3250:
3320:
31:
1204:
must satisfy, and argue that there is at least one set satisfying this condition. If it is not possible to define a unique example of such a set at each stage, then it may be necessary to invoke (some form of) the axiom of choice to select one such at each step. For inductions and recursions of
1009:
to produce a well-ordered relation that can be treated by transfinite induction. However, if the relation in question is already well-ordered, one can often use transfinite induction without invoking the axiom of choice. For example, many results about
595:
All three cases are identical except for the type of ordinal considered. They do not formally need to be considered separately, but in practice the proofs are typically so different as to require separate presentations. Zero is sometimes considered a
1080:
939:
766:
673:
1217:
of interest to set theorists that satisfy the axiom of dependent choice but not the full axiom of choice, the knowledge that a particular proof only requires dependent choice can be useful.
1169:
The above argument uses the axiom of choice in an essential way at the very beginning, in order to well-order the reals. After that step, the axiom of choice is not used again.
589:
476:
234:
59:
790:
1629:
392:
1355:
534:
611:
is similar to transfinite induction; however, instead of proving that something holds for all ordinal numbers, we construct a sequence of objects, one for each ordinal.
421:
263:
155:
1384:
563:
450:
357:
208:
3784:
2304:
505:
179:
79:
1325:
1305:
1014:
are proved by transfinite induction on the ordinal rank of the set; these ranks are already well-ordered, so the axiom of choice is not needed to well-order them.
963:
to be broad enough to make the above properties meaningful. The uniqueness of the sequence satisfying these properties can be proved using transfinite induction.
322:
283:
2387:
1528:
1395:
A class function is a rule (specifically, a logical formula) assigning each element in the lefthand class to an element in the righthand class. It is not a
2701:
804:
As in the case of induction, we may treat different types of ordinals separately: another formulation of transfinite recursion is the following:
2859:
1647:
3473:
3286:
2714:
2037:
3801:
2299:
1040:
2719:
2709:
2446:
1652:
1172:
Other uses of the axiom of choice are more subtle. For example, a construction by transfinite recursion frequently will not specify a
2197:
1643:
2855:
1464:
887:
2952:
2696:
1521:
3779:
2257:
1950:
1691:
1214:
117:
3934:
3373:
3213:
2915:
2678:
2673:
2498:
1919:
1603:
721:
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3208:
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2429:
1671:
633:
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2279:
1408:
In fact, the domain of the relation does not even need to be a set. It can be a proper class, provided that the relation
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2616:
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1613:
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1450:
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3230:
3179:
3076:
2574:
2535:
2012:
1210:
3071:
1686:
3939:
3279:
3001:
2540:
2392:
2375:
2098:
1578:
3898:
3816:
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3643:
3457:
3380:
2903:
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2841:
2727:
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2314:
2234:
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2022:
1635:
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3193:
2920:
2898:
2865:
2758:
2604:
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2562:
2513:
2397:
2332:
2157:
2123:
2118:
1992:
1823:
1800:
1396:
1226:
158:
101:
3766:
3680:
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3558:
3123:
2976:
2768:
2486:
2222:
2128:
1987:
1972:
1853:
1828:
1241:
1034:
967:
568:
3249:
455:
213:
37:
1481:
775:
3840:
3830:
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3548:
3488:
3368:
3096:
3058:
2935:
2739:
2579:
2503:
2481:
2309:
2267:
2166:
2133:
1997:
1785:
1696:
705:
615:
3835:
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3649:
3463:
3405:
3336:
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3225:
3116:
3101:
