408:
1073:
981:
33:
475:, was not accepted by all mathematicians, mostly because the axiom of choice was a paradigm of non-constructive mathematics. In 1908, Zermelo succeeded in producing an improved proof making use of Dedekind's notion of the "chain" of a set, which became more widely accepted; this was mainly because that same year he also offered an
525:
Without considering external forces such as current and wind, the optimal control is for the boat to always head towards D. Its path then is a line segment from O to D, which is trivially optimal. With consideration of current and wind, if the combined force applied to the boat is non-zero, the
319:
521:
problem. The problem deals with a boat navigating on a body of water, originating from a point O to a destination point D. The boat is capable of a certain maximum speed, and we want to derive the best possible control to reach D in the least possible time.
327:
1730:
482:
Zermelo began to axiomatize set theory in 1905; in 1908, he published his results despite his failure to prove the consistency of his axiomatic system. See the article on
1745:
805:
1740:
223:
1720:
1710:
574:, Schriften der Mathematisch-Naturwissenschaftlichen Klasse der Heidelberger Akademie der Wissenschaften, vol. 23, Berlin: Springer-Verlag,
540:, Schriften der Mathematisch-Naturwissenschaftlichen Klasse der Heidelberger Akademie der Wissenschaften, vol. 21, Berlin: Springer-Verlag,
1530:
451:
Zermelo began to work on the problems of set theory under
Hilbert's influence and in 1902 published his first work concerning the addition of
1725:
432:, a list of 23 unsolved fundamental questions which mathematicians should attack during the coming century. The first of these, a problem of
332:
1755:
1695:
1012:
631:
1930. "On boundary numbers and domains of sets: new investigations in the foundations of set theory" in Ewald, William B., ed., 1996.
1690:
709:
421:
1001:
1735:
962:
750:
587:
553:
1219:
1039:
162:
459:. In 1904, he succeeded in taking the first step suggested by Hilbert towards the continuum hypothesis when he proved the
1547:
916:
Zermelo, Ernst (1929). "Die
Berechnung der Turnier-Ergebnisse als ein Maximumproblem der Wahrscheinlichkeitsrechnung".
141:
828:
699:
307:
1700:
1525:
514:
498:
299:
126:
121:
1405:
1299:
1178:
467:). This result brought fame to Zermelo, who was appointed Professor in Göttingen, in 1905. His proof of the
1750:
1705:
1542:
1535:
1173:
1136:
674:
295:
1190:
624:
1913. "On an
Application of Set Theory to the Theory of the Game of Chess" in Rasmusen E., ed., 2001.
378:
1224:
1116:
1104:
1099:
918:
861:
1032:
1006:
361:. He finished his doctorate in 1894 at the University of Berlin, awarded for a dissertation on the
282:
1644:
1562:
1437:
1389:
1203:
1126:
996:
840:
738:
636:
381:, at that time the leading centre for mathematical research in the world, where he completed his
572:
Ernst
Zermeloâcollected works. Vol. II. Calculus of variations, applied mathematics, and physics
190:
1715:
1596:
1477:
1289:
1109:
856:
393:
362:
358:
486:
for an outline of this paper, together with the original axioms, with the original numbering.
411:
Ernst
Zermelo tombstone in Friedhof GĂŒnterstal, in GĂŒnterstal district of Freiburg im Breisgau
306:. Furthermore, his 1929 work on ranking chess players is the first description of a model for
1512:
1482:
1426:
1346:
1326:
1304:
726:
536:
Zermelo, Ernst (2013), Ebbinghaus, Heinz-Dieter; Fraser, Craig G.; Kanamori, Akihiro (eds.),
468:
460:
445:
429:
303:
228:
136:
1685:
1680:
1586:
1576:
1410:
1341:
1294:
1234:
1121:
788:
704:
694:
597:
563:
502:
437:
389:
350:
103:
78:
310:
that continues to have a profound impact on various applied fields utilizing this method.
