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271:. His mother was Caroline Henriette Dedekind (née Emperius), the daughter of a professor at the Collegium. Richard Dedekind had three older siblings. As an adult, he never used the names Julius Wilhelm. He was born in Braunschweig (often called "Brunswick" in English), which is where he lived most of his life and died. His body rests at
983:
443:
defines all the nonnegative numbers whose squares are less than 2 and the negative numbers into the lesser class, and the positive numbers whose squares are greater than 2 into the greater class. Every location on the number line continuum contains either a rational or an irrational number. Thus
380:(Institute of Technology) in 1862, Dedekind returned to his native Braunschweig, where he spent the rest of his life, teaching at the Institute. He retired in 1894, but did occasional teaching and continued to publish. He never married, instead living with his sister Julia.
444:
there are no empty locations, gaps, or discontinuities. Dedekind published his thoughts on irrational numbers and
Dedekind cuts in his pamphlet "Stetigkeit und irrationale Zahlen" ("Continuity and irrational numbers"); in modern terminology,
581:
Although the book is assuredly based on
Dirichlet's lectures, and although Dedekind himself referred to the book throughout his life as Dirichlet's, the book itself was entirely written by Dedekind, for the most part after Dirichlet's
1939:
694:. Thus began an enduring relationship of mutual respect, and Dedekind became one of the first mathematicians to admire Cantor's work concerning infinite sets, proving a valued ally in Cantor's disputes with
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407:
1979:
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1989:
1954:
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was still teaching, although mostly at an elementary level, and
Dedekind became his last student. Dedekind received his doctorate in 1852, for a thesis titled
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1944:
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1969:
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1984:
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439:), with all the numbers of one class (greater) being strictly greater than all the numbers of the other (lesser) class. For example, the
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509: 1 2 3 4 5 6 7 8 9 10 ...
140:
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313:, was the main facility for mathematical research in Germany. Thus Dedekind went to Berlin for two years of study, where he and
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632:. (Thus Dedekind can be said to have been Kummer's most important disciple.) In an 1882 article, Dedekind and
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339:, and they became good friends. Because of lingering weaknesses in his mathematical knowledge, he studied
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651:("What are numbers and what are they good for?" Ewald 1996: 790), which included his definition of an
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of the secondary literature on
Dedekind. Also consult Stillwell's "Introduction" to Dedekind (1996).
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coefficients. The concept underwent further development in the hands of
Hilbert and, especially, of
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351:. About this time, he became one of the first people to understand the importance of the notion of
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465:: a set is infinite when it is "similar to a proper part of itself," in modern terminology, is
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302:"). This thesis did not display the talent evident in Dedekind's subsequent publications.
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966:, published by Open Court Publishing Company, translated by Wooster Woodruff Beman.
922:, published by Open Court Publishing Company, translated by Wooster Woodruff Beman.
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Labyrinth of
Thought: A history of set theory and its role in modern mathematics
1004:
The
Mystery of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity
461:
between them. He invoked similarity to give the first precise definition of an
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173:
152:
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427:), now a standard definition of the real numbers. The idea of a cut is that an
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989:
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605:
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242:
896:
Ewald, William B., ed. (1996) "Continuity and irrational numbers", p. 766 in
559:. Dedekind's study of Lejeune Dirichlet's work led him to his later study of
278:
He first attended the
Collegium Carolinum in 1848 before transferring to the
1646:
1609:
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1458:
659:
foundation for the natural numbers, whose primitive notions were the number
383:
Dedekind was elected to the
Academies of Berlin (1880) and Rome, and to the
371:
533:
332:
251:
156:
729:. Beman, W. W., ed. and trans. Dover. Contains English translations of
596:
included supplements introducing the notion of an ideal, fundamental to
263:
Dedekind's father was Julius Levin Ulrich
Dedekind, an administrator of
17:
613:
356:
108:
1671:
1493:
898:
From Kant to Hilbert: A Source Book in the Foundations of Mathematics
758:
From Kant to Hilbert: A Source Book in the Foundations of Mathematics
470:
763:
1854. "On the introduction of new functions in mathematics," 754â61.
577:("Lectures on Number Theory") about which it has been written that:
766:
1872. "Continuity and irrational numbers," 765â78. (translation of
1543:
1303:
917:. Dover. 1963 . Part III, Paragraph 32 – via Google Books â.
656:
406:
797:(Complete mathematical works, Vol. 1â3). Retrieved 5 August 2009.