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3038:
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2801:
2746:
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2471:
2419:
2187:
2176:
1848:
1748:
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362:
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3023:
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2971:
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2733:
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2525:
2508:
2461:
2274:
2183:
2017:
2002:
1962:
1914:
1899:
1887:
1843:
1818:
1588:
1537:
1485:
1456:
1334:
1236:
701:
510:
2751:
2207:
397:
239:
131:
1360:
1153:
sequence that does not have a rational difference with any element thus far constructed in the
539:
426:
336:
184:
3615:
3452:
3444:
3415:
3385:
3309:
3189:
2996:
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2688:
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2404:
2380:
2161:
2145:
2050:
2027:
1904:
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1838:
1733:
1568:
1489:
1460:
1231:
490:
331:
164:
64:
3903:
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3741:
3395:
3203:
3198:
3091:
3048:
2870:
2831:
2826:
2811:
2637:
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2491:
2289:
2239:
1813:
1775:
1310:
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3710:
3528:
3341:
3184:
3174:
3128:
3111:
3066:
3028:
2930:
2850:
2657:
2584:
2557:
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2451:
2365:
2339:
2294:
2262:
2063:
1865:
1808:
1758:
1723:
1681:
1492:
1281:
1146:
1006:
982:
298:
113:
1246:
1021:
shows one way that the axiom of choice can be used in a proof by transfinite induction:
30:
3908:
3705:
3686:
3590:
3575:
3532:
3468:
3410:
3169:
3148:
3106:
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2836:
2434:
2424:
2414:
2409:
2343:
2217:
2093:
1982:
1977:
1955:
1556:
1446:
1328:
268:
109:
3928:
3913:
3883:
3715:
3629:
3624:
3143:
2821:
2328:
2113:
2103:
2073:
2058:
1728:
1206:
597:
485:
17:
3863:
3858:
3676:
3605:
3563:
3422:
3319:
3043:
2890:
2791:
2783:
2663:
2611:
2520:
2456:
2439:
2370:
2229:
2088:
1790:
1573:
978:
619:
3888:
3523:
3153:
3033:
2212:
2202:
2149:
1833:
1753:
1738:
1618:
1563:
1030:
627:
600:
and then may sometimes be treated in proofs in the same case as limit ordinals.
3868:
3736:
3639:
3295:
2083:
1938:
1909:
1715:
1026:
1018:
1011:
105:
3671:
3634:
3585:
3483:
3235:
3138:
2191:
2108:
2068:
2032:
1968:
1780:
1770:
1743:
1497:
678:
More formally, we can state the
Transfinite Recursion Theorem as follows:
3220:
3018:
2466:
2171:
1765:
1266:
2816:
1608:
966:
More generally, one can define objects by transfinite recursion on any
3696:
3518:
1506:
1005:
Proofs or constructions using induction and recursion often use the
622:
can be created by starting with the empty set and for each ordinal
3568:
3328:
2360:
1706:
1551:
1075:{\displaystyle \langle r_{\alpha }\mid \alpha <\beta \rangle }
3268:
1510:
3264:
934:{\displaystyle F(\lambda )=G_{3}(F\upharpoonright \lambda )}
89:(used for those ordinals which have a predecessor), and a
265:
is also true. Then transfinite induction tells us that
1149:. Continue; at each step use the least real from the
761:{\displaystyle F(\alpha )=G(F\upharpoonright \alpha )}
1363:
1337:
1313:
1284:
1043:
890:
778:
724:
636:
571:
542:
513:
493:
458:
429:
400:
365:
339:
301:
271:
242:
216:
187:
167:
134:
67:
40:
668:{\displaystyle \{v_{\beta }\mid \beta <\alpha \}}
3849:
3812:
3724:
3614:
3502:
3443:
3327:
3302:
3162:
3057:
2889:
2782:
2634:
2327:
2250:
2144:
2048:
1937:
1864:
1799:
1714:
1705:
1627:
1544:
1278:It is not necessary here to assume separately that
715:(where Ord is the class of all ordinals) such that
675:. This process stops when no vector can be chosen.
288:Usually the proof is broken down into three cases:
93:(used for ordinals which don't have a predecessor).
1378:
1349:
1319:
1299:
1074:
1033:(this is where the axiom of choice enters via the
933:
784:
760:
667:
583:
557:
528:
499:
470:
444:
415:
386:
351:
316:
277:
257:
228:
202:
173:
149:
73:
61:. Each turn of the spiral represents one power of
53:
1157:sequence. Continue until all the reals in the
3280:
1522:
1399:because its domain and codomain are not sets.
