803:, would be in scope for articles on order theory, or (in that context) as a construction of the integers from the natural numbers, as ordered sets. It's out of scope for ordinals since the structures it produces are not regarded as ordinals (in mathematics, which is where he proposed it, right?) and nobody adopted Russell's idea of calling these things "negative ordinals". Russell misapplied or misunderstood the term "negative" here; the point in mathematics is to create a group from a semigroup, not a minus sign as a reverse direction.
436:
426:
405:
372:
859:
constructing integers from positive integers is create a homogeneous structure (transitive symmetry group), not an artificial left-right symmetry put in by hand by creating a duplicate mirrored copy. Russell's construction does not really sew the left and right sides together properly, in that whether you have unique successor or unique predecessor depends on which half you are in; it's not a single structure working differently than the sum of its parts.
251:
515:
1149:. As for 1.: oops, sorry! As for 2.: I now darkened the colors from RGB (100%,0%,0%) for red (similar for green, blue) and (80%,80%,0%) for yellow (similar for cyan, magenta) to (80%,0%,0%) and (50%,50%,0%). I find the result still hard to read. On the other hand, I'd like to keep the "rainbow" effect which makes it easier to recognize which label belongs to which spiral turn. Moreover, inspired by the current image
219:
363:
996:
until it reaches the center which would be ω. The spoke beginning with 1 would continue with ω·2, ω·2 ω·2, etc.. Between these, new spokes should begin in the second turn, ω+1 would continue with ω+ω. Clearly, most of these numbers would have to be replaced with just tick-marks and as you proceed around the spiral, and most of those would have to be omitted altogether.
1142:
1281:
After recently looking at the
Ordinal number article, I see that you changed the black and white picture to a color one (March of last year). Is that really an improvement? The old version seemed a lot easier to look at. The new layout is good, but the lack of contrast with the washed-out tones of
995:
The outermost ring of the spiral should begin with 0 at the 12:00 high position and continue with the natural numbers coming closer together until the first turn is completed. At that point it should meet ω which should be directly under 0. The spoke beginning with 0 would continue with ω, ω, ω, etc.
858:
In over a century, this definition of
Russell's never caught on. You can "double" any linear (or partial) order that has a minimum element, putting in a reflection symmetry by fiat, but what does that accomplish? Apparently not much, since Russell's terminology did not gain acceptance. The gain in
898:
I think, each stick denotes one number, so the sticks at the left end denote the numbers 0, 1, 2, 3, 4, ... . The leftmost "completely black triangle" (appearance to the human eye) then has ω at its tip. The next sticks right to it denote ω+1, ω+2, ω+3, ω+4, ... . The rightmost end of the picture
647:
Hmm, I suppose we could have a brief mention of the surreals (if we don't already; haven't checked) but the
Russell thing is kind of anachronistic for that, maybe a precursor but not using terminology that has survived (come to think of it, you could say that about much of Russell's work in math
882:
It's probably my own confusion, but the caption makes little sense to me. Assuming a "matchstick" is a single vertical line, shouldn't the caption read "... ω^3 . Each stick corresponds to ... ω." ? In any event, isn't the "+n" in the quoted caption useless?
919:
I am no expert, to be sure, on ordinals but can we please finally dump these attempts at graphical representations? the 'spiral' and 'matchstick' images -- they add nothing to the explanation and as far as I can see are not even discussed in the article text.
819:
There are not, in fact, "negative ordinals", not in the sense intended in this article. This article is about a specific structure in set theory, which has no negative ordinals. Just because a reliable source uses the same
153:
935:
The images are intended to illustrate the structure of (initial segments of) the ordinal numbers. The spiral form is just a hack to avoid an extreme aspect ratio. Do you have suggestion about alternatives? -
972:
It seems that the Svg file can be edited with an ordinary text editor, so I can offer to change the labels according to your suggestions. However, I'm afraid I didn't understand them. It seems that after
711:, and naturally with scientific notation there are negative Ns. Thus it seems necessary to acknowledge negative ordinals to properly document these order of magnitude articles in that category. —
1207:
492:
879:
There is an image with the caption "A graphical "matchstick" representation of the ordinal ω². Each stick corresponds to an ordinal of the form ω·m+n where m and n are natural numbers."