8:
1581:
1492:
1400:
1395:
1209:
1151:
1089:
1025:
992:
603:
444:
in 1878, and in the course of its statement
Hilbert mentioned also the need to prove the
382:
354:
1504:
1499:
1284:
1239:
1146:
935:
890:
882:
792:
483:
116:
844:
1361:
1198:
1161:
1131:
1062:
958:
939:
796:
746:
669:
583:
549:
369:). Zermelo remained at the University of Berlin, where he was appointed assistant to
388:
In 1910, Zermelo left Göttingen upon being appointed to the chair of mathematics at
1649:
1639:
1624:
1619:
1487:
1141:
985:
927:
894:
874:
866:
820:
776:
760:
575:
541:
490:
452:
248:
197:
764:
1518:
1456:
1274:
1094:
784:
689:
684:
664:
593:
559:
518:
472:
456:
404:
and at his request, Zermelo was reinstated to his honorary position in
Freiburg.
210:
131:
497:
independently improved
Zermelo's axiom system. The resulting system, now called
407:
1654:
1451:
1432:
1336:
1321:
1278:
1214:
1156:
679:
494:
476:
579:
545:
1674:
1461:
1375:
1370:
780:
425:
374:
291:
205:
93:
60:
1629:
1609:
1604:
1422:
1351:
1309:
1168:
1072:
824:
570:
Zermelo, Ernst (2013), Ebbinghaus, Heinz-Dieter; Kanamori, Akihiro (eds.),
441:
401:
397:
82:
392:, which he resigned in 1916. He was appointed to an honorary chair at the
1634:
1269:
338:
180:
1614:
1385:
1048:
931:
433:
370:
346:
326:
Ernst
Zermelo graduated from Berlin's LuisenstÀdtisches Gymnasium (now
886:
878:
32:
1417:
1380:
1331:
1229:
648:
Zermelo's Axiom of Choice, Its
Origins, Development, & Influence,
633:
From Kant to Hilbert: A Source Book in the Foundations of Mathematics
870:
713:
619:
1908. "Investigations in the foundations of set theory I," 199â215.
608:
From Frege to Gödel: A Source Book in Mathematical Logic, 1879â1931
526:
control for no current and wind does not yield the optimal path.
342:
318:
616:
1908. "A new proof of the possibility of well-ordering," 183â98.
1442:
1264:
56:
538:
Ernst Zermeloâcollected works. Vol. I. Set theory, miscellanea
1314:
1081:
1017:
957:. Providence: American Mathematical Society. pp. 36â37.
396:
in 1926, which he resigned in 1935 because he disapproved of
287:
652:
Studies in the History of Mathematics and Physical Sciences,
254:
269:
263:
286:; 27 July 1871 – 21 May 1953) was a German
613:
1904. "Proof that every set can be well-ordered," 139â41.
428:
challenged the mathematical community with his famous
251:
455:. By that time he had also discovered the so-called
266:
1731:
Martin Luther University of Halle-Wittenberg alumni
838:
260:
257:
990:
1672:
725:
743:Ernst Zermelo: An Approach to His Life and Work
508:
501:(ZF), is now the most commonly used system for
806:"Zermelo and the Early History of Game Theory"
803:
1746:Academic staff of the University of Göttingen
1033:
16:German logician and mathematician (1871â1953)
1741:Academic staff of the University of Freiburg
294:, whose work has major implications for the
731:The Search for Mathematical Roots 1870â1940
1721:Academic staff of the University of Zurich
1040:
1026:
737:
31:
952:
860:
415:
373:, under whose guidance he began to study
298:. He is known for his role in developing
759:
422:International Congress of Mathematicians
420:In 1900, in the Paris conference of the
406:
317:
1711:People from the Province of Brandenburg
1002:MacTutor History of Mathematics Archive
915:
804:Schwalbe, Ulrich; Walker, Paul (2001).
569:
535:
1673:
471:, based on the powerset axiom and the
1021:
367:Untersuchungen zur Variationsrechnung
300:ZermeloâFraenkel axiomatic set theory
281:
1726:Humboldt University of Berlin alumni
650:Gregory H. Moore, being Volume 8 of
13:
1756:20th-century German mathematicians
1696:19th-century German mathematicians
14:
1767:
975:
626:Readings in Games and Information
243:Ernst Friedrich Ferdinand Zermelo
1691:20th-century German philosophers
1071:
845:"Zermelo and the Skolem Paradox"
654:Springer Verlag, New York, 1982.