8:
1069:
1044:
662:
637:
34:Representation of the ordinal numbers up to
1213:is sufficient. Because there are models of
81:. Transfinite induction requires proving a
3287:
3273:
3265:
2348:
1943:
1711:
1529:
1515:
1507:
1362:
1336:
1312:
1283:
1051:
1042:
910:
889:
808:Transfinite Recursion Theorem (version 2)
777:
723:
682:Transfinite Recursion Theorem (version 1)
644:
635:
570:
541:
512:
492:
457:
428:
399:
364:
338:
300:
270:
241:
215:
186:
166:
133:
66:
45:
39:
29:
1258:
1165:sequence will enumerate the Vitali set.
1082:, where β is an ordinal with the
618:for a (possibly infinite-dimensional)
626:choosing a vector that is not in the
7:
977:need not even be a set; it can be a
949:Note that we require the domains of
704:of all sets), there exists a unique
1161:sequence are exhausted. The final
1001:Relationship to the axiom of choice
1017:The following construction of the
584:{\displaystyle \beta <\lambda }
116:. Its correctness is a theorem of
25:
831:, there exists a unique function
471:{\displaystyle \beta <\alpha }
229:{\displaystyle \beta <\alpha }
54:{\displaystyle \omega ^{\omega }}
3318:
3248:
796:s domain to ordinals <
785:{\displaystyle \upharpoonright }
1373:
1367:
1294:
1288:
928:
922:
916:
900:
894:
779:
755:
749:
743:
734:
728:
552:
546:
523:
517:
439:
433:
410:
404:
381:
369:
311:
305:
252:
246:
197:
191:
144:
138:
1:
3209:History of mathematical logic
3134:Primitive recursive function
1084:cardinality of the continuum
387:{\displaystyle P(\alpha +1)}
1350:{\displaystyle \beta <0}
1215:ZermeloâFraenkel set theory
1186:, given the sequence up to
792:denotes the restriction of
529:{\displaystyle P(\lambda )}
3961:
3785:von NeumannâBernaysâGödel
2198:SchröderâBernstein theorem
1925:Monadic predicate calculus
1584:Foundations of mathematics
1190:, but will specify only a
416:{\displaystyle P(\alpha )}
285:is true for all ordinals.
258:{\displaystyle P(\alpha )}
150:{\displaystyle P(\alpha )}
3586:One-to-one correspondence
3316:
3244:
3231:Philosophy of mathematics
3180:Automated theorem proving
2351:
2305:Von NeumannâBernaysâGödel
1946:
1379:{\displaystyle P(\beta )}
1307:is true. As there is no
1211:axiom of dependent choice
684:. Given a class function
558:{\displaystyle P(\beta )}
445:{\displaystyle P(\beta )}
352:{\displaystyle \alpha +1}
203:{\displaystyle P(\beta )}
161:defined for all ordinals
108:, for example to sets of
1416:, the collection of all
989:, the collection of all
500:{\displaystyle \lambda }
181:. Suppose that whenever
2881:Self-verifying theories
2702:Tarski's axiomatization
1653:Tarski's undefinability
1648:incompleteness theorems
1493:"Transfinite Induction"
1449:(1972), "Section 7.1",
174:{\displaystyle \alpha }
74:{\displaystyle \omega }
3935:Mathematical induction
3544:Constructible universe
3364:Constructibility (V=L)
3255:Mathematics portal
2866:Proof of impossibility
2514:propositional variable
1824:Propositional calculus
1380:
1351:
1321:
1320:{\displaystyle \beta }
1301:
1269:. Accessed 2022-03-24.
