Julius König

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 mathematical
journals, he called himself “Julius”.

Julius (Gyula) König was literary and mathematically highly
gifted. He studied medicine in Vienna and, from 1868 on, in
Heidelberg. After having worked, instructed by Hermann von Helmholtz,
about electrical stimulation of nerves, he switched to mathematics and
obtained his doctorate under the supervision of Leo Königsberger, a
very famous mathematician at those times. His thesis

Zur Theorie
der Modulargleichungen der elliptischen Functionen

covers 24
pages. As a post-doc he completed his mathematical studies in Berlin
attending lessons by

Leopold Kronecker

and

Karl Weierstrass

. He then
returned to Budapest where he was appointed as a dozent at the
University in 1871. He became a professor at the Teacher’s College in
Budapest in 1873 and, in the following year, was appointed professor
at the Technical University of Budapest. He remained with the
university for the rest of his life. He was on three occasions Dean of
the Engineering Faculty and also on three occasions he was Rector of
the University. In 1889 he was elected a member of the Hungarian
Academy of Sciences. In 1905 he retired but continued to give lessons
on topics of his interest. His son Dénes also became a distinguished
mathematician.

0.2

Works

König worked in many mathematical fields. His work on

polynomial

ideals,

discriminants

and elimination theory can be considered as a
link between Leopold Kronecker and

David Hilbert

as well as

Emmy
Noether

. Later on his ideas were grossly simplified. So they are only
of historical interest today.

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

“The

foundations

of

set theory

are a formalization and legalization
of facts which are taken from the internal view of our consciousness,
such that our ’scientific thinking’ itself is an object of scientific
thinking.”

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

0.3

König and Set Theory

One of the greatest achievements of Georg Cantor was the construction
of a

one-to-one correspondence

between the points of a square and the
points of one of its edges by means of

continued fractions

. König
found a simple method involving

decimal numbers

which had escaped
Cantor.

1904, on the III. international mathematical congress at Heidelberg
König gave a talk to disprove Cantor’s

continuum hypothesis

. The
announcement was a sensation and was widely reported by the press. All

section

meetings were cancelled so that everyone could hear his
contribution.

König applied a

theorem

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

Nevertheless König continued his efforts to disprove parts of set
theory. In 1905 he published a paper proving that not all sets could
be

well-ordered

. It is easy to show that the finitely defined elements
of the

continuum

form a subset of the continuum of

cardinality

ℵ

0

. The reason is that such a definition must be given
completely by a finite number of letters and punctuation marks, only a
finite 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

ℵ

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 generally
accepted. Contrary to Cantor, presently the majority of mathematicians
considers undefinable numbers not as absurdities. This assumption
leads, according to König, “in a strangely

simple way

to the result
that the continuum cannot get well-ordered. If we imagine the elements
of the continuum as a well-ordered set, those

elements

which cannot be
finitely defined form a subset of that well-ordered set which
certainly contains elements of the continuum so. Hence in this
well-order there should be a first not finitely definable element,
following after all finitely definable numbers. This is
impossible. This number has just been finitely defined by the last

sentence

. The assumption that the continuum could be well-ordered has
led to a

contradiction

.”

Königs

conclusion

is not stringent. His

argument

rests upon a change
of

language

.

The last part of his life König spent working on his own approach to
set theory, logic and

arithmetic

, which was published in 1914, one
year after his death. When he died he had been working on the final
chapter of the book.

0.4

About König

At first Georg Cantor highly esteemed König. In a letter to Philip
Jourdain in 1905 he wrote: “You certainly heard that Mr. Julius

König

of

Budapest

was lead astray, by a theorem of
Mr.

Bernstein

which in

general is wrong

, to give a talk
at Heidelberg, on the

international congress of mathematicians

,
opposing my theorem according to which every set, i.e., every

consistent

multitude can be assigned an aleph. Anyway, the

positive

contributions from König himself are well done.” Later on Cantor
changed his attitude: “What

Kronecker

and his pupils as well
as

Gordan

have said against set theory, what

König

,

Poincaré

, and

Borel

have written against it, soon will
be recognized by

all

as a

rubbish

” (Letter to Hilbert,
1912). “Then it will show up that

Poincaré’s

and

König’s

attacks against set theory are nonsense” (Letter to
Schwarz, 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-Euclidischen
Geometrie

, Nachrichten von der König. Gesellschaft der
Wissenschaften und der Georg-August-Universität zu Göttingen, No. 9
(1872) 157-164.

Einleitung in die allgemeine Theorie der Algebraischen
Groessen

, 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

, Leipzig
1914.

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.

Title

Julius König

Canonical name

JuliusKonig

Date of creation

2013-03-22 17:02:43

Last modified on

2013-03-22 17:02:43

Owner

WM (16977)

Last modified by

WM (16977)

Numerical id

19

Author

WM (16977)

Entry type

Biography

Classification

msc 01A60

Classification

msc 01A55

Synonym

Gyula König

Synonym

König Gyula

Related topic

BijectionBetweenUnitIntervalAndUnitSquare

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