Julius König

Julius König, born 16 December 1849 in Györ, Hungary,died 8 April 1913 in Budapest, was a Hungarian mathematician.

0.1 Biography

The Hungarian translation of the Latin name “Julius” is“Gyula”. But when König contributed to German mathematicaljournals, he called himself “Julius”.

Julius (Gyula) König was literary and mathematically highlygifted. He studied medicine in Vienna and, from 1868 on, inHeidelberg. After having worked, instructed by Hermann von Helmholtz,about electrical stimulation of nerves, he switched to mathematics andobtained his doctorate under the supervision of Leo Königsberger, avery famous mathematician at those times. His thesis Zur Theorieder Modulargleichungen der elliptischen Functionen covers 24pages. As a post-doc he completed his mathematical studies in Berlinattending lessons by Leopold Kronecker and Karl Weierstrass. He thenreturned to Budapest where he was appointed as a dozent at theUniversity in 1871. He became a professor at the Teacher’s College inBudapest in 1873 and, in the following year, was appointed professorat the Technical University of Budapest. He remained with theuniversity for the rest of his life. He was on three occasions Dean ofthe Engineering Faculty and also on three occasions he was Rector ofthe University. In 1889 he was elected a member of the HungarianAcademy of Sciences. In 1905 he retired but continued to give lessonson topics of his interest. His son Dénes also became a distinguishedmathematician.

0.2 Works

König worked in many mathematical fields. His work on polynomialideals, discriminants and elimination theory can be considered as alink between Leopold Kronecker and David Hilbert as well as EmmyNoether. Later on his ideas were grossly simplified. So they are onlyof historical interest today.

König already considered material influences on scientific thinkingand the mechanisms which stand behind thinking.

“The foundations of set theory are a formalization and legalizationof facts which are taken from the internal view of our consciousness,such that our ’scientific thinking’ itself is an object of scientificthinking.”

But mainly he is remembered for his contributions to and hisopposition against set theory.

0.3 König and Set Theory

One of the greatest achievements of Georg Cantor was the constructionof a one-to-one correspondence between the points of a square and thepoints of one of its edges by means of continued fractions. Königfound a simple method involving decimal numbers which had escapedCantor.

1904, on the III. international mathematical congress at HeidelbergKönig gave a talk to disprove Cantor’s continuum hypothesis. Theannouncement was a sensation and was widely reported by the press. Allsection meetings were cancelled so that everyone could hear hiscontribution.

König applied a theorem proved in the thesis of Felix Bernstein,alas this theorem was not as generally valid as Bernstein hadclaimed. Ernst Zermelo, the later editor of Cantor’s collected works,found the error already the next day. In 1905 there appeared shortnotes by Bernstein, correcting his theorem, and König, withdrawinghis claim.

Nevertheless König continued his efforts to disprove parts of settheory. In 1905 he published a paper proving that not all sets couldbe well-ordered. It is easy to show that the finitely defined elementsof the continuum form a subset of the continuum of cardinality $\\aleph$ ${}_{0}$ . The reason is that such a definition must be givencompletely by a finite number of letters and punctuation marks, only afinite number of which is available.

This statement was doubted by Cantor in a letter to Hilbert in 1906: “Infinite definitions (which are not possible in finite time) are absurdities. If König’s claim concerning the cardinality $\\aleph$ ${}_{0}$ of all ’finitely definable’ real numbers was correct, it would imply that the whole of real numbers was countable; this is most certainly wrong. Therefore König’s assumption must be in error. Am I wrong or am I right?”

Cantor was wrong. Today König’s assumption is generallyaccepted. Contrary to Cantor, presently the majority of mathematiciansconsiders undefinable numbers not as absurdities. This assumptionleads, according to König, “in a strangely simple way to the resultthat the continuum cannot get well-ordered. If we imagine the elementsof the continuum as a well-ordered set, those elements which cannot befinitely defined form a subset of that well-ordered set whichcertainly contains elements of the continuum so. Hence in thiswell-order there should be a first not finitely definable element,following after all finitely definable numbers. This isimpossible. This number has just been finitely defined by the lastsentence. The assumption that the continuum could be well-ordered hasled to a contradiction.”

Königs conclusion is not stringent. His argument rests upon a changeof language.

The last part of his life König spent working on his own approach toset theory, logic and arithmetic, which was published in 1914, oneyear after his death. When he died he had been working on the finalchapter of the book.

