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Contributions to the Founding of the Theory of Transfinite Numbers (Dover Books on Mathematics) by Georg Cantor (1955) Paperback
Contributions to the Founding of the Theory of Transfinite Numbers by Cantor,Georg. [1955] Paperback
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Published January 1, 1955
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October 8, 2024
TWO IMPORTANT ARTICLES BY THE INVENTOR OF SET THEORY
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 some 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 'The Basic Laws of Arithmetic,' 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 'The Principles of Mathematics' 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" (particular 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 some 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 'The Basic Laws of Arithmetic,' 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 'The Principles of Mathematics' 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" (particular the historical development of it), you will appreciate (and perhaps even treasure!) this book. More "casual" readers should probably avoid it, however.
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