. I believe that these ideas are every bit as important as the Calculus and that they should be on every mathematician’s shelf in some form. Why not from the horse’s mouth? Perhaps because it is a bit heavy going when there are clearer modern treatments available. Still very much worth having.

⭐集合論の濃度と順序数に関するCantorによる1895年と1897年の論文をJourdainが英訳して、82頁にわたる歴史的解説を加えたものである。整然とした議論の進め方は、原論文と知らされなければ、集合論の教科書を読んでいるように感じるほどで、完成度が高いと思う。わたくしは物理学科出身の非専門家で集合論を深く学んだことはないので、数学科の方の意見を聞きたいところである。有名な論文、よく引用される論文はめったに読まれることがないというのは通説となっている。Dover版は廉価かつ入手しやすいので、英訳とはいえ原論文に目を通すよい機会となろう。Cantorの生の考え方に触れるのは刺激的で得るところが多いに違いない。集合論を学んでいる方、すでに学んだ方にぜひ読んでいただきたい。Dedekindの論文とともに最高の古典だ。集合論は入門書でちょっとかじっただけですが、原著は堂々たる定理群とその証明から構築されていて、こういう体系を初めて作り出したのは、さすが天才だなあと感服しました。 私のような入門書レベルの頭で考えると、正整数の集合と正の偶数の集合とは、要素数が同じようにも見えるし、前者が二倍のようにも見えますが、原著は無限集合の要素数が多いとか少ないとかには触れないで、「二つの集合について、（何らかの工夫により）要素を一対一に対応させることができるとき、二つの集合は同等であるという」と定義しているだけです。入門書ではよく「無限集合では、部分が全体に等しいことがある」などと説かれていますが、原著はそんなことはいっていません。 集合論については、へたに入門書をかじるより、この本を読む方がずっといいと思います。

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