1408:
1098:
210:
1121:
147:
1314:
for comparison purposes. The colors are intended to ease distinguishing the different turns of the spiral and to ease recognition of which label belongs to which spoke.
1123:; the latter is unusual in my experience and the 2 looks like it could be a misplaced superscript or subscript. I agree about the yellow; that's pretty hard to see. --
1398:
1413:
376:
1423:
482:
79:
778:
of this article? So what that the transfinite ordinals are the "interesting ones", the encyclopedia is to serve all readers, young and old. The article
1393:
1318:
648:
logic). It's not really even clear from the excerpt given just what
Russell was talking about -- does anyone have it handy to give more context? --
1403:
1311:
458:
44:
1418:
85:
842:
Should we disregard one old source by
Russell or thousands of more recent articles which use "ordinal" in a sense which excludes negatives?
633:
At least the latter fact is worth mentioning in the article; and why not give
Russell's reference (but not the full quotation) then, too? -
921:
1339:
is the best of the three. The colors make it easier to distinguish the turns, and it's also easier to read without the fade-out effect.
1025:
277:
262:
770:
Bertrand
Russell wrote his book in response to continental developements in logic and set theory. How dare editors remove the on-topic
860:
804:
272:
449:
410:
1242:
1150:
608:
The "negative fourth magnitude" has no apparent connection to the topic of this article. It might have some connection with the
782:
is your focus, but the general reader needs to know that there are negative ordinals, especially for the reason given above. —
1388:
1383:
1224:
1029:
268:
99:
30:
1010:
I started to redesign the picture according to your suggestion; it will take a few days to get the limit ordinals arranged. -
593:
was reverted as "surreal"! Now an illustration, as Venus has negative fourth apparent magnitude, is appended. Any comments? —
899:
then denotes ω^2. Maybe, the phrase "matchstick representation" should be explained in a footnote (along the above lines). -
168:
104:
20:
135:
544:
540:
536:
532:
528:
206:
202:
198:
194:
190:
74:
1024:
It took me longer than expected, but now I came up with a first version that meets the above requirements (I hope), see
385:
574:
65:
755:. The interesting ones are infinite; finite ordinals are not in any essential way different from natural numbers. --
1156:
1326:
1214:
1037:
1015:
986:
941:
904:
675:
638:
218:
185:
250:
1241:
I think you put the wrong link. We don't want a PDF as an image (not sure that even works). Probably you meant
553:
229:
129:
925:
888:
109:
1336:
703:
nearly all the uses refer to a heirachial organization where N corresponds to an ordinal number such as at
267:, but it has been removed from the list. There are suggestions below for improving the article to meet the
125:
864:
808:
775:
1322:
1278:
1210:
1033:
1011:
982:
937:
900:
671:
634:
391:
435:
957:
corresponds to one turn of the spiral. If I could, I would change it so that that function is α -: -->
175:
741:
for which your comments might make a certain amount of sense; that sense is discussed at our article
362:
1344:
1301:
1287:
1264:
1250:
1232:
1128:
1077:
1065:
1001:
963:
847:
833:
760:
700:
692:
653:
624:
161:
55:
457:
on
Knowledge. If you would like to participate, please visit the project page, where you can join
884:
779:
708:
704:
696:
559:
441:
234:
70:
425:
404:
1282:
the colors makes the whole thing hard to follow and understand what is going on. Any thoughts?
695:
a x 10 there is some discussion concerning the label for N beyond mere exponent. Some usage of
787:
716:
598:
51:
1103:
1317:
I also have uploaded a version using the (darker) spoke colors for label coloring, too, see
555:
514:
231:
141:
954:
742:
667:
616:
609:
1340:
1337:
https://upload.wikimedia.org/wikipedia/commons/a/af/Omega-exp-omega-normal-dark_svg.svg
1297:
1283:
1260:
1246:
1228:
1124:
1061:
997:
959:
843:
829:
756:
663:
649:
620:
24:
1060:
2. Please use darker colors. Yellow on white is virtually illegible (to me at least).
1377:
783:
771:
752:
728:
712:
594:
950:
The matchsticks are good. They clearly show how ω can be embedded into the reals.
