42:
1117:
391:
employed
Bernays to assist him with his investigations of the foundation of arithmetic. Bernays also lectured on other areas of mathematics at the University of Göttingen. In 1918, that university awarded him a second habilitation for a thesis on the axiomatics of the
410:
1739:
450:) published in 1934 and 1939, which is discussed in Sieg and Ravaglia (2005). A proof in this work that a sufficiently strong consistent theory cannot contain its own reference
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492:
1734:
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715:
41:
1784:
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638:
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160:Über die Darstellung von positiven, ganzen Zahlen durch die primitiven, binären quadratischen Formen einer nicht-quadratischen Diskriminante
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429:
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455:
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1217:
1180:
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982:
Sieg, Wilfried; Ravaglia, Mark (2005), "Chapter 77. David
Hilbert and Paul Bernays, Grundlagen der Mathematik", in
865:
425:
405:
In 1922, Göttingen appointed
Bernays extraordinary professor without tenure. His most successful student there was
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1160:
1148:
1143:
463:
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467:(republished in MĂĽller 1976), Bernays set out an axiomatic set theory whose starting point was a related theory
1076:
1047:
720:
266:
Bernays was born into a distinguished German-Jewish family of scholars and businessmen. His great-grandfather,
274:
1688:
1606:
1481:
1433:
1247:
1170:
267:
194:
1764:
1640:
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1333:
1153:
476:
393:
349:
1556:
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1390:
1370:
1348:
1053:
983:
472:
398:
353:
1729:
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1454:
1385:
1338:
1278:
1165:
357:
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123:
109:
1625:
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1449:
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1439:
1253:
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1133:
1069:
1033:
706:
369:
220:
1548:
1543:
1328:
1283:
1190:
1037:
945:
882:
765:
757:
710:
680:
480:
294:
240:
119:
960:, Studies in Logic and the Foundations of Mathematics, vol. 84, Amsterdam: North-Holland,
1405:
1242:
1234:
1205:
1175:
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999:
961:
929:
634:
608:
567:
519:
484:
210:
1693:
1683:
1668:
1663:
1531:
1185:
991:
919:
874:
853:
829:
749:
672:
577:
529:
468:
365:
175:
1013:
975:
941:
648:
622:
589:
541:
1562:
1500:
1318:
1138:
1009:
971:
937:
644:
618:
585:
581:
563:
537:
533:
515:
406:
206:
593:
545:
377:
376:. Bernays was Privatdozent at the University of Zurich, 1912–1917, where he came to know
190:
738:"Completeness Before Post: Bernays, Hilbert, and the Development of Propositional Logic"
1698:
1495:
1476:
1380:
1365:
1322:
1258:
1200:
995:
897:. Van Nostrand. Dover reprint, 2001. A gentle introduction to some of the ideas in the
338:
306:
66:
1718:
1703:
1673:
1505:
1419:
1414:
949:
905:
684:
562:, Die Grundlehren der mathematischen Wissenschaften, vol. 50, Berlin, New York:
553:
514:, Die Grundlehren der mathematischen Wissenschaften, vol. 40, Berlin, New York:
505:
488:
388:
381:
373:
322:
318:
286:
255:
202:
198:
180:
886:
834:
769:
416:
After working privately for
Hilbert for six months, Bernays and his family moved to
1653:
1466:
1395:
1353:
1212:
1116:
733:
361:
326:
17:
607:, Studies in Logic and the Foundations of Mathematics, Amsterdam: North-Holland,
602:
1678:
1313:
417:
330:
302:
298:
282:
236:
144:
99:
88:
1658:
1526:
1429:
1092:
421:
348:
awarded him a Ph.D. in mathematics for a thesis, supervised by Landau, on the
310:
247:
1026:
933:
878:
1461:
1424:
1375:
1273:
783:
84:
334:
857:
440:
Bernays's collaboration with
Hilbert culminated in the two volume work,
924:
676:
451:
761:
413:
in 1933, the university fired
Bernays because of his Jewish ancestry.
1486:
1308:
737:
656:
151:
62:
420:, whose nationality he had inherited from his father, and where the
753:
471:
had set out in the 1920s. Von
Neumann's theory took the notions of
1358:
1125:
244:
1065:
1061:
633:(in German), Darmstadt: Wissenschaftliche Buchgesellschaft,
479:
as primitive. Bernays recast von
Neumann's theory so that
411:
Law for the
Restoration of the Professional Civil Service
487:
were primitive. Bernays's theory, with modifications by
461:
In seven papers, published between 1937 and 1954 in the
273:
Bernays spent his childhood in Berlin, and attended the
270:, served as chief rabbi of Hamburg from 1821 to 1849.