0.4 About König

At first Georg Cantor highly esteemed König. In a letter to PhilipJourdain in 1905 he wrote: “You certainly heard that Mr. JuliusKönig of Budapest was lead astray, by a theorem ofMr. Bernstein which in general is wrong, to give a talkat Heidelberg, on the international congress of mathematicians,opposing my theorem according to which every set, i.e., everyconsistent multitude can be assigned an aleph. Anyway, the positivecontributions from König himself are well done.” Later on Cantorchanged his attitude: “What Kronecker and his pupils as wellas Gordan have said against set theory, what König,Poincaré, and Borel have written against it, soon willbe recognized by all as a rubbish” (Letter to Hilbert,1912). “Then it will show up that Poincaré’s andKönig’s attacks against set theory are nonsense” (Letter toSchwarz, 1913).

0.5 Some Papers and Books by König

Zur Theorie der Modulargleichungen der elliptischen Functionen,Thesis, Heidelberg 1870.

Über eine reale Abbildung der s.g. Nicht-EuclidischenGeometrie, Nachrichten von der König. Gesellschaft derWissenschaften und der Georg-August-Universität zu Göttingen, No. 9(1872) 157-164.

Einleitung in die allgemeine Theorie der AlgebraischenGroessen, Leipzig 1903.

Zum Kontinuum-Problem’, Mathematische Annalen 60 (1905)177-180.

Über die Grundlagen der Mengenlehre und das Kontinuumproblem,Mathematische Annalen 61 (1905) 156-160.

Über die Grundlagen der Mengenlehre und das Kontinuumproblem(Zweite Mitteilung), Mathematische Annalen 63 (1907) 217-221.

Neue Grundlagen der Logik, Arithmetik und Mengenlehre, Leipzig1914.

0.6 Literature and Websources

Brockhaus: Die Enzyklopädie, 20th ed. vol. 12, Leipzig 1996, p. 148.

W. Burau: Dictionary of Scientific Biography vol. 7, New York 1973, p. 444.

J. J. O’Connor, E. F. Robertson: The MacTutor History of Mathematics archive.

H. Meschkowski, W. Nilson (eds.): Georg Cantor Briefe, Berlin 1991.

W. Mückenheim: Die Mathematik des Unendlichen, Aachen 2006.

B. Szénássy, History of Mathematics in Hungary until the 20th Century, Berlin 1992.

English Wikipedia: Article Julius König.