387:(1900). He received honorary doctorates from the universities of
374:(now ETH ZĂŒrich). When the Collegium Carolinum was upgraded to a
961:. Dover. 1963 . Part V, Paragraph 64 – via Google Books â.
1243:
1080:. Vol. 4. Detroit: Charles Scribner's Sons. pp. 1â5.
1074:
Biermann, Kurt-R (2008). "Dedekind, (Julius Wilhelm) Richard".
347:. Yet he was also the first at Göttingen to lecture concerning
241:. He is also considered a pioneer in the development of modern
457:
Dedekind defined two sets to be "similar" when there exists a
1239:
1007:, Pocket Books nonfiction, Simon and Schuster, p. 102,
1232:
Dedekind's Contributions to the Foundations of Mathematics
1143:
Frege, Dedekind, and Peano on the foundations of arithmetic
604:, does not appear in Dedekind's work.) Dedekind defined an
1940:
Academic staff of the Technical University of Braunschweig
784:
1872â82, 1899. Correspondence with Cantor, 843â77, 930â40.
682:. For instance, around 1900, he wrote the first papers on
749:, ed. and trans. Cambridge Uni. Press. A translation of
671:, citing Dedekind, formulated an equivalent but simpler
567:. In 1863, he published Lejeune Dirichlet's lectures on
1234:
http://plato.stanford.edu/entries/dedekind-foundations/
321:
in 1854. Dedekind returned to Göttingen to teach as a
419:
school, Dedekind developed the notion now known as a
628:, devised as part of Kummer's 1843 attempt to prove
517: 1 4 9 16 25 36 49 64 81 100 ...
1824:
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233:. His best known contribution is the definition of
183:
169:
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114:
104:
82:
57:
41:
1116:
415:While teaching calculus for the first time at the
213:; 6 October 1831 â 12 February 1916) was a German
520:Dedekind's work in this area anticipated that of
1035:Richard Dedekind. Gesammelte mathematische Werke
751:Ăber die Theorie der ganzen algebraischen Zahlen
317:were contemporaries; they were both awarded the
647:In 1888, he published a short monograph titled
579:
532:anticipated later works by major proponents of
30:"Dedekind" redirects here. For other uses, see
720:A Source Book in Mathematical Logic, 1879â1931
698:, who was philosophically opposed to Cantor's
1950:Academic staff of the University of Göttingen
1255:
1224:Open Court Publishing Company, Chicago, 1901.
1042:Bulletin of the American Mathematical Society
608:as a subset of a set of numbers, composed of
8:
524:, who is commonly considered the founder of
481:can be shown to be similar to the subset of
1160:The Search for Mathematical Roots 1870â1940
1077:Complete Dictionary of Scientific Biography
876:. Cambridge University Press. p. 196.
808:List of things named after Richard Dedekind
1262:
1248:
1240:
1104:, 1983, "Dedekind's invention of ideals,"
49:
38:
1980:Members of the French Academy of Sciences
1054:
775:What are numbers and what should they be?
296:Ăber die Theorie der Eulerschen Integrale
600:. (The word "Ring", introduced later by
1191:MacTutor History of Mathematics Archive
864:
612:that satisfy polynomial equations with
547:Dedekind edited the collected works of
1152:. Basel: BirkhÀuser, chap. 3, 4 and 7.
1990:Mathematicians from the German Empire
957:"The Nature and Meaning of Numbers".
913:"The Nature and Meaning of Numbers".
678:Dedekind made other contributions to
528:. Likewise, his contributions to the
208:
7:
1955:Humboldt University of Berlin alumni
282:in 1850. There, Dedekind was taught
217:who made important contributions to
978:Was sind und was sollen die Zahlen?
900:, 2 vols. Oxford University Press.
737:Was sind und was sollen die Zahlen?
649:Was sind und was sollen die Zahlen?
640:, giving an algebraic proof of the
513:
231:axiomatic foundations of arithmetic
1975:People from the Duchy of Brunswick
1930:20th-century German mathematicians
1920:19th-century German mathematicians
1211:Works by or about Richard Dedekind
592:The 1879 and 1894 editions of the
366:In 1858, he began teaching at the
25:
1145:. Assen, Netherlands: Van Gorcum.
732:Stetigkeit und irrationale Zahlen
1925:19th-century German philosophers
1293:
1222:Essays on the Theory of Numbers.
932:Moore, G.H. (17 November 1982).
