228:
showed there was a model of ZFC in which every ordinal-definable set of reals is measurable, Solovay showed there is a model of ZF + DC in which there is some translation-invariant extension of
Lebesgue measure to all subsets of the reals, and
223:
Solovay suggested in his paper that the use of an inaccessible cardinal might not be necessary. Several authors proved weaker versions of
Solovay's result without assuming the existence of an inaccessible cardinal. In particular
153:
is a model of ZFC with the property that every set of reals that is definable over a countable sequence of ordinals is
Lebesgue measurable, and has the Baire and perfect set properties. (This includes all definable and
259:
showed that consistency of an inaccessible cardinal is also necessary for constructing a model in which all sets of reals are
Lebesgue measurable. More precisely he showed that if every
252:. Combined with Solovay's result, this shows that the statements "There is an inaccessible cardinal" and "Every set of reals has the perfect set property" are equiconsistent over ZF.
181:
satisfying ZF + DC such that every set of reals is
Lebesgue measurable, has the perfect set property, and has the Baire property. The proof of this uses the fact that every real in
162:
the notion of a definable set of reals cannot be defined in the language of set theory, while the notion of a set of reals definable over a countable sequence of ordinals can be.)
85:(Zermelo–Fraenkel set theory plus the axiom of choice), the axiom of choice is essential, at least granted that the existence of an inaccessible cardinal is consistent with ZFC.
277:
is inaccessible in the constructible universe, so that the condition about an inaccessible cardinal cannot be dropped from
Solovay's theorem. Shelah also showed that the
331:), the constructible sets generated by the reals, is Lebesgue measurable and has the Baire property; this includes every "reasonably definable" set of reals.
233:
showed that there is a model in which all sets of reals have the Baire property (so that the inaccessible cardinal is indeed unnecessary in this case).
342:
Krivine, Jean-Louis (1969), "Modèles de ZF + AC dans lesquels tout ensemble de réels définissable en termes d'ordinaux est mesurable-Lebesgue",
391:
17:
159:
289:
condition is close to the best possible by constructing a model (without using an inaccessible cardinal) in which all
436:
Raisonnier, Jean (1984), "A mathematical proof of S. Shelah's theorem on the measure problem and related results.",
240:, who showed (in ZF) that if every set of reals has the perfect set property and the first uncountable cardinal ℵ
51:
520:
473:
438:
518:(1990), "Large cardinals imply that every reasonably definable set of reals is Lebesgue measurable",
400:
Miller, Arnold W. (1989), "Review of "Can You Take
Solovay's Inaccessible Away? by Saharon Shelah"",
94:
718:
100:
Solovay's theorem is as follows. Assuming the existence of an inaccessible cardinal, there is an
713:
260:
249:
563:
320:
215:, consisting of the constructible closure of the real numbers, which has similar properties.
71:
663:
639:
600:
551:
504:
461:
363:
112:
8:
708:
67:
588:
425:
78:
371:
651:
627:
580:
558:
539:
492:
417:
387:
351:
105:
59:
43:
561:(1970), "A model of set-theory in which every set of reals is Lebesgue measurable",
619:
572:
529:
482:
447:
409:
379:
659:
635:
596:
547:
500:
457:
359:
173:
that are hereditarily definable over a countable sequence of ordinals. The model
55:
511:
468:
155:
116:
690:
Handbook of the
History of Logic: Sets and Extensions in the Twentieth Century
149:
for the notion of forcing that collapses all cardinals less than κ to ω. Then
702:
655:
631:
623:
607:
584:
543:
496:
421:
355:
135:
39:
610:(1957), "Zur Axiomatik der Mengenlehre (Fundierungs- und Auswahlaxiom)",
515:
101:
63:
685:
592:
534:
487:
452:
429:
383:
31:
576:
413:
77:
In this way
Solovay showed that in the proof of the existence of a
127:
Solovay constructed his model in two steps, starting with a model
612:
Zeitschrift fĂĽr
Mathematische Logik und Grundlagen der Mathematik
378:, Lecture Notes in Mathematics, vol. 179, pp. 187–197,
372:"Théorèmes de consistance en théorie de la mesure de R. Solovay"
273:
set of reals is measurable then the first uncountable cardinal ℵ
185:
is definable over a countable sequence of ordinals, and hence
692:, ed. A. Kanamori, D. M. Gabbay, T. Thagard, J. Woods (2011).
111:
such that every set of reals is Lebesgue measurable, has the
201:
93:
ZF stands for Zermelo–Fraenkel set theory, and DC for the
82:
344:
Comptes Rendus de l'Académie des Sciences, Série A et B
471:(1984), "Can you take Solovay's inaccessible away?",
646:Stern, Jacques (1985), "Le problème de la mesure",
236:The case of the perfect set property was solved by
70:. The construction relies on the existence of an
700:
165:The second step is to construct Solovay's model
376:Séminaire Bourbaki vol. 1968/69 Exposés 347-363
408:(2), Association for Symbolic Logic: 633–635,
131:of ZFC containing an inaccessible cardinal Îş.
510:
316:
200:, one can also use the smaller inner model
435:
302:
533:
486:
451:
158:of reals; however for reasons related to
606:
557:
369:
341:
237:
225:
47:
14:
701:
467:
399:
310:
256:
230:
645:
306:
313:for expositions of Shelah's result.