745:, not here. That's fundamentally a linguistic rather than mathematical concept.
454:
431:
557:
233:
1348:
1330:
1305:
1291:
1268:
1254:
1236:
1218:
1132:
1069:
1041:
1019:
1005:
990:
967:
945:
929:
908:
892:
868:
851:
837:
812:
791:
764:
720:
679:
657:
642:
628:
602:
977:
turns, we are at ω; but I'm not sure whether your α corresponds to my
619:, which embed a copy of the ordinals, and which have negatives.) --
800:
707:. A number of these orders of magnitude have been assembled in a
1259:
No, Trovatore. That is not it. That is an old incorrect version.
560:
508:
356:
235:
15:
733:
are you sure you've understood what this article is actually
271:. Once these issues have been addressed, the article can be
1321:; it has a reduced fade-out effect; I' prefer this one. -
737:? I grant you that there's a different sense of the term
958:ω·(1 + α). However, I do not know how to edit the image.
699:
has been noted but also that at the disambiguation page
326:
307:
953:
The spiral could be improved by defining clearly what
160:
1227:
should replace what is in the article. Thanks again.
1159:
1106:
1080:
453:, a collaborative effort to improve the coverage of
281:
of the decision if they believe there was a mistake.
751:article is about the ordinal numbers introduced by
1201:
1115:
1092:
824:does not imply that it's talking about the same
615:(The "surreal" comment is probably a pun on the
33:for general discussion of the article's subject.
1409:Knowledge level-5 vital articles in Mathematics
1202:{\displaystyle \omega ,\omega +1,...,\omega +4}
828:, or that it is on-topic for this article. --
799:Russell's construction, in the generality of
568:This page has archives. Sections older than
174:
8:
1296:Yes, please turn off the fade-out effect.
1153:, I'd added some fade-out effect, e.g. at
399:
286:
245:
1158:
1105:
1079:
1046:Thank you for your effort. Two problems:
1319:File:Omega-exp-omega-normal-dark svg.svg
1399:Knowledge vital articles in Mathematics
401:
360:
1312:File:Omega-exp-omega-normal-bw svg.svg
578:when more than 5 sections are present.
1414:B-Class vital articles in Mathematics
7:
915:Get rid of pictures, or replace them
447:This article is within the scope of
1026:File:Omega-exp-omega-normal svg.svg
390:It is of interest to the following
23:for discussing improvements to the
1424:High-priority mathematics articles
670:under "See also") is just fine. -
14:
572:may be automatically archived by
467:Knowledge:WikiProject Mathematics
1394:Knowledge level-5 vital articles
1243:File:Omega-exp-omega-labeled.svg
1151:File:Omega-exp-omega-labeled.svg
1140:
513:
470:Template:WikiProject Mathematics
434:
424:
403:
370:
361:
249:
217:
45:Click here to start a new topic.
1310:I've uploaded a b/w version at
1225:File:Omega-exp-omega-normal.pdf
1030:File:Omega-exp-omega-normal.pdf
487:This article has been rated as
1404:B-Class level-5 vital articles
1093:{\displaystyle \omega \cdot 2}
1032:. Comments are appreciated. -
909:20:41, 21 September 2021 (UTC)
893:00:23, 20 September 2021 (UTC)
1:
1269:14:48, 10 February 2023 (UTC)
1255:01:52, 10 February 2023 (UTC)
991:13:12, 31 December 2022 (UTC)
968:00:35, 31 December 2022 (UTC)
946:19:07, 30 December 2022 (UTC)
930:13:20, 30 December 2022 (UTC)
869:06:10, 29 November 2020 (UTC)
852:06:04, 29 November 2020 (UTC)
838:05:56, 29 November 2020 (UTC)
813:06:31, 29 November 2020 (UTC)
792:04:32, 29 November 2020 (UTC)
765:04:33, 28 November 2020 (UTC)
721:01:35, 28 November 2020 (UTC)
680:19:26, 27 November 2020 (UTC)
658:19:09, 27 November 2020 (UTC)
643:10:00, 27 November 2020 (UTC)
629:04:01, 27 November 2020 (UTC)
603:03:26, 27 November 2020 (UTC)
589:A section quoting Russell in
461:and see a list of open tasks.