714:1890. "Letter to Keferstein" in
27:German mathematician (1831â1916)
1056:10.1090/S0002-9904-1933-05535-0
959:Essays on the Theory of Numbers
915:Essays on the Theory of Numbers
727:Essays on the Theory of Numbers
710:Primary literature in English:
686:. In 1872, while on holiday in
202:Julius Wilhelm Richard Dedekind
1945:University of Göttingen alumni
795:Gesammelte mathematische Werke
790:Primary literature in German:
756:Ewald, William B., ed., 1996.
722:. Harvard Univ. Press: 98â103.
574:Vorlesungen ĂŒber Zahlentheorie
337:Peter Gustav Lejeune Dirichlet
335:. He studied for a while with
1:
760:, 2 vols. Oxford Uni. Press.
1970:Scientists from Braunschweig
1935:Academic staff of ETH Zurich
743:Theory of Algebraic Integers
1985:Philosophers of mathematics
1141:Gillies, Douglas A., 1982.
1125:University of Chicago Press
777:, 787â832. (translation of
2006:
1760:von NeumannâBernaysâGödel
1106:Bull. London Math. Soc. 15
530:foundations of mathematics
385:French Academy of Sciences
273:Braunschweig Main Cemetery
29:
1561:One-to-one correspondence
1291:
1202:Works by Richard Dedekind
976:Richard Dedekind (1888).
934:Zermeloâs Axiom of Choice
874:Remarkable Mathematicians
675:, now the standard ones.
459:one-to-one correspondence
247:philosophy of mathematics
195:
178:Philosophy of mathematics
162:
48:
1196:University of St Andrews
1960:German number theorists
1162:. Princeton Uni. Press.
1148:Ferreirós, José, 2007.
1001:Aczel, Amir D. (2001),
980:. Braunschweig: Vieweg.
561:algebraic number fields
280:University of Göttingen
149:Algebraic number theory
123:University of Göttingen
1519:Constructible universe
1339:Constructibility (V=L)
936:. New York: Springer.
848:Dedekind zeta function
655:. He also proposed an
590:
485:whose members are the
412:
237:through the notion of
1742:Principia Mathematica
1576:Transfinite induction
1435:(i.e. set difference)
1156:Ivor Grattan-Guinness
982:Online available at:
838:Dedekind psi function
828:Dedekind-infinite set
823:Dedekind eta function
634:Heinrich Martin Weber
630:Fermat's Last Theorem
411:Dedekind, before 1886
410:
377:Technische Hochschule
210:[ËdeËdÉËkÉȘnt]
137:Dedekind-Peano axioms
1816:Burali-Forti paradox
1571:Set-builder notation
1524:Continuum hypothesis
1464:Symmetric difference
1182:Robertson, Edmund F.
872:James, Ioan (2002).
642:RiemannâRoch theorem
620:. Ideals generalize
327:, giving courses on
307:University of Berlin
190:Carl Friedrich Gauss
1777:TarskiâGrothendieck
1220:Dedekind, Richard,
1180:O'Connor, John J.;
1168:online bibliography
853:Ideal (ring theory)
716:Jean van Heijenoort
700:transfinite numbers
622:Ernst Eduard Kummer
489:of every member of
298:("On the Theory of
265:Collegium Carolinum
119:Collegium Carolinum
1366:Limitation of size
1186:"Richard Dedekind"
665:successor function
636:applied ideals to
610:algebraic integers
435:into two classes (
413:
305:At that time, the
300:Eulerian integrals
141:Dedekind's theorem
76:Duchy of Brunswick
32:Dedekind (surname)
1897:
1896:
1806:Russell's paradox
1755:ZermeloâFraenkel
1656:Dedekind-infinite
1529:Diagonal argument
1428:Cartesian product
1285:Set (mathematics)
1206:Project Gutenberg
1119:The First Moderns
1087:978-0-684-31559-1
943:978-0-387-90670-6
883:978-0-521-52094-2
696:Leopold Kronecker
667:. The next year,
549:Lejeune Dirichlet
429:irrational number
345:abelian functions
199:
198:
164:Scientific career
16:(Redirected from
1997:
1879:Bertrand Russell
1869:John von Neumann
1854:Abraham Fraenkel
1849:Richard Dedekind
1811:Suslin's problem
1722:Cantor's theorem
1439:De Morgan's laws
1297:
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1228:Internet Archive
1215:Internet Archive
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441:square root of 2
433:rational numbers
315:Bernhard Riemann
223:abstract algebra
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185:Doctoral advisor
145:Abstract algebra
89:
86:12 February 1916
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779:Was sind und...