1242:Well-founded induction
1227:Mathematical induction
1076:
935:
817:, and class functions
786:
762:
669:
585:
559:
530:
501:
472:
446:
417:
388:
353:
318:
279:
259:
230:
204:
175:
151:
102:mathematical induction
94:
75:
55:
3767:Principia Mathematica
3601:Transfinite induction
3460:(i.e. set difference)
3124:Kolmogorov complexity
3077:Computably enumerable
2977:Model complete theory
2769:Principia Mathematica
1829:Propositional formula
1658:BanachâTarski paradox
1412:is set-like: for any
1381:
1352:
1322:
1302:
1077:
1037:), giving a sequence
1035:well-ordering theorem
968:well-founded relation
936:
787:
763:
670:
609:Transfinite recursion
604:Transfinite recursion
586:
560:
531:
502:
473:
447:
418:
389:
354:
319:
280:
260:
231:
205:
176:
152:
98:Transfinite induction
76:
56:
33:
18:Transfinite iteration
3841:Burali-Forti paradox
3596:Set-builder notation
3549:Continuum hypothesis
3489:Symmetric difference
3072:ChurchâTuring thesis
3059:Computability theory
2268:continuum hypothesis
1786:Square of opposition
1644:Gödel's completeness
1452:Axiomatic set theory
1361:
1335:
1311:
1300:{\displaystyle P(0)}
1282:
1041:
888:
776:
722:
706:transfinite sequence
634:
569:
540:
511:
491:
456:
427:
423:(and, if necessary,
398:
363:
337:
317:{\displaystyle P(0)}
299:
269:
240:
214:
185:
165:
132:
65:
38:
27:Mathematical concept
3802:TarskiâGrothendieck
3226:Mathematical object
3117:P versus NP problem
3082:Computable function
2876:Reverse mathematics
2802:Logical consequence
2679:primitive recursive
2674:elementary function
2447:Free/bound variable
2300:TarskiâGrothendieck
1819:Logical connectives
1749:Logical equivalence
1599:Logical consequence
1327:less than 0, it is
1209:length, the weaker
1138: −
1126:is least such that
981:, provided it is a
484:Prove that for any
330:Prove that for any
100:is an extension of
3391:Limitation of size
3024:Transfer principle
2987:Semantics of logic
2972:Categorical theory
2948:Non-standard model
2462:Logical connective
1589:Information theory
1538:Mathematical logic
1490:Weisstein, Eric W.
1457:Dover Publications
1376:
1347:
1317:
1297:
1267:Ordinal Arithmetic
1237:Transfinite number
1072:
931:
782:
758:
665:
581:
555:
526:
497:
468:
442:
413:
384:
349:
314:
275:
255:
226:
200:
171:
147:
124:Induction by cases
95:
71:
51:
3922:
3921:
3831:Russell's paradox
3780:ZermeloâFraenkel
3681:Dedekind-infinite
3554:Diagonal argument
3453:Cartesian product
3310:Set (mathematics)
3262:
3261:
3194:Abstract category
2997:Theories of truth
2807:Rule of inference
2797:Natural deduction
2778:
2777:
2323:
2322:
2028:Cartesian product
1933:
1932:
1839:Many-valued logic
1814:Boolean functions
1697:Russell's paradox
1672:diagonal argument
1569:First-order logic
1482:Emerson, Jonathan
983:set-like relation
768:for all ordinals
614:As an example, a
332:successor ordinal
278:{\displaystyle P}
210:is true for all
106:well-ordered sets
16:(Redirected from
3952:
3904:Bertrand Russell
3894:John von Neumann
3879:Abraham Fraenkel
3874:Richard Dedekind
3836:Suslin's problem
3747:Cantor's theorem
3464:De Morgan's laws
3322:
3289:
3282:
3275:
3266:
3253:
3252:
3204:History of logic
3199:Category of sets
3092:Decision problem
2871:Ordinal analysis
2812:Sequent calculus
2710:Boolean algebras
2650:
2649:
2624:
2595:logical/constant
2349:
2335:
2258:ZermeloâFraenkel
2009:Set operations:
1944:
1881:
1712:
1692:LöwenheimâSkolem
1579:Formal semantics
1531:
1524:
1517:
1508:
1503:
1502:
1469:
1433:
1406:
1400:
1393:
1387:
1385:
1383:
1382:
1377:
1356:
1354:
1353:
1348:
1326:
1324:
1323:
1318:
1306:
1304:
1303:
1298:
1276:
1270:
1263:
1081:
1079:
1078:
1073:
1056:
1055:
941:, for all limit
940:
938:
937:
932:
915:
914:
791:
789:
788:
783:
767:
765:
764:
759:
674:
672:
671:
666:
649:
648:
590:
588:
587:
582:
564:
562:
561:
556:
535:
533:
532:
527:
506:
504:
503:
498:
477:
475:
474:
469:
451:
449:
448:
443:
422:
420:
419:
414:
393:
391:
390:
385:
358:
356:
355:
350:
323:
321:
320:
315:
284:
282:
281:
276:
264:
262:
261:
256:
235:
233:
232:
227:
209:
207:
206:
201:
180:
178:
177:
172:
156:
154:
153:
148:
114:cardinal numbers
85:(used for 0), a
80:
78:
77:
72:
60:
58:
57:
52:
50:
49:
21:
3960:
3959:
3955:
3954:
3953:
3951:
3950:
3949:
3940:Ordinal numbers
3925:
3924:
3923:
3918:
3845:
3824:
3808:
3773:New Foundations
3720:
3610:
3529:Cardinal number
3512:
3498:
3439:
3323:
3314:
3298:
3293:
3263:
3258:
3247:
3240:
3185:Category theory
3175:Algebraic logic
3158:
3129:Lambda calculus
3067:Church encoding
3053:
3029:Truth predicate
2885:
2851:Complete theory
2774:
2643:
2639:
2635:
2630:
2622:
2342: and
2338:
2333:
2319:
2295:New Foundations
2263:axiom of choice
2246:
2208:Gödel numbering
2148: and
2140:
2044:
1929:
1879:
1860:
1809:Boolean algebra
1795:
1759:Equiconsistency
1724:Classical logic
1701:
1682:Halting problem
1670: and
1646: and
1634: and
1633:
1628:Theorems (
1623:
1540:
1535:
1480:
1479:
1476:
1467:
1447:Suppes, Patrick
1445:
1442:
1437:
1436:
1407:
1403:
1394:
1390:
1359:
1358:
1333:
1332:
1309:
1308:
1280:
1279:
1277:
1273:
1264:
1260:
1255:
1223:
1203:
1185:
1147:rational number
1144:
1137:
1136:
1125:
1118:
1117:
1106:
1099:
1092:
1047:
1039:
1038:
1007:axiom of choice
1003:
985:; i.e. for any
962:
955:
906:
886:
885:
869:
852:
830:
823:
816:
774:
773:
720:
719:
640:
632:
631:
630:of the vectors
606:
567:
566:
538:
537:
509:
508:
489:
488:
454:
453:
425:
424:
396:
395:
361:
360:
335:
334:
328:Successor case:
297:
296:
267:
266:
238:
237:
212:
211:
183:
182:
163:
162:
130:
129:
126:
110:ordinal numbers
63:
62:
41:
36:
35:
28:
23:
22:
15:
12:
11:
5:
3958:
3956:
3948:
3947:
3942:
3937:
3927:
3926:
3920:
3919:
3917:
3916:
3911:
3909:Thoralf Skolem
3906:
3901:
3896:
3891:
3886:
3881:
3876:
3871:
3866:
3861:
3855:
3853:
3847:
3846:
3844:
3843:
3838:
3833:
3827:
3825:
3823:
3822:
3819:
3813:
3810:
3809:
3807:
3806:
3805:
3804:
3799:
3794:
3793:
3792:
3777:
3776:
3775:
3763:
3762:
3761:
3750:
3749:
3744:
3739:
3734:
3728:
3726:
3722:
3721:
3719:
3718:
3713:
3708:
3703:
3694:
3689:
3684:
3674:
3669:
3668:
3667:
3662:
3657:
3647:
3637:
3632:
3627:
3621:
3619:
3612:
3611:
3609:
3608:
3603:
3598:
3593:
3591:Ordinal number
3588:
3583:
3578:
3573:
3572:
3571:
3566:
3556:
3551:
3546:
3541:
3536:
3526:
3521:
3515:
3513:
3511:
3510:
3507:
3503:
3500:
3499:
3497:
3496:
3491:
3486:
3481:
3476:
3471:
3469:Disjoint union
3466:
3461:
3455:
3449:
3447:
3441:
3440:
3438:
3437:
3436:
3435:
3430:
3419:
3418:
3416:Martin's axiom
3413:
3408:
3403:
3398:
3393:
3388:
3383:
3381:Extensionality
3378:
3377:
3376:
3366:
3361:
3360:
3359:
3354:
3349:
3339:
3333:
3331:
3325:
3324:
3317:
3315:
3313:
3312:
3306:
3304:
3300:
3299:
3294:
3292:
3291:
3284:
3277:
3269:
3260:
3259:
3245:
3242:
3241:
3239:
3238:
3233:
3228:
3223:
3218:
3217:
3216:
3206:
3201:
3196:
3187:
3182:
3177:
3172:
3170:Abstract logic
3166:
3164:
3160:
3159:
3157:
3156:
3151:
3149:Turing machine
3146:
3141:
3136:
3131:
3126:
3121:
3120:
3119:
3114:
3109:
3104:
3099:
3089:
3087:Computable set
3084:
3079:
3074:
3069:
3063:
3061:
3055:
3054:
3052:
3051:
3046:
3041:
3036:
3031:
3026:
3021:
3016:
3015:
3014:
3009:
3004:
2994:
2989:
2984:
2982:Satisfiability
2979:
2974:
2969:
2968:
2967:
2957:
2956:
2955:
2945:
2944:
2943:
2938:
2933:
2928:
2923:
2913:
2912:
2911:
2906:
2899:Interpretation
2895:
2893:
2887:
2886:
2884:
2883:
2878:
2873:
2868:
2863:
2853:
2848:
2847:
2846:
2845:
2844:
2834:
2829:
2819:
2814:
2809:
2804:
2799:
2794:
2788:
2786:
2780:
2779:
2776:
2775:
2773:
2772:
2764:
2763:
2762:
2761:
2756:
2755:
2754:
2749:
2744:
2724:
2723:
2722:
2720:minimal axioms
2717:
2706:
2705:
2704:
2693:
2692:
2691:
2686:
2681:
2676:
2671:
2666:
2653:
2651:
2632:
2631:
2629:
2628:
2627:
2626:
2614:
2609:
2608:
2607:
2602:
2597:
2592:
2582:
2577:
2572:
2567:
2566:
2565:
2560:
2550:
2549:
2548:
2543:
2538:
2533:
2523:
2518:
2517:
2516:
2511:
2506:
2496:
2495:
2494:
2489:
2484:
2479:
2474:
2469:
2459:
2454:
2449:
2444:
2443:
2442:
2437:
2432:
2427:
2417:
2412:
2410:Formation rule
2407:
2402:
2401:
2400:
2395:
2385:
2384:
2383:
2373:
2368:
2363:
2358:
2352:
2346:
2329:Formal systems
2325:
2324:
2321:
2320:
2318:
2317:
2312:
2307:
2302:
2297:
2292:
2287:
2282:
2277:
2272:
2271:
2270:
2265:
2254:
2252:
2248:
2247:
2245:
2244:
2243:
2242:
2232:
2227:
2226:
2225:
2218:Large cardinal
2215:
2210:
2205:
2200:
2195:
2181:
2180:
2179:
2174:
2169:
2154:
2152:
2142:
2141:
2139:
2138:
2137:
2136:
2131:
2126:
2116:
2111:
2106:
2101:
2096:
2091:
2086:
2081:
2076:
2071:
2066:
2061:
2055:
2053:
2046:
2045:
2043:
2042:
2041:
2040:
2035:
2030:
2025:
2020:
2015:
2007:
2006:
2005:
2000:
1990:
1985:
1983:Extensionality
1980:
1978:Ordinal number
1975:
1965:
1960:
1959:
1958:
1947:
1941:
1935:
1934:
1931:
1930:
1928:
1927:
1922:
1917:
1912:
1907:
1902:
1897:
1896:
1895:
1885:
1884:
1883:
1870:
1868:
1862:
1861:
1859:
1858:
1857:
1856:
1851:
1846:
1836:
1831:
1826:
1821:
1816:
1811:
1805:
1803:
1797:
1796:
1794:
1793:
1788:
1783:
1778:
1773:
1768:
1763:
1762:
1761:
1751:
1746:
1741:
1736:
1731:
1726:
1720:
1718:
1709:
1703:
1702:
1700:
1699:
1694:
1689:
1684:
1679:
1674:
1662:Cantor's
1660:
1655:
1650:
1640:
1638:
1625:
1624:
1622:
1621:
1616:
1611:
1606:
1601:
1596:
1591:
1586:
1581:
1576:
1571:
1566:
1561:
1560:
1559:
1548:
1546:
1542:
1541:
1536:
1534:
1533:
1526:
1519:
1511:
1505:
1504:
1475:
1474:External links
1472:
1471:
1470:
1465:
1441:
1438:
1435:
1434:
1432:must be a set.
