406:
187:. Many theorems related to these cardinals have generalizations to their unfoldable or strongly unfoldable counterparts. For example, the existence of a strongly unfoldable implies the consistency of a slightly weaker version of the
199:
Assuming V = L, the least unfoldable cardinal is greater than the least indescribable cardinal. Assuming a Ramsey cardinal exists, it is less than the least Ramsey cardinal.
447:
221:
240:
440:
142:
74:
158:
471:
289:
184:
466:
433:
217:
209:
In L, any unfoldable cardinal is strongly unfoldable; thus unfoldable and strongly unfoldable have the same
206:
is unfoldable and will be strongly unfoldable in L. It may fail to be strongly unfoldable in V, however.
180:
210:
188:
131:
63:
380:
331:
319:
306:
275:
267:
249:
235:
341:
298:
259:
353:
379:
Villaveces, Andres (1996). "Chains of End
Elementary Extensions of Models of Set Theory".
349:
203:
176:
100:
32:
417:
93:
25:
460:
310:
279:
322:(2006). "Diamond (on the regulars) can fail at any strongly unfoldable cardinal".
287:
Johnstone, Thomas A. (2008). "Strongly unfoldable cardinals made indestructible".
108:
40:
17:
345:
413:
216:
A cardinal k is κ-strongly unfoldable, and κ-unfoldable, if and only if it is
302:
119:
51:
130:
contains all its sequences of length less than κ, there is a non-trivial
62:
contains all its sequences of length less than κ, there is a non-trivial
405:
271:
385:
336:
254:
224:
and preceded by a stationary set of totally indescribable cardinals.
263:
115:
47:
421:
172:
if and only if it is strongly λ-unfoldable for all λ.
175:
These properties are essentially weaker versions of
374:
372:
441:
92:if and only if it is an λ-unfoldable for all
8:
238:(2001). "Unfoldable cardinals and the GCH".
195:Relations between large cardinal properties
448:
434:
384:
335:
253:
165:contains all its sequences of length λ.
368:
141:into a transitive model "N" with the
7:
402:
400:
14:
153:(κ) ≥ λ, and V(λ) is a subset of
73:into a transitive model with the
404:
324:Annals of Pure and Applied Logic
220:. A κ+ω-unfoldable cardinal is
1:
241:The Journal of Symbolic Logic
420:. You can help Knowledge by
183:cardinals, consistent with
488:
399:
346:10.1016/j.apal.2006.05.001
161:, we can demand also that
159:Without loss of generality
290:Journal of Symbolic Logic
107:if and only if for every
39:if and only if for every
168:Likewise, a cardinal is
416:-related article is a
303:10.2178/jsl/1230396915
105:strongly λ-unfoldable
24:is a certain kind of
211:consistency strength
189:proper forcing axiom
132:elementary embedding
114:of cardinality κ of
64:elementary embedding
46:of cardinality κ of
320:Hamkins, Joel David
236:Hamkins, Joel David
170:strongly unfoldable
22:unfoldable cardinal
122:such that κ is in
54:such that κ is in
429:
428:
318:Džamonja, Mirna;
479:
472:Set theory stubs
450:
443:
436:
408:
401:
391:
390:
388:
376:
357:
339:
314:
297:(4): 1215–1248.
283:
257:
248:(3): 1186–1198.
109:transitive model
41:transitive model
487:
486:
482:
481:
480:
478:
477:
476:
467:Large cardinals
457:
456:
455:
454:
397:
395:
394:
378:
377:
370:
365:
360:
317:
286:
264:10.2307/2695100
234:
230:
204:Ramsey cardinal
197:
101:cardinal number
33:cardinal number
12:
11:
5:
485:
483:
475:
474:
469:
459:
458:
453:
452:
445:
438:
430:
427:
426:
409:
393:
392:
367:
366:
364:
361:
359:
358:
330:(1–3): 83–95.
315:
284:
231:
229:
226:
218:weakly compact
196:
193:
143:critical point
88:A cardinal is
75:critical point
26:large cardinal
13:
10:
9:
6:
4:
3:
2:
484:
473:
470:
468:
465:
464:
462:
451:
446:
444:
439:
437:
432:
431:
425:
423:
419:
415:
410:
407:
403:
398:
387:
382:
375:
373:
369:
362:
355:
351:
347:
343:
338:
333:
329:
325:
321:
316:
312:
308:
304:
300:
296:
292:
291:
285:
281:
277:
273:
269:
265:
261:
256:
251:
247:
243:
242:
237:
233:
232:
227:
225:
223:
222:indescribable
219:
214:
212:
207:
205:
200:
194:
192:
190:
186:
182:
178:
173:
171:
166:
164:
160:
156:
152:
148:
144:
140:
136:
133:
129:
125:
121:
117:
113:
110:
106:
102:
97:
95:
91:
86:
84:
80:
76:
72:
68:
65:
61:
57:
53:
49:
45:
42:
38:
34:
29:
27:
23:
19:
422:expanding it
411:
396:
386:math/9611209
337:math/0409304
327:
323:
294:
288:
255:math/9909029
245:
239:
215:
208:
201:
198:
181:supercompact
174:
169:
167:
162:
154:
150:
146:
138:
134:
127:
123:
111:
104:
98:
89:
87:
82:
81:being κ and
78:
70:
66:
59:
55:
43:
37:λ-unfoldable
36:
31:Formally, a
30:
21:
15:
18:mathematics
461:Categories
414:set theory
228:References
90:unfoldable
363:Citations
149:being κ,
120:power set
85:(κ) ≥ λ.
52:power set
311:30534686
94:ordinals
28:number.
354:2279655
280:6269487
272:2695100
118:-minus-
50:-minus-
352:
309:
278:
270:
177:strong
412:This
381:arXiv
332:arXiv
307:S2CID
276:S2CID
268:JSTOR
250:arXiv
185:V = L
103:κ is
35:κ is
20:, an
418:stub
179:and
126:and
58:and
342:doi
328:144
299:doi
260:doi
145:of
137:of
116:ZFC
96:λ.
77:of
69:of
48:ZFC
16:In
463::
371:^
350:MR
348:.
340:.
326:.
305:.
295:73
293:.
274:.
266:.
258:.
246:66
244:.
213:.
202:A
191:.
157:.
99:A
449:e
442:t
435:v
424:.
389:.
383::
356:.
344::
334::
313:.
301::
282:.
262::
252::
163:N
155:N
151:j
147:j
139:M
135:j
128:M
124:M
112:M
83:j
79:j
71:M
67:j
60:M
56:M
44:M
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.