747:Stillwell, John
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479:natural numbers
473:. Thus the set
446:VollstÀndigkeit
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1689:
1687:
1684:
1682:
1679:
1677:
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1668:
1665:
1663:
1660:
1657:
1653:
1650:
1648:
1645:
1641:
1638:
1636:
1633:
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1620:
1616:
1613:
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1601:
1598:
1597:
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1588:
1582:
1579:
1577:
1574:
1572:
1569:
1567:
1564:
1562:
1559:
1557:
1554:
1552:
1549:
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1532:
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1527:
1525:
1522:
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1517:
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1440:
1437:
1434:
1431:
1429:
1426:
1425:
1423:
1421:
1417:
1409:
1408:specification
1406:
1404:
1401:
1400:
1399:
1396:
1395:
1392:
1389:
1387:
1384:
1382:
1379:
1377:
1374:
1372:
1369:
1367:
1364:
1362:
1359:
1357:
1354:
1350:
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1345:
1342:
1340:
1337:
1333:
1330:
1328:
1325:
1323:
1320:
1319:
1318:
1315:
1313:
1310:
1309:
1307:
1305:
1301:
1296:
1286:
1283:
1282:
1280:
1276:
1272:
1265:
1260:
1258:
1253:
1251:
1246:
1245:
1242:
1235:
1231:
1229:
1225:
1223:
1218:
1216:
1212:
1209:
1207:
1203:
1200:
1197:
1193:
1192:
1187:
1183:
1178:
1177:
1173:
1171:
1169:
1161:
1157:
1154:
1151:
1147:
1144:
1140:
1136:
1134:0-226-22480-5
1130:
1126:
1121:
1120:
1114:
1110:
1107:
1103:
1100:
1099:
1095:
1089:
1083:
1079:
1078:
1072:
1071:
1067:
1057:
1052:
1048:
1044:
1043:
1038:
1036:
1030:
1024:
1021:
1016:
1014:9780743422994
1010:
1006:
1005:
997:
994:
991:
988:
985:
979:
972:
969:
965:
960:
953:
950:
945:
939:
935:
928:
925:
921:
916:
909:
906:
903:
899:
893:
890:
885:
879:
875:
868:
865:
858:
854:
851:
849:
846:
844:
841:
839:
836:
834:
831:
829:
826:
824:
821:
819:
816:
814:
811:
809:
806:
805:
801:
796:
793:
792:
791:
783:
780:
776:
772:
769:
768:Stetigkeit...
765:
762:
761:
759:
755:
752:
748:
744:
740:
738:
734:
733:
728:
725:1963 (1901).
724:
721:
717:
713:
712:
711:
705:
703:
701:
697:
693:
689:
685:
681:
676:
674:
673:set of axioms
670:
666:
662:
658:
654:
650:
645:
643:
639:
635:
631:
627:
626:ideal numbers
623:
619:
615:
611:
607:
603:
599:
595:
587:Edwards, 1983
583:
578:
576:
575:
570:
569:number theory
566:
562:
558:
554:
550:
545:
543:
539:
538:Gottlob Frege
535:
531:
527:
523:
516:
508:
504:
502:
496:
492:
488:
484:
480:
476:
472:
468:
464:
460:
455:
453:
452:
447:
442:
438:
434:
430:
426:
422:
418:
409:
402:
400:
398:
394:
390:
386:
381:
379:
378:
373:
369:
364:
362:
358:
354:
350:
349:Galois theory
346:
342:
338:
334:
330:
326:
325:
320:
316:
312:
308:
303:
301:
297:
293:
289:
286:by professor
285:
284:number theory
281:
276:
274:
270:
266:
258:
256:
254:
253:
248:
244:
240:
236:
232:
228:
224:
220:
219:number theory
216:
215:mathematician
211:
203:
194:
191:
188:
186:
182:
179:
175:
172:
168:
165:
161:
158:
154:
150:
146:
142:
138:
134:
131:
127:
124:
120:
117:
113:
110:
107:
103:
99:
98:German Empire
95:
85:
81:
77:
73:
60:
56:
52:
47:
40:
37:
33:
19:
1848:
1839:Georg Cantor
1834:Paul Bernays
1765:MorseâKelley
1740:
1673:
1672:Subset
1619:hereditarily
1581:Venn diagram
1539:ordered pair
1454:Intersection
1398:Axiom schema
1221:
1189:
1166:There is an
1165:
1159:
1149:
1142:
1118:
1105:
1075:
1046:
1040:
1034:
1023:
1003:
996:
977:
971:
964:1901 edition
958:
952:
933:
927:
920:1901 edition
914:
908:
897:
892:
873:
867:
843:Dedekind sum
813:Dedekind cut
789:
778:
774:
767:
757:
750:
742:
736:
730:
726:
719:
709:
706:Bibliography
692:Georg Cantor
677:
653:infinite set
648:
646:
618:Emmy Noether
593:
591:
580:
572:
546:
522:Georg Cantor
519:
514:
506:
500:
494:
490:
482:
474:
467:equinumerous
463:infinite set
456:
451:completeness
449:
445:
431:divides the
424:
421:Dedekind cut
414:
397:Braunschweig
382:
375:
365:
324:Privatdozent
322:
319:habilitation
304:
295:
288:Moritz Stern
277:
269:Braunschweig
262:
250:
239:Dedekind cut
235:real numbers
201:
200:
163:
153:Real numbers
133:Dedekind cut
94:Braunschweig
88:(1916-02-12)
72:Braunschweig
36:
1965:Algebraists
1915:1916 deaths
1910:1831 births
1864:Thomas Jech
1707:Alternative
1686:Uncountable
1640:Ultrafilter
1499:Cardinality
1403:replacement
1344:Determinacy
1123:. Chicago:
1029:Bell, E. T.