322:Ernst Zermelo in Freiburg (1953)
247:
163:AckermannâTeubner Memorial Award
982:Works by or about Ernst Zermelo
529:
377:. In 1897, Zermelo went to the
142:Zermelo's theorem (game theory)
1047:
946:
909:
849:The Bulletin of Symbolic Logic
769:The Bulletin of Symbolic Logic
328:Heinrich-Schliemann-Oberschule
154:Gertrud Seekamp (1944 â death)
1:
1736:University of Freiburg alumni
733:. Princeton University Press.
719:
465:every set can be well ordered
955:Set Theory and Metric Spaces
902:
515:Zermelo's navigation problem
509:Zermelo's navigation problem
127:Zermelo's navigation problem
7:
813:Games and Economic Behavior
675:Axiom of limitation of size
658:
337:) in 1889. He then studied
122:ZermeloâFraenkel set theory
10:
1772:
1531:von NeumannâBernaysâGödel
953:Kaplansky, Irving (2020).
296:foundations of mathematics
37:Ernst Zermelo in the 1900s
1595:
1558:
1470:
1360:
1332:One-to-one correspondence
1248:
1189:
1080:
1069:
1055:
919:Mathematische Zeitschrift
628:, Wiley-Blackwell: 79â82.
580:10.1007/978-3-540-70856-8
546:10.1007/978-3-540-79384-7
400:'s regime. At the end of
236:
219:
196:
186:
176:
169:
158:
150:
109:
99:
89:
67:
42:
30:
23:
1007:University of St Andrews
841:Ebbinghaus, Heinz-Dieter
765:"Zermelo and set theory"
739:Ebbinghaus, Heinz-Dieter
637:Oxford University Press
610:. Harvard Univ. Press.
499:ZermeloâFraenkel axioms
379:University of Göttingen
313:
1701:Mathematical logicians
1290:Constructible universe
1117:Constructibility (V=L)
825:10.1006/game.2000.0794
781:10.2178/bsl/1102083759
727:Grattan-Guinness, Ivor
513:Proposed in 1931, the
416:Research in set theory
412:
394:University of Freiburg
363:calculus of variations
359:University of Freiburg
323:
1513:Principia Mathematica
1347:Transfinite induction
1206:(i.e. set difference)
469:well-ordering theorem
461:well-ordering theorem
453:transfinite cardinals
446:well-ordering theorem
410:
321:
304:well-ordering theorem
302:and his proof of the
283:[tsÉÉÌŻËmeËlo]
1587:Burali-Forti paradox
1342:Set-builder notation
1295:Continuum hypothesis
1235:Symmetric difference
993:Robertson, Edmund F.
705:Von Neumann universe
695:Cumulative hierarchy
503:axiomatic set theory
438:continuum hypothesis
351:University of Berlin
191:University of ZĂŒrich
104:University of Berlin
79:Freiburg im Breisgau
1751:German male writers
1706:Writers from Berlin
1548:TarskiâGrothendieck
991:O'Connor, John J.;
700:Pairwise comparison
604:Jean van Heijenoort
383:habilitation thesis
355:University of Halle
308:pairwise comparison
1137:Limitation of size
1013:Zermelo Navigation
932:10.1007/BF01180541
484:Zermelo set theory
430:Hilbert's problems
413:
324:
224:Stefan Straszewicz
117:Zermelo set theory
1668:
1667:
1577:Russell's paradox
1526:ZermeloâFraenkel
1427:Dedekind-infinite
1300:Diagonal argument
1199:Cartesian product
1063:Set (mathematics)
964:978-1-4704-6384-7
839:Van Dalen, Dirk;
761:Kanamori, Akihiro
752:978-3-642-08050-0
670:Axiom of infinity
643:Works by others:
589:978-3-540-70855-1
555:978-3-540-79383-0
390:Zurich University
240:
239:
220:Doctoral students
171:Scientific career
137:Zermelo's theorem
1763:
1650:Bertrand Russell
1640:John von Neumann
1625:Abraham Fraenkel
1620:Richard Dedekind
1582:Suslin's problem
1493:Cantor's theorem
1210:De Morgan's laws
1075:
1042:
1035:
1028:
1019:
1018:
1009:
986:Internet Archive
969:
968:
950:
944:
943:
913:
898:
864:
835:
834:on 1 April 2017.
