346:
277:
129:
220:
42:. The ordinals 1, 2, and 3 are the first three successor ordinals and the ordinals ω+1, ω+2 and ω+3 are the first three infinite successor ordinals.
291:
226:
84:
458:
466:
435:
63:
192:
488:
452:
425:
183:
8:
401:
391:
179:
462:
431:
379:
375:
28:
341:{\displaystyle \alpha +\lambda =\bigcup _{\beta <\lambda }(\alpha +\beta )}
482:
396:
51:
66:(the standard model of the ordinals used in set theory), the successor
20:
430:, Springer Undergraduate Mathematics Series, Springer, p. 46,
150:, it is immediate that there is no ordinal number between α and
50:
Every ordinal other than 0 is either a successor ordinal or a
366:. Multiplication and exponentiation are defined similarly.
454:
134:
Since the ordering on the ordinal numbers is given by
378:
of the class of ordinal numbers, with respect to the
294:
229:
195:
87:
38:. An ordinal number that is a successor is called a
272:{\displaystyle \alpha +S(\beta )=S(\alpha +\beta )}
124:{\displaystyle S(\alpha )=\alpha \cup \{\alpha \}.}
340:
271:
214:
123:
211:
34:is the smallest ordinal number greater than
480:
178:The successor operation can be used to define
115:
109:
419:
417:
57:
423:
481:
461:, Springer, Exercise 3C, p. 100,
450:
414:
374:The successor points and zero are the
215:{\displaystyle \alpha +0=\alpha \!}
173:
13:
459:Undergraduate Texts in Mathematics
14:
500:
444:
335:
323:
266:
254:
245:
239:
97:
91:
1:
407:
158:), and it is also clear that
64:von Neumann's ordinal numbers
45:
16:Operation on ordinal numbers
7:
385:
369:
10:
505:
427:Sets, Logic and Categories
424:Cameron, Peter J. (1999),
282:and for a limit ordinal
78:is given by the formula
74:) of an ordinal number
451:Devlin, Keith (1993),
342:
273:
216:
125:
58:In Von Neumann's model
343:
274:
217:
184:transfinite recursion
126:
292:
227:
193:
85:
402:Successor cardinal
392:Ordinal arithmetic
338:
322:
269:
212:
121:
307:
40:successor ordinal
496:
473:
471:
448:
442:
440:
421:
365:
347:
345:
344:
339:
321:
278:
276:
275:
270:
221:
219:
218:
213:
180:ordinal addition
174:Ordinal addition
162: <
138: <
130:
128:
127:
122:
504:
503:
499:
498:
497:
495:
494:
493:
489:Ordinal numbers
479:
478:
477:
476:
469:
449:
445:
438:
422:
415:
410:
388:
376:isolated points
372:
352:
351:In particular,
311:
293:
290:
289:
228:
225:
224:
194:
191:
190:
182:rigorously via
176:
142:if and only if
86:
83:
82:
60:
48:
17:
12:
11:
5:
502:
492:
491:
475:
474:
467:
443:
436:
412:
411:
409:
406:
405:
404:
399:
394:
387:
384:
380:order topology
371:
368:
349:
348:
337:
334:
331:
328:
325:
320:
317:
314:
310:
306:
303:
300:
297:
280:
279:
268:
265:
262:
259:
256:
253:
250:
247:
244:
241:
238:
235:
232:
222:
210:
207:
204:
201:
198:
175:
172:
132:
131:
120:
117:
114:
111:
108:
105:
102:
99:
96:
93:
90:
59:
56:
47:
44:
29:ordinal number
15:
9:
6:
4:
3:
2:
501:
490:
487:
486:
484:
470:
468:9780387940946
464:
460:
456:
455:
447:
439:
437:9781852330569
433:
429:
428:
420:
418:
413:
403:
400:
398:
397:Limit ordinal
395:
393:
390:
389:
383:
381:
377:
367:
363:
359:
355:
332:
329:
326:
318:
315:
312:
308:
304:
301:
298:
295:
288:
287:
286:
285:
263:
260:
257:
251:
248:
242:
236:
233:
230:
223:
208:
205:
202:
199:
196:
189:
188:
187:
185:
181:
171:
169:
165:
161:
157:
153:
149:
146: ∈
145:
141:
137:
118:
112:
106:
103:
100:
94:
88:
81:
80:
79:
77:
73:
69:
65:
55:
53:
52:limit ordinal
43:
41:
37:
33:
30:
26:
22:
453:
446:
426:
373:
361:
357:
353:
350:
283:
281:
186:as follows:
177:
167:
163:
159:
155:
151:
147:
143:
139:
135:
133:
75:
71:
67:
61:
49:
39:
35:
31:
24:
18:
408:References
46:Properties
21:set theory
333:β
327:α
319:λ
313:β
309:⋃
302:λ
296:α
264:β
258:α
243:β
231:α
209:α
197:α
113:α
107:∪
104:α
95:α
25:successor
483:Category
386:See also
370:Topology
465:
434:
62:Using
27:of an
23:, the
463:ISBN
432:ISBN
360:) =
316:<
364:+ 1
170:).
19:In
485::
457:,
416:^
382:.
54:.
472:.
441:.
362:α
358:α
356:(
354:S
336:)
330:+
324:(
305:=
299:+
284:λ
267:)
261:+
255:(
252:S
249:=
246:)
240:(
237:S
234:+
206:=
203:0
200:+
168:α
166:(
164:S
160:α
156:α
154:(
152:S
148:β
144:α
140:β
136:α
119:.
116:}
110:{
101:=
98:)
92:(
89:S
76:α
72:α
70:(
68:S
36:α
32:α
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