42:Put new text under old text.
1419:B-Class mathematics articles
1237:19:46, 9 February 2023 (UTC)
1219:12:08, 9 February 2023 (UTC)
1209:; should I turn that off? -
1133:01:44, 9 February 2023 (UTC)
1070:01:00, 9 February 2023 (UTC)
1042:18:14, 8 February 2023 (UTC)
1020:17:19, 23 January 2023 (UTC)
875:Faulty caption on pictorial?
774:? Why are negative ordinals
1057:ω. 2ω = ω < ω + ω = ω2 .
1006:00:05, 1 January 2023 (UTC)
50:New to Knowledge? Welcome!
1440:
691:By way of explanation: In
275:. Editors may also seek a
591:Principles of Mathematics
486:
419:
398:
347:
334:Good article reassessment
289:
285:
263:Mathematics good articles
80:Be welcoming to newcomers
1349:18:18, 22 May 2024 (UTC)
1331:07:51, 19 May 2024 (UTC)
1306:04:55, 15 May 2024 (UTC)
1292:04:49, 15 May 2024 (UTC)
1116:{\displaystyle \omega 2}
493:project's priority scale
450:WikiProject Mathematics
1389:B-Class vital articles
1384:Delisted good articles
1203:
1117:
1094:
666:' recent edit (adding
575:Lowercase sigmabot III
75:avoid personal attacks
1204:
1118:
1095:
377:level-5 vital article
350:Delisted good article
269:good article criteria
211:Auto-archiving period
100:Neutral point of view
1157:
1104:
1078:
662:Ok, I see. For now,
473:mathematics articles
315:Good article nominee
105:No original research
701:order (mathematics)
693:scientific notation
1223:Yes, I agree that
1199:
1113:
1090:
780:transfinite number
705:decade (log scale)
697:order of magnitude
442:Mathematics portal
386:content assessment
290:Article milestones
86:dispute resolution
47:
585:Negative ordinals
582:
581:
507:
506:
503:
502:
499:
498:
355:
354:
343:
342:
308:September 3, 2006
242:
241:
66:Assume good faith
43:
1431:
1323:Jochen Burghardt
1279:Jochen Burghardt
1211:Jochen Burghardt
1208:
1206:
1205:
1200:
1148:
1144:
1143:
1122:
1120:
1119:
1114:
1099:
1097:
1096:
1091:
1034:Jochen Burghardt
1012:Jochen Burghardt
983:Jochen Burghardt
938:Jochen Burghardt
901:Jochen Burghardt
732:
672:Jochen Burghardt
635:Jochen Burghardt
577:
561:
517:
509:
475:
474:
471:
468:
465:
444:
439:
438:
428:
421:
420:
415:
407:
400:
383:
374:
373:
366:
365:
357:
348:Current status:
329:
310:
287:
253:
246:
236:
222:
221:
212:
179:
178:
164:
95:Article policies
16:
1439:
1438:
1434:
1433:
1432:
1430:
1429:
1428:
1374:
1373:
1155:
1154:
1141:
1139:
1102:
1101:
1076:
1075:
955:normal function
917:
877:
776:WP:Out of scope
743:ordinal numeral
726:
668:surreal numbers
617:surreal numbers
610:ordinal numeral
587:
573:
562:
556:
522:
472:
469:
466:
463:
462:
440:
433:
413:
384:on Knowledge's
381:
371:
325:
306:
259:was one of the
238:
237:
232:
209:
121:
116:
115:
114:
91:
61:
12:
11:
5:
1437:
1435:
1427:
1426:
1421:
1416:
1411:
1406:
1401:
1396:
1391:
1386:
1376:
1375:
1372:
1371:
1370:
1369:
1368:
1367:
1366:
1365:
1364:
1363:
1362:
1361:
1360:
1359:
1358:
1357:
1356:
1355:
1354:
1353:
1352:
1351:
1315:
1294:
1275:
1274:
1273:
1272:
1271:
1198:
1195:
1192:
1189:
1186:
1183:
1180:
1177:
1174:
1171:
1168:
1165:
1162:
1137:
1136:
1135:
1112:
1109:
1089:
1086:
1083:
1058:
951:
922:71.139.124.132
916:
913:
912:
911:
876:
873:
872:
871:
856:
855:
854:
817:
816:
815:
768:
767:
746:
739:ordinal number
689:
688:
687:
686:
685:
684:
683:
682:
613:
586:
583:
580:
579:
567:
564:
563:
558:
554:
552:
549:
548:
524:
523:
518:
512:
505:
504:
501:
500:
497:
496:
485:
479:
478:
476:
459:the discussion
446:
445:
429:
417:
416:
408:
396:
395:
389:
367:
353:
352:
345:
344:
341:
340:
337:
330:
322:
321:
318:
311:
303:
302:
299:
296:
292:
291:
283:
282:
257:Ordinal number
254:
240:
239:
230:
228:
227:
224:
223:
181:
180:
118:
117:
113:
112:
107:
102:
93:
92:
90:
89:
82:
77:
68:
62:
60:
59:
48:
39:
38:
35:
34:
28:
25:Ordinal number
13:
10:
9:
6:
4:
3:
2:
1436:
1425:
1422:
1420:
1417:
1415:
1412:
1410:
1407:
1405:
1402:
1400:
1397:
1395:
1392:
1390:
1387:
1385:
1382:
1381:
1379:
1350:
1346:
1342:
1338:
1335:I agree that
1334:
1333:
1332:
1328:
1324:
1320:
1316:
1313:
1309:
1308:
1307:
1303:
1299:
1295:
1293:
1289:
1285:
1280:
1276:
1270:
1266:
1262:
1258:
1257:
1256:
1252:
1248:
1244:
1240:
1239:
1238:
1234:
1230:
1226:
1222:
1221:
1220:
1216:
1212:
1196:
1193:
1190:
1187:
1184:
1181:
1178:
1175:
1172:
1169:
1166:
1163:
1160:
1152:
1147:
1138:
1134:
1130:
1126:
1110:
1107:
1087:
1084:
1081:
1073:
1072:
1071:
1067:
1063:
1059:
1056:
1052:
1048:
1047:
1045:
1044:
1043:
1039:
1035:
1031:
1027:
1023:
1022:
1021:
1017:
1013:
1009:
1008:
1007:
1003:
999:
994:
993:
992:
988:
984:
980:
976:
971:
970:
969:
965:
961:
956:
952:
949:
948:
947:
943:
939:
934:
933:
932:
931:
927:
923:
914:
910:
906:
902:
897:
896:
895:
894:
890:
886:
885:Jamesdowallen
880:
874:
870:
866:
862:
857:
853:
849:
845:
841:
840:
839:
835:
831:
827:
823:
818:
814:
810:
806:
802:
798:
797:
796:
795:
794:
793:
789:
785:
781:
777:
773:
766:
762:
758:
754:
750:
747:
744:
740:
736:
730:
725:
724:
723:
722:
718:
714:
710:
706:
702:
698:
694:
681:
677:
673:
669:
665:
661:
660:
659:
655:
651:
646:
645:
644:
640:
636:
632:
631:
630:
626:
622:
618:
614:
611:
607:
606:
605:
604:
600:
596:
592:
584:
576:
571:
566:
565:
551:
550:
547:
546:
542:
538:
534:
530:
526:
525:
521:
516:
511:
510:
494:
490:
489:High-priority
484:
481:
480:
477:
460:
456:
452:
451:
443:
437:
432:
430:
427:
423:
422:
418:
414:High‑priority
412:
409:
406:
402:
397:
393:
387:
379:
378:
368:
364:
359:
358:
351:
346:
338:
336:
335:
331:
328:
324:
323:
319:
317:
316:
312:
309:
305:
304:
300:
297:
294:
293:
288:
284:
280:
279:
274:
270:
266:
265:
264:
258:
255:
252:
248:
247:
244:
226:
225:
220:
216:
208:
204:
200:
196:
192:
189:
187:
183:
182:
177:
173:
170:
167:
163:
159:
155:
152:
149:
146:
143:
140:
137:
134:
131:
127:
124:
123:Find sources:
120:
119:
111:
110:Verifiability
108:
106:
103:
101:
98:
97:
96:
87:
83:
81:
78:
76:
72:
69:
67:
64:
63:
57:
53:
52:Learn to edit
49:
46:
41:
40:
37:
36:
32:
26:
22:
18:
17:
1145:
1100:rather than
1074:I would use
1054:
1050:
978:
974:
918:
881:
878:
861:73.89.25.252
825:
821:
805:73.89.25.252
769:
753:Georg Cantor
748:
738:
734:
690:
590:
588:
569:
527:
519:
488:
448:
392:WikiProjects
375:
349:
333:
332:
327:June 6, 2009
314:
313:
278:reassessment
276:
261:
260:
256:
243:
214:
184:
171:
165:
157:
150:
144:
138:
132:
122:
94:
19:This is the
1053:instead of
709:WP:Category
464:Mathematics
455:mathematics
411:Mathematics
273:renominated
148:free images
31:not a forum
1378:Categories
1341:PatrickR2
1298:PatrickR2
1284:PatrickR2
1261:JRSpriggs
1247:Trovatore
1229:JRSpriggs
1125:Trovatore
1062:JRSpriggs
998:JRSpriggs
960:JRSpriggs
844:JRSpriggs
830:Trovatore
757:Trovatore
664:JRSpriggs
650:Trovatore
621:Trovatore
380:is rated
88:if needed
71:Be polite
21:talk page
1049:1. Use ω
612:article.
570:180 days
520:Archives
339:Delisted
215:180 days
186:Archives
56:get help
29:This is
27:article.
784:Rgdboer
729:Rgdboer
713:Rgdboer
595:Rgdboer
491:on the
382:B-class
298:Process
154:WP refs
142:scholar
801:posets
388:scale.
320:Listed
301:Result
126:Google
826:thing
772:WP:RS
735:about
369:This
169:JSTOR
130:books
84:Seek
1345:talk
1327:talk
1302:talk
1288:talk
1265:talk
1251:talk
1245:. --
1233:talk
1215:talk
1146:Done
1129:talk
1066:talk
1038:talk
1028:and
1016:talk
1002:talk
987:talk
981:. -
964:talk
942:talk
926:talk
905:talk
889:talk
865:talk
848:talk
834:talk
822:word
809:talk
788:talk
761:talk
749:This
717:talk
676:talk
654:talk
639:talk
625:talk
599:talk
483:High
295:Date
162:FENS
136:news
73:and
176:TWL
1380::
1347:)
1329:)
1304:)
1290:)
1267:)
1253:)
1235:)
1217:)
1191:ω
1167:ω
1161:ω
1131:)
1108:ω
1085:⋅
1082:ω
1068:)
1040:)
1018:)
1004:)
989:)
966:)
944:)
928:)
907:)
891:)
867:)
850:)
836:)
811:)
790:)
763:)
719:)
678:)
656:)
641:)
627:)
601:)
543:,
539:,
535:,
531:,
213::
205:,
201:,
197:,
193:,
156:)
54:;
1343:(
1325:(
1300:(
1286:(
1277:@
1263:(
1249:(
1231:(
1213:(
1197:4
1194:+
1188:,
1185:.
1182:.
1179:.
1176:,
1173:1
1170:+
1164:,
1127:(
1111:2
1088:2
1064:(
1055:n
1051:n
1036:(
1014:(
1000:(
985:(
979:n
975:n
962:(
940:(
924:(
903:(
887:(
863:(
846:(
832:(
807:(
786:(
759:(
731::
727:@
715:(
674:(
652:(
637:(
623:(
597:(
545:5
541:4
537:3
533:2
529:1
495:.
394::
207:5
203:4
199:3
195:2
191:1
188::
172:·
166:·
158:·
151:·
145:·
139:·
133:·
128:(
58:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