833:
827:. Archived from
810:
800:
756:
734:
600:
566:
491:Abraham Fraenkel
336:
285:
280:
276:
275:
272:
271:
268:
265:
262:
259:
256:
253:
232:
198:Doctoral advisor
74:
52:
50:
35:
21:
20:
1771:
1770:
1766:
1765:
1764:
1762:
1761:
1760:
1671:
1670:
1669:
1664:
1591:
1570:
1554:
1519:New Foundations
1466:
1356:
1275:Cardinal number
1258:
1244:
1185:
1076:
1067:
1051:
1046:
997:"Ernst Zermelo"
978:
973:
972:
965:
951:
947:
914:
910:
905:
862:10.1.1.137.3354
831:
808:
753:
722:
690:Choice function
685:Boltzmann brain
665:Axiom of choice
661:
590:
556:
532:
519:optimal control
511:
479:of set theory.
473:axiom of choice
457:Russell paradox
418:
330:
316:
278:
250:
246:
226:
215:
211:Hermann Schwarz
146:
132:Zermelo ordinal
100:Alma mater
85:
76:
72:
63:
54:
48:
46:
38:
26:
17:
12:
11:
5:
1769:
1759:
1758:
1753:
1748:
1743:
1738:
1733:
1728:
1723:
1718:
1713:
1708:
1703:
1698:
1693:
1688:
1683:
1666:
1665:
1663:
1662:
1657:
1655:Thoralf Skolem
1652:
1647:
1642:
1637:
1632:
1627:
1622:
1617:
1612:
1607:
1601:
1599:
1593:
1592:
1590:
1589:
1584:
1579:
1573:
1571:
1569:
1568:
1565:
1559:
1556:
1555:
1553:
1552:
1551:
1550:
1545:
1540:
1539:
1538:
1523:
1522:
1521:
1509:
1508:
1507:
1496:
1495:
1490:
1485:
1480:
1474:
1472:
1468:
1467:
1465:
1464:
1459:
1454:
1449:
1440:
1435:
1430:
1420:
1415:
1414:
1413:
1408:
1403:
1393:
1383:
1378:
1373:
1367:
1365:
1358:
1357:
1355:
1354:
1349:
1344:
1339:
1337:Ordinal number
1334:
1329:
1324:
1319:
1318:
1317:
1312:
1302:
1297:
1292:
1287:
1282:
1272:
1267:
1261:
1259:
1257:
1256:
1253:
1249:
1246:
1245:
1243:
1242:
1237:
1232:
1227:
1222:
1217:
1215:Disjoint union
1212:
1207:
1201:
1195:
1193:
1187:
1186:
1184:
1183:
1182:
1181:
1176:
1165:
1164:
1162:Martin's axiom
1159:
1154:
1149:
1144:
1139:
1134:
1129:
1127:Extensionality
1124:
1119:
1114:
1113:
1112:
1107:
1102:
1092:
1086:
1084:
1078:
1077:
1070:
1068:
1066:
1065:
1059:
1057:
1053:
1052:
1045:
1044:
1037:
1030:
1022:
1016:
1015:
1010:
988:
977:
976:External links
974:
971:
970:
963:
945:
926:(1): 436â460.
907:
906:
904:
901:
900:
899:
871:10.2307/421203
855:(2): 145â161.
836:
819:(1): 123â137.
801:
775:(4): 487â553.
