384:
260:
219:
178:
293:
118:
95:
49:
425:
368:
352:
344:
228:
187:
146:
449:
418:
459:
274:
454:
411:
310:
305:
140:
129:
103:
80:
34:
399:
21:
8:
444:
333:
364:
348:
340:
125:
121:
320:
315:
266:
71:
221:. Since there are only countably many computable relations, there are also only
395:
60:
29:
438:
270:
222:
181:
357:
17:
391:
75:
63:
25:
136:
67:
363:. Perspectives in mathematical logic, Springer-Verlag, 1990.
265:
The computable ordinals are exactly the ordinals that have an
383:
337:
The Theory of
Recursive Functions and Effective Computability
184:. An ordinal is computable if and only if it is smaller than
277:
231:
190:
149:
106:
83:
37:
287:
254:
213:
172:
112:
89:
43:
143:, the first nonrecursive ordinal, and denoted by
436:
124:of a computable ordinal is computable, and the
419:
426:
412:
255:{\displaystyle \omega _{1}^{\mathsf {CK}}}
214:{\displaystyle \omega _{1}^{\mathsf {CK}}}
173:{\displaystyle \omega _{1}^{\mathsf {CK}}}
139:of all computable ordinals is called the
437:
246:
243:
205:
202:
164:
161:
378:
339:, 1967. Reprinted 1987, MIT Press,
13:
280:
14:
471:
180:. The Church–Kleene ordinal is a
382:
225:many computable ordinals. Thus,
288:{\displaystyle {\mathcal {O}}}
128:of all computable ordinals is
1:
326:
398:. You can help Knowledge by
7:
299:
10:
476:
377:
100:It is easy to check that
361:Higher Recursion Theory
311:Large countable ordinal
113:{\displaystyle \omega }
90:{\displaystyle \alpha }
44:{\displaystyle \alpha }
394:-related article is a
306:Arithmetical hierarchy
289:
256:
215:
174:
114:
91:
45:
290:
257:
216:
175:
141:Church–Kleene ordinal
115:
92:
46:
450:Computability theory
275:
229:
188:
147:
104:
81:
35:
251:
210:
169:
120:is computable. The
334:Hartley Rogers Jr.
285:
252:
232:
211:
191:
170:
150:
110:
87:
41:
407:
406:
467:
460:Set theory stubs
428:
421:
414:
386:
379:
321:Ordinal notation
316:Ordinal analysis
294:
292:
291:
286:
284:
283:
267:ordinal notation
261:
259:
258:
253:
250:
249:
240:
220:
218:
217:
212:
209:
208:
199:
179:
177:
176:
171:
168:
167:
158:
119:
117:
116:
111:
96:
94:
93:
88:
66:of a computable
50:
48:
47:
42:
475:
474:
470:
469:
468:
466:
465:
464:
455:Ordinal numbers
435:
434:
433:
432:
375:
373:
329:
302:
279:
278:
276:
273:
272:
242:
241:
236:
230:
227:
226:
201:
200:
195:
189:
186:
185:
160:
159:
154:
148:
145:
144:
105:
102:
101:
82:
79:
78:
72:natural numbers
36:
33:
32:
20:, specifically
12:
11:
5:
473:
463:
462:
457:
452:
447:
431:
430:
423:
416:
408:
405:
404:
387:
372:
371:
355:
330:
328:
325:
324:
323:
318:
313:
308:
301:
298:
282:
262:is countable.
248:
245:
239:
235:
207:
204:
198:
194:
166:
163:
157:
153:
109:
86:
59:if there is a
51:is said to be
40:
9:
6:
4:
3:
2:
472:
461:
458:
456:
453:
451:
448:
446:
443:
442:
440:
429:
424:
422:
417:
415:
410:
409:
403:
401:
397:
393:
388:
385:
381:
380:
376:
370:
369:0-387-19305-7
366:
362:
359:
356:
354:
353:0-07-053522-1
350:
347:(paperback),
346:
345:0-262-68052-1
342:
338:
335:
332:
331:
322:
319:
317:
314:
312:
309:
307:
304:
303:
297:
295:
268:
263:
237:
233:
224:
196:
192:
183:
182:limit ordinal
155:
151:
142:
138:
133:
132:downwards.
131:
127:
123:
107:
98:
84:
77:
73:
69:
65:
64:well-ordering
62:
58:
54:
38:
31:
27:
23:
22:computability
19:
400:expanding it
389:
374:
360:
358:Gerald Sacks
336:
264:
134:
99:
56:
52:
15:
74:having the
18:mathematics
445:Set theory
439:Categories
392:set theory
327:References
76:order type
61:computable
53:computable
26:set theory
271:Kleene's
234:ω
223:countably
193:ω
152:ω
122:successor
108:ω
85:α
57:recursive
39:α
300:See also
137:supremum
70:of the
30:ordinal
367:
351:
343:
130:closed
68:subset
390:This
28:, an
396:stub
365:ISBN
349:ISBN
341:ISBN
135:The
24:and
296:.
269:in
126:set
55:or
16:In
441::
97:.
427:e
420:t
413:v
402:.
281:O
247:K
244:C
238:1
206:K
203:C
197:1
165:K
162:C
156:1
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