895:
Mathematical Logic and the
Foundation of Mathematics
239:
mathematician who made significant contributions to
1639:
1602:
1514:
1404:
1292:
1233:
1124:
1099:
988:
Landmark writings in western mathematics 1640--1940
216:
186:
174:
150:
140:
115:
105:
95:
73:
48:
32:
659:Zum Entscheidungsproblem der mathematischen Logik
254:. He was an assistant and close collaborator of
911:Zeitschrift fĂĽr allgemeine Wissenschaftstheorie
1740:Institute for Advanced Study visiting scholars
990:, Elsevier B. V., Amsterdam, pp. 981–99,
424:employed him on occasion. He also visited the
1077:
958:Sets and classes. On the work by Paul Bernays
8:
235:(17 October 1888 – 18 September 1977) was a
631:Abhandlungen zur Philosophie der Mathematik
1084:
1070:
1062:
655:Bernays, Paul; Schonfinkel, Moses (1928),
40:
29:
923:
833:
815:, Volume I. By D. Hilbert and P. Bernays"
1043:MacTutor History of Mathematics Archive
716:MacTutor History of Mathematics Archive
697:
908:(1978), "Paul Bernays (1888--1977)",
7:
493:von Neumann–Bernays–Gödel set theory
380:. His collected communications with
428:and was a visiting scholar at the
25:
432:in 1935–36 and again in 1959–60.
409:. After Nazi Germany enacted the
1115:
996:10.1016/B978-044450871-3/50158-3
1735:20th-century Swiss philosophers
1055:Paul Bernays: A Short Biography
835:10.1090/s0002-9904-1935-06048-3
317:, he studied mathematics under
281:, he studied mathematics under
27:Swiss mathematician (1888–1977)
1:
956:MĂĽller, Gert H., ed. (1976),
559:Grundlagen der Mathematik. II
1785:Academic staff of ETH Zurich
788:Institute for Advanced Study
511:Grundlagen der Mathematik. I
430:Institute for Advanced Study
1760:Philosophers of mathematics
217:Other notable students
1801:
1575:von Neumann–Bernays–Gödel
893:Kneebone, Geoffrey, 1963.
866:Bulletin of Symbolic Logic
809:MacLane, Saunders (1935).
742:Bulletin of Symbolic Logic
448:Foundations of Mathematics
426:University of Pennsylvania
333:; physics under Voigt and
1376:One-to-one correspondence
1113:
899:Grundlagen der Mathematik
813:Grundlagen der Mathematik
464:Journal of Symbolic Logic
443:Grundlagen der Mathematik
291:Ferdinand Georg Frobenius
252:philosophy of mathematics
226:
133:
128:Philosophy of mathematics
39:
1048:University of St Andrews
858:"Bernays and Set Theory"
721:University of St Andrews
556:; Bernays, Paul (1939),
508:; Bernays, Paul (1934),
1027:Hilbert Bernays Project
456:Hilbert–Bernays paradox
337:; and philosophy under
315:University of Göttingen
268:Isaac ben Jacob Bernays
1755:Mathematical logicians
1334:Constructible universe
1161:Constructibility (V=L)
984:Grattan-Guinness, Ivor
879:10.2178/bsl/1231081769
629:Bernays, Paul (1976),
601:Bernays, Paul (1958),
394:propositional calculus
356:. That same year, the
354:binary quadratic forms
350:analytic number theory
1557:Principia Mathematica
1391:Transfinite induction
1250:(i.e. set difference)
822:Bull. Amer. Math. Soc
665:Mathematische Annalen
399:Principia Mathematica
1775:Swiss mathematicians
1770:Swiss Ashkenazi Jews
1631:Burali-Forti paradox
1386:Set-builder notation
1339:Continuum hypothesis
1279:Symmetric difference
1034:Robertson, Edmund F.
711:"Paul Isaac Bernays"
707:Robertson, Edmund F.
604:Axiomatic Set Theory
358:University of Zurich
346:University of Berlin
309:; and physics under
279:University of Berlin
277:, 1895–1907. At the
275:Köllnische Gymnasium
195:Julius Richard BĂĽchi
124:Axiomatic set theory
110:University of Berlin
1745:Jewish philosophers
1592:Tarski–Grothendieck
1032:O'Connor, John J.;
705:O'Connor, John J.;
384:span many decades.