757:
751:
735:
721:
718:
717:
716:
707:
702:
697:
692:
687:
682:
680:Axiom of union
677:
672:
667:
660:
657:
656:
655:
641:
640:
629:
622:
621:
620:
617:
614:
601:
588:
567:
554:
531:
528:
510:
507:
495:Thoralf Skolem
477:axiomatization
440:introduced by
417:
414:
315:
312:
238:
237:
234:
233:
221:
217:
216:
214:
213:
208:
202:
200:
194:
193:
188:
184:
183:
178:
174:
173:
167:
166:
160:
156:
155:
152:
148:
147:
145:
144:
139:
134:
129:
124:
119:
113:
111:
110:Known for
107:
106:
101:
97:
96:
91:
87:
86:
77:
75:(aged 81)
69:
65:
64:
55:
44:
40:
39:
36:
28:
27:
24:
15:
9:
6:
4:
3:
2:
1768:
1757:
1754:
1752:
1749:
1747:
1744:
1742:
1739:
1737:
1734:
1732:
1729:
1727:
1724:
1722:
1719:
1717:
1716:Set theorists
1714:
1712:
1709:
1707:
1704:
1702:
1699:
1697:
1694:
1692:
1689:
1687:
1684:
1682:
1679:
1678:
1676:
1661:
1660:Ernst Zermelo
1658:
1656:
1653:
1651:
1648:
1646:
1645:Willard Quine
1643:
1641:
1638:
1636:
1633:
1631:
1628:
1626:
1623:
1621:
1618:
1616:
1613:
1611:
1608:
1606:
1603:
1602:
1600:
1598:
1597:Set theorists
1594:
1588:
1585:
1583:
1580:
1578:
1575:
1574:
1572:
1566:
1564:
1561:
1560:
1557:
1549:
1546:
1544:
1543:KripkeâPlatek
1541:
1537:
1534:
1533:
1532:
1529:
1528:
1527:
1524:
1520:
1517:
1516:
1515:
1514:
1510:
1506:
1503:
1502:
1501:
1498:
1497:
1494:
1491:
1489:
1486:
1484:
1481:
1479:
1476:
1475:
1473:
1469:
1463:
1460:
1458:
1455:
1453:
1450:
1448:
1446:
1441:
1439:
1436:
1434:
1431:
1428:
1424:
1421:
1419:
1416:
1412:
1409:
1407:
1404:
1402:
1399:
1398:
1397:
1394:
1391:
1387:
1384:
1382:
1379:
1377:
1374:
1372:
1369:
1368:
1366:
1363:
1359:
1353:
1350:
1348:
1345:
1343:
1340:
1338:
1335:
1333:
1330:
1328:
1325:
1323:
1320:
1316:
1313:
1311:
1308:
1307:
1306:
1303:
1301:
1298:
1296:
1293:
1291:
1288:
1286:
1283:
1280:
1276:
1273:
1271:
1268:
1266:
1263:
1262:
1260:
1254:
1251:
1250:
1247:
1241:
1238:
1236:
1233:
1231:
1228:
1226:
1223:
1221:
1218:
1216:
1213:
1211:
1208:
1205:
1202:
1200:
1197:
1196:
1194:
1192:
1188:
1180:
1179:specification
1177:
1175:
1172:
1171:
1170:
1167:
1166:
1163:
1160:
1158:
1155:
1153:
1150:
1148:
1145:
1143:
1140:
1138:
1135:
1133:
1130:
1128:
1125:
1123:
1120:
1118:
1115:
1111:
1108:
1106:
1103:
1101:
1098:
1097:
1096:
1093:
1091:
1088:
1087:
1085:
1083:
1079:
1074:
1064:
1061:
1060:
1058:
1054:
1050:
1043:
1038:
1036:
1031:
1029:
1024:
1023:
1020:
1014:
1011:
1008:
1004:
1003:
998:
994:
989:
987:
983:
980:
979:
966:
960:
956:
949:
941:
937:
933:
929:
925:
922:(in German).
921:
920:
912:
908:
896:
892:
888:
884:
880:
876:
872:
868:
863:
858:
854:
850:
846:
843:(June 2000).
842:
837:
830:
826:
822:
818:
814:
807:
802:
798:
794:
790:
786:
782:
778:
774:
770:
766:
762:
758:
754:
748:
744:
740:
736:
732:
728:
724:
723:
715:
711:
710:14990 Zermelo
708:
706:
703:
701:
698:
696:
693:
691:
688:
686:
683:
681:
678:
676:
673:
671:
668:
666:
663:
662:
653:
649:
646:
645:
644:
638:
634:
630:
627:
623:
618:
615:
612:
611:
609:
605:
602:
599:
595:
591:
585:
581:
577:
573:
568:
565:
561:
557:
551:
547:
543:
539:
534:
533:
527:
523:
520:
517:is a classic
516:
506:
504:
500:
496:
492:
487:
485:
480:
478:
474:
470:
466:
462:
458:
454:
449:
447:
443:
439:
435:
431:
427:
426:David Hilbert
423:
409:
405:
403:
399:
395:
391:
386:
384:
380:
376:
375:hydrodynamics
372:
368:
364:
360:
356:
352:
348:
344:
340:
334:
329:
320:
311:
309:
305:
301:
297:
293:
292:mathematician
289:
284:
274:
244:
235:
230:
225:
222:
218:
212:
209:
207:
206:Lazarus Fuchs
204:
203:
201:
199:
195:
192:
189:
185:
182:
179:
175:
172:
168:
164:
161:
157:
153:
149:
143:
140:
138:
135:
133:
130:
128:
125:
123:
120:
118:
115:
114:
112:
108:
105:
102:
98:
95:
92:
88:
84:
80:
70:
66:
62:
61:German Empire
58:
45:
41:
34:
29:
25:Ernst Zermelo
22:
19:
1659:
1610:Georg Cantor
1605:Paul Bernays
1536:MorseâKelley
1511:
1444:
1443:Subset
1390:hereditarily
1352:Venn diagram
1310:ordered pair
1225:Intersection
1169:Axiom schema
1000:
954:
948:
923:
917:
911:
852:
848:
829:the original
816:
812:
772:
768:
745:. Springer.