598:ring theory
594:Vorlesungen
417:Polytechnic
368:Polytechnic
329:probability
245:and of the
229:), and the
227:ring theory
174:Mathematics
105:Nationality
1904:Categories
1859:Kurt Gödel
1844:Paul Cohen
1681:Transitive
1449:Identities
1433:Complement
1420:Operations
1381:Regularity
1349:projective
1312:Adjunction
1271:Set theory
1068:References
688:Interlaken
536:, such as
526:set theory
370:school in
361:arithmetic
243:set theory
64:1831-10-06
1792:Paradoxes
1712:Axiomatic
1691:Universal
1667:Singleton
1662:Recursive
1605:Countable
1600:Amorphous
1459:Power set
1376:Power set
1327:dependent
1322:countable
1049:: 16â17.
902:full text
657:axiomatic
423:(German:
311:Göttingen
249:known as
1796:Problems
1700:Theories
1676:Superset
1652:Infinite
1481:Concepts
1361:Infinity
1278:Overview
1158:, 2000.
1115:(1998).
1031:(1933).
802:See also
718:, 1967.
663:and the
585:â
534:logicism
341:elliptic
333:geometry
252:logicism
157:Logicism
18:Dedekind
1734:General
1729:Zermelo
1635:subbase
1617: (
1556:Forcing
1534:Element
1506: (
1484:Methods
1371:Pairing
1226:at the
1213:at the
1108:: 8â17.
680:algebra
614:integer
602:Hilbert
557:Riemann
487:squares
425:Schnitt
357:algebra
206:German:
1625:Filter
1615:Finite
1551:Family
1494:Almost
1332:global
1317:Choice
1304:Axioms
1131:
1084:
1011:
940:
880:
773:1888.
741:1996.
582:death.
565:ideals
555:, and
395:, and
393:Zurich
372:ZĂŒrich
353:groups
309:, not
170:Fields
109:German
1717:Naive
1647:Fuzzy
1610:Empty
1593:types
1544:tuple
1514:Class
1508:large
1469:Union
1386:Union
984:MPIWG
859:Notes
606:ideal
553:Gauss
292:Gauss
1630:base
1129:ISBN
1082:ISBN
1009:ISBN
938:ISBN
878:ISBN
735:and
563:and
540:and
437:sets
403:Work
389:Oslo
359:and
355:for
343:and
331:and
259:Life
83:Died
58:Born
1591:Set
1204:at
1051:doi
990:UBS
987:GDZ
661:one
624:'s
571:as
503:):
493:, (
477:of
267:in
1906::
1194:,
1188:,
1184:,
1127:.
1047:39
1045:.
1039:.
745:.
702:.
644:.
551:,
544:.
511:â
498:â
454:.
448:,
399:.
391:,
363:.
290:.
275:.
255:.
221:,
96:,
74:,
1674:·
1658:)
1654:(
1621:)
1510:)
1263:e
1256:t
1249:v
1236:.
1137:.
1090:.
1059:.
1053::
1037:"
1018:.
946:.
886:.
781:)
770:)
753:.
515:N
507:N
501:N
495:N
491:N
483:N
475:N
204:(
66:)
62:(
34:.
20:)
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