372:. The examiner was
297:; philosophy under
1780:Swiss philosophers
1181:Limitation of size
925:10.1007/BF01801939
677:10.1007/BF01459101
387:Starting in 1917,
295:Friedrich Schottky
241:mathematical logic
233:Paul Isaac Bernays
120:Mathematical logic
18:Paul Isaac Bernays
1750:Jewish scientists
1712:
1711:
1621:Russell's paradox
1570:Zermelo–Fraenkel
1471:Dedekind-infinite
1344:Diagonal argument
1243:Cartesian product
1107:Set (mathematics)
1005:978-0-444-50871-3
967:978-0-444-10907-1
854:Kanamori, Akihiro
640:978-3-534-06706-0
614:978-0-486-66637-2
573:978-3-540-05110-7
525:978-3-540-04134-4
436:Mathematical work
230:
229:
211:Saunders Mac Lane
187:Doctoral students
135:Scientific career
77:18 September 1977
16:(Redirected from
1792:
1694:Bertrand Russell
1684:John von Neumann
1669:Abraham Fraenkel
1664:Richard Dedekind
1626:Suslin's problem
1537:Cantor's theorem
1254:De Morgan's laws
1119:
1086:
1079:
1072:
1063:
1050:
1016:
978:
952:
927:
889:
862:
840:
839:
837:
819:
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724:
723:
702:
687:
651:
625:
597:
592:, archived from
549:
544:, archived from
469:John von Neumann
454:is known as the
370:Picard's theorem
366:complex analysis
364:for a thesis on
176:Doctoral advisor
165:
80:
58:
56:
44:
30:
21:
1800:
1799:
1795:
1794:
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1790:
1789:
1715:
1714:
1713:
1708:
1635:
1614:
1598:
1563:New Foundations
1510:
1400:
1319:Cardinal number
1302:
1288:
1229:
1120:
1111:
1095:
1090:
1031:
1023:
1006:
981:
968:
955:
904:
860:
852:
849:
844:
843:
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807:
803:
793:
791:
782:
781:
777:
732:
731:
727:
704:
703:
699:
694:
671:(99): 342–372,
654:
641:
628:
615:
600:
574:
564:Springer-Verlag
552:
526:
516:Springer-Verlag
504:
501:
438:
407:Gerhard Gentzen
264:
209:
207:Gerhard Gentzen
205:
201:
197:
193:
170:
163:
126:
122:
106:Alma mater
91:
82:
78:
69:
60:
59:17 October 1888
54:
52:
35:
28:
23:
22:
15:
12:
11:
5:
1798:
1796:
1788:
1787:
1782:
1777:
1772:
1767:
1762:
1757:
1752:
1747:
1742:
1737:
1732:
1727:
1717:
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1710:
1709:
1707:
1706:
1701:
1699:Thoralf Skolem
1696:
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1613:
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1609:
1603:
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1597:
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1584:
1583:
1582:
1567:
1566:
1565:
1553:
1552:
1551:
1540:
1539:
1534:
1529:
1524:
1518:
1516:
1512:
1511:
1509:
1508:
1503:
1498:
1493:
1484:
1479:
1474:
1464:
1459:
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1457:
1452:
1447:
1437:
1427:
1422:
1417:
1411:
1409:
1402:
1401:
1399:
1398:
1393:
1388:
1383:
1381:Ordinal number
1378:
1373:
1368:
1363:
1362:
1361:
1356:
1346:
1341:
1336:
1331:
1326:
1316:
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1297:
1293:
1290:
1289:
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1286:
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1276:
1271:
1266:
1261:
1259:Disjoint union
1256:
1251:
1245:
1239:
1237:
1231:
1230:
1228:
1227:
1226:
1225:
1220:
1209:
1208:
1206:Martin's axiom
1203:
1198:
1193:
1188:
1183:
1178:
1173:
1171:Extensionality
1168:
1163:
1158:
1157:
1156:
1151:
1146:
1136:
1130:
1128:
1122:
1121:
1114:
1112:
1110:
1109:
1103:
1101:
1097:
1096:
1091:
1089:
1088:
1081:
1074:
1066:
1060:
1059:
1051:
1038:"Paul Bernays"
1029:
1022:
1021:External links
1019:
1018:
1017:
1004:
979:
966:
953:
906:Lauener, Henri
902:
891:
848:
845:
842:
841:
828:(3): 162–165.