742:
730:
651:
647:
642:
632:
625:
607:
571:
537:
530:Publications
524:
512:
488:
481:
464:
450:
419:
402:World War II
398:Adolf Hitler
387:
366:
325:
242:
241:
187:Institutions
170:
83:West Germany
73:(1953-05-21)
53:27 July 1871
18:
1686:1953 deaths
1681:1871 births
1635:Thomas Jech
1478:Alternative
1457:Uncountable
1411:Ultrafilter
1270:Cardinality
1174:replacement
1122:Determinacy
339:mathematics
331: [
227: [
181:Mathematics
90:Nationality
71:21 May 1953
1675:Categories
1630:Kurt Gödel
1615:Paul Cohen
1452:Transitive
1220:Identities
1204:Complement
1191:Operations
1152:Regularity
1090:Adjunction
1049:Set theory
879:1874/27769
720:References
639:: 1219â33.
635:, 2 vols.
436:, was the
434:set theory
357:, and the
347:philosophy
49:1871-07-27
1563:Paradoxes
1483:Axiomatic
1462:Universal
1438:Singleton
1433:Recursive
1376:Countable
1371:Amorphous
1230:Power set
1147:Power set
1105:dependent
1100:countable
940:122877703
903:Citations
857:CiteSeerX
797:231795240
489:In 1922,
385:in 1899.
1567:Problems
1471:Theories
1447:Superset
1423:Infinite
1252:Concepts
1132:Infinity
1056:Overview
763:(2004).
741:(2007).
729:(2000).
714:asteroid
659:See also
606:, 1967.
288:logician
1505:General
1500:Zermelo
1406:subbase
1388: (
1327:Forcing
1305:Element
1277: (
1255:Methods
1142:Pairing
984:at the
895:8530810
789:2136635
598:3137671
564:2640544
349:at the
343:physics
279:German:
1396:Filter
1386:Finite
1322:Family
1265:Almost
1110:global
1095:Choice
1082:Axioms
961:
938:
893:
887:421203
885:
859:
795:
787:
749:
596:
586:
562:
552:
442:Cantor
371:Planck
353:, the
177:Fields
165:(1916)
159:Awards
151:Spouse
94:German
57:Berlin
1488:Naive
1418:Fuzzy
1381:Empty
1364:types
1315:tuple
1285:Class
1279:large
1240:Union
1157:Union
936:S2CID
891:S2CID
883:JSTOR
832:(PDF)
809:(PDF)
793:S2CID
335:]
231:]
1401:base
959:ISBN
747:ISBN
584:ISBN
550:ISBN
493:and
345:and
314:Life
290:and
68:Died
43:Born
1362:Set
928:doi
875:hdl
867:doi
821:doi
777:doi
576:doi
542:doi
255:ÉËr
1677::
1005:,
999:,
995:,
934:.
924:29
889:.
881:.
873:.
865:.
851:.
847:.
817:34
815:.
811:.
791:.
785:MR
783:.
773:10
771:.
767:.
712:,
594:MR
592:,
582:,
560:MR
558:,
548:,
505:.
448:.
424:,
341:,
333:de
277:,
270:oÊ
229:pl
81:,
59:,
1445:·
1429:)
1425:(
1392:)
1281:)
1041:e
1034:t
1027:v
967:.
942:.
930::
897:.
877::
869::
853:6
823::
799:.
779::
755:.
578::
544::
463:(
365:(
273:/
267:l
264:É
261:m
258:Ë
252:z
249:/
245:(
51:)
47:(
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