24:
301:sets of reals are measurable. See
25:
730:
323:exist then every set of reals in
196:Instead of using Solovay's model
50:) in which all of the axioms of
160:Tarski's undefinability theorem
122:
678:
218:
13:
1:
521:Israel Journal of Mathematics
474:Israel Journal of Mathematics
439:Israel Journal of Mathematics
402:The Journal of Symbolic Logic
334:
30:In the mathematical field of
686:Large Cardinals with Forcing
671:
370:Krivine, Jean-Louis (1971),
169:as the class of all sets in
134:The first step is to take a
88:
54:(ZF) hold, exclusive of the
7:
52:Zermelo–Fraenkel set theory
10:
735:
317:Shelah & Woodin (1990)
104:of ZF + DC of a suitable
95:axiom of dependent choice
624:10.1002/malq.19570031302
145:by adding a generic set
248:is inaccessible in the
27:Set theory construction
321:supercompact cardinals
250:constructible universe
564:Annals of Mathematics
193:have the same reals.
177:is an inner model of
72:inaccessible cardinal
44:Robert M. Solovay
113:perfect set property
18:Solovay's model
68:Lebesgue measurable
58:, but in which all
618:(13–20): 173–210,
559:Solovay, Robert M.
535:10.1007/BF02801471
488:10.1007/BF02760522
453:10.1007/BF02760523
384:10.1007/BFb0058812
244:is regular, then ℵ
79:non-measurable set
567:, Second Series,
393:978-3-540-05356-9
303:Raisonnier (1984)
106:forcing extension
16:(Redirected from
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693:
682:
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650:(121): 325–346,
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271:
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21:
734:
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729:
728:
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724:
723:
719:Large cardinals
699:
698:
697:
696:
683:
679:
674:
669:
577:10.2307/1970696
512:Shelah, Saharon
469:Shelah, Saharon
414:10.2307/2274892
394:
337:
319:showed that if
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295:
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293:
286:
283:
282:
281:
276:
269:
266:
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156:projective sets
125:
91:
56:axiom of choice
42:constructed by
28:
23:
22:
15:
12:
11:
5:
732:
722:
721:
716:
714:Measure theory
711:
695:
694:
684:A. Kanamori, "
676:
675:
673:
670:
668:
667:
643:
608:Specker, Ernst
604:
555:
528:(3): 381–394,
508:
465:
433:
397:
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367:
338:
336:
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296:
284:
274:
267:
245:
241:
238:Specker (1957)
226:Krivine (1969)
220:
217:
124:
121:
117:Baire property
115:, and has the
90:
87:
26:
9:
6:
4:
3:
2:
731:
720:
717:
715:
712:
710:
707:
706:
704:
691:
687:
681:
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665:
661:
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649:
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641:
637:
633:
629:
625:
621:
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598:
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586:
582:
578:
574:
570:
566:
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560:
556:
553:
549:
545:
541:
536:
531:
527:
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517:
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463:
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353:
350:: A549–A552,
349:
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332:
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318:
314:
312:
311:Miller (1989)
308:
304:
292:
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263:
258:
257:Shelah (1984)
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239:
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231:Shelah (1984)
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136:Levy collapse
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110:
107:
103:
98:
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86:
84:
80:
75:
73:
69:
65:
61:
57:
53:
49:
45:
41:
37:
36:Solovay model
33:
19:
689:
680:
647:
615:
611:
568:
562:
525:
519:
516:Woodin, Hugh
478:
472:
443:
437:
405:
401:
375:
347:
343:
328:
324:
315:
307:Stern (1985)
290:
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150:
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138:
133:
128:
126:
123:Construction
108:
99:
92:
76:
64:real numbers
35:
29:
571:(1): 1–56,
481:(1): 1–47,
219:Complements
102:inner model
709:Set theory
703:Categories
648:Astérisque
335:References
32:set theory
672:Citations
656:0303-1179
632:0044-3050
585:0003-486X
544:0021-2172
497:0021-2172
446:: 48–56,
422:0022-4812
356:0151-0509
255:Finally,
89:Statement
664:0768968
640:0099297
601:0265151
593:1970696
552:1074499
505:0768264
462:0768265
430:2274892
364:0253894
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688:". In
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630:
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591:
583:
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354:
34:, the
589:JSTOR
426:JSTOR
81:from
40:model
38:is a
652:ISSN
628:ISSN
581:ISSN
540:ISSN
493:ISSN
418:ISSN
388:ISBN
352:ISSN
309:and
305:and
189:and
66:are
60:sets
48:1970
620:doi
573:doi
530:doi
483:doi
448:doi
410:doi
380:doi
348:269
211:of
141:of
83:ZFC
62:of
705::
660:MR
658:,
636:MR
634:,
626:,
614:,
597:MR
595:,
587:,
579:,
569:92
548:MR
546:,
538:,
526:70
524:,
514:;
501:MR
499:,
491:,
479:48
477:,
458:MR
456:,
444:48
442:,
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416:,
406:54
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360:MR
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329:R
327:(
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291:Δ
285:3
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262:ÎŁ
246:1
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213:M
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207:R
205:(
203:L
198:N
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187:N
183:M
179:M
175:N
171:M
167:N
151:M
147:G
143:M
139:M
129:M
109:V
20:)
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