单词	JuliusKonig
释义	Julius König Julius König, born 16 December 1849 in Györ, Hungary,died 8 April 1913 in Budapest, was a Hungarian mathematician. 0.1 Biography The Hungarian translation of the Latin name “Julius” is“Gyula”. But when König contributed to German mathematicaljournals, he called himself “Julius”. Julius (Gyula) König was literary and mathematically highlygifted. He studied medicine in Vienna and, from 1868 on, inHeidelberg. After having worked, instructed by Hermann von Helmholtz,about electrical stimulation of nerves, he switched to mathematics andobtained his doctorate under the supervision of Leo Königsberger, avery famous mathematician at those times. His thesis Zur Theorieder Modulargleichungen der elliptischen Functionen covers 24pages. As a post-doc he completed his mathematical studies in Berlinattending lessons by Leopold Kronecker and Karl Weierstrass. He thenreturned to Budapest where he was appointed as a dozent at theUniversity in 1871. He became a professor at the Teacher’s College inBudapest in 1873 and, in the following year, was appointed professorat the Technical University of Budapest. He remained with theuniversity for the rest of his life. He was on three occasions Dean ofthe Engineering Faculty and also on three occasions he was Rector ofthe University. In 1889 he was elected a member of the HungarianAcademy of Sciences. In 1905 he retired but continued to give lessonson topics of his interest. His son Dénes also became a distinguishedmathematician. 0.2 Works König worked in many mathematical fields. His work on polynomialideals, discriminants and elimination theory can be considered as alink between Leopold Kronecker and David Hilbert as well as EmmyNoether. Later on his ideas were grossly simplified. So they are onlyof historical interest today. König already considered material influences on scientific thinkingand the mechanisms which stand behind thinking. “The foundations of set theory are a formalization and legalizationof facts which are taken from the internal view of our consciousness,such that our ’scientific thinking’ itself is an object of scientificthinking.” But mainly he is remembered for his contributions to and hisopposition against set theory. 0.3 König and Set Theory One of the greatest achievements of Georg Cantor was the constructionof a one-to-one correspondence between the points of a square and thepoints of one of its edges by means of continued fractions. Königfound a simple method involving decimal numbers which had escapedCantor. 1904, on the III. international mathematical congress at HeidelbergKönig gave a talk to disprove Cantor’s continuum hypothesis. Theannouncement was a sensation and was widely reported by the press. Allsection meetings were cancelled so that everyone could hear hiscontribution. König applied a theorem proved in the thesis of Felix Bernstein,alas this theorem was not as generally valid as Bernstein hadclaimed. Ernst Zermelo, the later editor of Cantor’s collected works,found the error already the next day. In 1905 there appeared shortnotes by Bernstein, correcting his theorem, and König, withdrawinghis claim. Nevertheless König continued his efforts to disprove parts of settheory. In 1905 he published a paper proving that not all sets couldbe well-ordered. It is easy to show that the finitely defined elementsof the continuum form a subset of the continuum of cardinality $\\aleph$ ${}_{0}$ . The reason is that such a definition must be givencompletely by a finite number of letters and punctuation marks, only afinite number of which is available. This statement was doubted by Cantor in a letter to Hilbert in 1906: “Infinite definitions (which are not possible in finite time) are absurdities. If König’s claim concerning the cardinality $\\aleph$ ${}_{0}$ of all ’finitely definable’ real numbers was correct, it would imply that the whole of real numbers was countable; this is most certainly wrong. Therefore König’s assumption must be in error. Am I wrong or am I right?” Cantor was wrong. Today König’s assumption is generallyaccepted. Contrary to Cantor, presently the majority of mathematiciansconsiders undefinable numbers not as absurdities. This assumptionleads, according to König, “in a strangely simple way to the resultthat the continuum cannot get well-ordered. If we imagine the elementsof the continuum as a well-ordered set, those elements which cannot befinitely defined form a subset of that well-ordered set whichcertainly contains elements of the continuum so. Hence in thiswell-order there should be a first not finitely definable element,following after all finitely definable numbers. This isimpossible. This number has just been finitely defined by the lastsentence. The assumption that the continuum could be well-ordered hasled to a contradiction.” Königs conclusion is not stringent. His argument rests upon a changeof language. The last part of his life König spent working on his own approach toset theory, logic and arithmetic, which was published in 1914, oneyear after his death. When he died he had been working on the finalchapter of the book. 0.4 About König At first Georg Cantor highly esteemed König. In a letter to PhilipJourdain in 1905 he wrote: “You certainly heard that Mr. JuliusKönig of Budapest was lead astray, by a theorem ofMr. Bernstein which in general is wrong, to give a talkat Heidelberg, on the international congress of mathematicians,opposing my theorem according to which every set, i.e., everyconsistent multitude can be assigned an aleph. Anyway, the positivecontributions from König himself are well done.” Later on Cantorchanged his attitude: “What Kronecker and his pupils as wellas Gordan have said against set theory, what König,Poincaré, and Borel have written against it, soon willbe recognized by all as a rubbish” (Letter to Hilbert,1912). “Then it will show up that Poincaré’s andKönig’s attacks against set theory are nonsense” (Letter toSchwarz, 1913). 0.5 Some Papers and Books by König Zur Theorie der Modulargleichungen der elliptischen Functionen,Thesis, Heidelberg 1870. Über eine reale Abbildung der s.g. Nicht-EuclidischenGeometrie, Nachrichten von der König. Gesellschaft derWissenschaften und der Georg-August-Universität zu Göttingen, No. 9(1872) 157-164. Einleitung in die allgemeine Theorie der AlgebraischenGroessen, Leipzig 1903. Zum Kontinuum-Problem’, Mathematische Annalen 60 (1905)177-180. Über die Grundlagen der Mengenlehre und das Kontinuumproblem,Mathematische Annalen 61 (1905) 156-160. Über die Grundlagen der Mengenlehre und das Kontinuumproblem(Zweite Mitteilung), Mathematische Annalen 63 (1907) 217-221. Neue Grundlagen der Logik, Arithmetik und Mengenlehre, Leipzig1914. 0.6 Literature and Websources Brockhaus: Die Enzyklopädie, 20th ed. vol. 12, Leipzig 1996, p. 148. W. Burau: Dictionary of Scientific Biography vol. 7, New York 1973, p. 444. J. J. O’Connor, E. F. Robertson: The MacTutor History of Mathematics archive. H. Meschkowski, W. Nilson (eds.): Georg Cantor Briefe, Berlin 1991. W. Mückenheim: Die Mathematik des Unendlichen, Aachen 2006. B. Szénássy, History of Mathematics in Hungary until the 20th Century, Berlin 1992. English Wikipedia: Article Julius König.
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