801:
784:"Paul Bernays"
775:
754:10.2307/421184
725:
696:
695:
693:
690:
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688:
652:
639:
626:
613:
598:
572:
554:Hilbert, David
550:
524:
506:Hilbert, David
500:
497:
491:, is known as
437:
434:
339:Leonard Nelson
307:Ernst Cassirer
263:
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81:(aged 88)
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67:United Kingdom
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1765:Set theorists
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1267:
1265:
1262:
1260:
1257:
1255:
1252:
1249:
1246:
1244:
1241:
1240:
1238:
1236:
1232:
1224:
1223:specification
1221:
1219:
1216:
1215:
1214:
1211:
1210:
1207:
1204:
1202:
1199:
1197:
1194:
1192:
1189:
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1179:
1177:
1174:
1172:
1169:
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1159:
1155:
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1123:
1118:
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1102:
1098:
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1087:
1082:
1080:
1075:
1073:
1068:
1067:
1064:
1058:
1056:
1052:
1049:
1045:
1044:
1039:
1035:
1030:
1028:
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1024:
1020:
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1001:
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993:
989:
985:
980:
977:
973:
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959:
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951:
947:
943:
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931:
926:
921:
917:
913:
912:
907:
903:
900:
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888:
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872:
868:
867:
859:
855:
851:
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836:
831:
827:
823:
816:
814:
805:
802:
789:
785:
779:
776:
771:
767:
763:
759:
755:
751:
748:(3): 331–66.
747:
743:
739:
735:
734:Zach, Richard
729:
726:
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718:
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712:
708:
701:
698:
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596:on 2011-05-17
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548:on 2011-05-17
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502:
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466:
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459:
457:
453:
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445:
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435:
433:
431:
427:
423:
419:
414:
412:
408:
403:
401:
400:
395:
390:
389:David Hilbert
385:
383:
379:
375:
374:Ernst Zermelo
371:
367:
363:
359:
355:
351:
347:
344:In 1912, the
342:
340:
336:
332:
328:
324:
323:Edmund Landau
320:
319:David Hilbert
316:
312:
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304:
300:
296:
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288:
287:Edmund Landau
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1654:Georg Cantor
1649:Paul Bernays
1648:
1580:Morse–Kelley
1555:
1488:
1487:Subset
1434:hereditarily
1396:Venn diagram
1354:ordered pair
1269:Intersection
1213:Axiom schema
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378:George PĂłlya
362:habilitation
360:awarded him
343:
327:Hermann Weyl
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191:Corrado Böhm
159:
134:
79:(1977-09-18)
34:Paul Bernays
1730:1977 deaths
1725:1888 births
1679:Thomas Jech
1522:Alternative
1501:Uncountable
1455:Ultrafilter
1314:Cardinality
1218:replacement
1166:Determinacy
418:Switzerland
331:Felix Klein
303:Carl Stumpf
299:Alois Riehl
283:Issai Schur
145:Mathematics
96:Nationality
89:Switzerland
1719:Categories
1674:Kurt Gödel
1659:Paul Cohen
1496:Transitive
1264:Identities
1248:Complement
1235:Operations
1196:Regularity
1134:Adjunction
1093:Set theory
847:References
582:65.0021.02
534:60.0017.02
489:Kurt Gödel
446:(English:
422:ETH Zurich
382:Kurt Gödel
311:Max Planck
250:, and the
248:set theory
55:1888-10-17
1607:Paradoxes
1527:Axiomatic
1506:Universal
1482:Singleton
1477:Recursive
1420:Countable
1415:Amorphous
1274:Power set
1191:Power set
1149:dependent
1144:countable
950:147959212
934:0044-2216
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811:"Review:
685:122312654
313:. At the
262:Biography
245:axiomatic
1611:Problems
1515:Theories
1491:Superset
1467:Infinite
1296:Concepts
1176:Infinity
1100:Overview
887:15567244
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477:argument
473:function
335:Max Born
221:Hao Wang
1549:General
1544:Zermelo
1450:subbase
1432: (
1371:Forcing
1349:Element
1321: (
1299:Methods
1186:Pairing
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986:(ed.),
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942:0546580
794:July 1,
649:0444417
623:0106178
590:0272596
542:0237246
481:classes
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1430:Finite
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883:S2CID
861:(PDF)
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237:Swiss
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1445:base
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