1401:
1388:
1375:
1372:
1369:
1366:
1346:
1343:
1340:
1329:vacuously true
1316:
1296:
1293:
1290:
1287:
1271:
1257:
1256:
1254:
1251:
1250:
1249:
1244:
1239:
1234:
1229:
1222:
1219:
1198:
1180:
1167:
1166:
1142:
1134:
1130:
1123:
1115:
1111:
1104:
1097:
1090:
1071:
1068:
1065:
1062:
1059:
1054:
1050:
1046:
1002:
999:
960:
953:
947:
946:
930:
927:
924:
921:
918:
913:
909:
905:
902:
899:
896:
893:
883:
867:
854:
850:
828:
821:
814:
810:. Given a set
802:
801:
781:
757:
754:
751:
748:
745:
742:
739:
736:
733:
730:
727:
664:
661:
658:
655:
652:
647:
643:
639:
605:
602:
593:
592:
580:
577:
574:
554:
551:
548:
545:
525:
522:
519:
516:
496:
479:
467:
464:
461:
441:
438:
435:
432:
412:
409:
406:
403:
383:
380:
377:
374:
371:
368:
348:
345:
342:
325:
313:
310:
307:
304:
274:
254:
251:
248:
245:
225:
222:
219:
199:
196:
193:
190:
170:
146:
143:
140:
137:
125:
122:
87:successor case
70:
48:
44:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3957:
3946:
3943:
3941:
3938:
3936:
3933:
3932:
3930:
3915:
3914:Ernst Zermelo
3912:
3910:
3907:
3905:
3902:
3900:
3899:Willard Quine
3897:
3895:
3892:
3890:
3887:
3885:
3882:
3880:
3877:
3875:
3872:
3870:
3867:
3865:
3862:
3860:
3857:
3856:
3854:
3852:
3851:Set theorists
3848:
3842:
3839:
3837:
3834:
3832:
3829:
3828:
3826:
3820:
3818:
3815:
3814:
3811:
3803:
3800:
3798:
3797:KripkeâPlatek
3795:
3791:
3788:
3787:
3786:
3783:
3782:
3781:
3778:
3774:
3771:
3770:
3769:
3768:
3764:
3760:
3757:
3756:
3755:
3752:
3751:
3748:
3745:
3743:
3740:
3738:
3735:
3733:
3730:
3729:
3727:
3723:
3717:
3714:
3712:
3709:
3707:
3704:
3702:
3700:
3695:
3693:
3690:
3688:
3685:
3682:
3678:
3675:
3673:
3670:
3666:
3663:
3661:
3658:
3656:
3653:
3652:
3651:
3648:
3645:
3641:
3638:
3636:
3633:
3631:
3628:
3626:
3623:
3622:
3620:
3617:
3613:
3607:
3604:
3602:
3599:
3597:
3594:
3592:
3589:
3587:
3584:
3582:
3579:
3577:
3574:
3570:
3567:
3565:
3562:
3561:
3560:
3557:
3555:
3552:
3550:
3547:
3545:
3542:
3540:
3537:
3534:
3530:
3527:
3525:
3522:
3520:
3517:
3516:
3514:
3508:
3505:
3504:
3501:
3495:
3492:
3490:
3487:
3485:
3482:
3480:
3477:
3475:
3472:
3470:
3467:
3465:
3462:
3459:
3456:
3454:
3451:
3450:
3448:
3446:
3442:
3434:
3433:specification
3431:
3429:
3426:
3425:
3424:
3421:
3420:
3417:
3414:
3412:
3409:
3407:
3404:
3402:
3399:
3397:
3394:
3392:
3389:
3387:
3384:
3382:
3379:
3375:
3372:
3371:
3370:
3367:
3365:
3362:
3358:
3355:
3353:
3350:
3348:
3345:
3344:
3343:
3340:
3338:
3335:
3334:
3332:
3330:
3326:
3321:
3311:
3308:
3307:
3305:
3301:
3297:
3290:
3285:
3283:
3278:
3276:
3271:
3270:
3267:
3257:
3256:
3251:
3243:
3237:
3234:
3232:
3229:
3227:
3224:
3222:
3219:
3215:
3212:
3211:
3210:
3207:
3205:
3202:
3200:
3197:
3195:
3191:
3188:
3186:
3183:
3181:
3178:
3176:
3173:
3171:
3168:
3167:
3165:
3161:
3155:
3152:
3150:
3147:
3145:
3144:Recursive set
3142:
3140:
3137:
3135:
3132:
3130:
3127:
3125:
3122:
3118:
3115:
3113:
3110:
3108:
3105:
3103:
3100:
3098:
3095:
3094:
3093:
3090:
3088:
3085:
3083:
3080:
3078:
3075:
3073:
3070:
3068:
3065:
3064:
3062:
3060:
3056:
3050:
3047:
3045:
3042:
3040:
3037:
3035:
3032:
3030:
3027:
3025:
3022:
3020:
3017:
3013:
3010:
3008:
3005:
3003:
3000:
2999:
2998:
2995:
2993:
2990:
2988:
2985:
2983:
2980:
2978:
2975:
2973:
2970:
2966:
2963:
2962:
2961:
2958:
2954:
2953:of arithmetic
2951:
2950:
2949:
2946:
2942:
2939:
2937:
2934:
2932:
2929:
2927:
2924:
2922:
2919:
2918:
2917:
2914:
2910:
2907:
2905:
2902:
2901:
2900:
2897:
2896:
2894:
2892:
2888:
2882:
2879:
2877:
2874:
2872:
2869:
2867:
2864:
2861:
2860:from ZFC
2857:
2854:
2852:
2849:
2843:
2840:
2839:
2838:
2835:
2833:
2830:
2828:
2825:
2824:
2823:
2820:
2818:
2815:
2813:
2810:
2808:
2805:
2803:
2800:
2798:
2795:
2793:
2790:
2789:
2787:
2785:
2781:
2771:
2770:
2766:
2765:
2760:
2759:non-Euclidean
2757:
2753:
2750:
2748:
2745:
2743:
2742:
2738:
2737:
2735:
2732:
2731:
2729:
2725:
2721:
2718:
2716:
2713:
2712:
2711:
2707:
2703:
2700:
2699:
2698:
2694:
2690:
2687:
2685:
2682:
2680:
2677:
2675:
2672:
2670:
2667:
2665:
2662:
2661:
2659:
2655:
2654:
2652:
2647:
2641:
2636:Example
2633:
2625:
2620:
2619:
2618:
2615:
2613:
2610:
2606:
2603:
2601:
2598:
2596:
2593:
2591:
2588:
2587:
2586:
2583:
2581:
2578:
2576:
2573:
2571:
2568:
2564:
2561:
2559:
2556:
2555:
2554:
2551:
2547:
2544:
2542:
2539:
2537:
2534:
2532:
2529:
2528:
2527:
2524:
2522:
2519:
2515:
2512:
2510:
2507:
2505:
2502:
2501:
2500:
2497:
2493:
2490:
2488:
2485:
2483:
2480:
2478:
2475:
2473:
2470:
2468:
2465:
2464:
2463:
2460:
2458:
2455:
2453:
2450:
2448:
2445:
2441:
2438:
2436:
2433:
2431:
2428:
2426:
2423:
2422:
2421:
2418:
2416:
2413:
2411:
2408:
2406:
2403:
2399:
2396:
2394:
2393:by definition
2391:
2390:
2389:
2386:
2382:
2379:
2378:
2377:
2374:
2372:
2369:
2367:
2364:
2362:
2359:
2357:
2354:
2353:
2350:
2347:
2345:
2341:
2336:
2330:
2326:
2316:
2313:
2311:
2308:
2306:
2303:
2301:
2298:
2296:
2293:
2291:
2288:
2286:
2283:
2281:
2280:KripkeâPlatek
2278:
2276:
2273:
2269:
2266:
2264:
2261:
2260:
2259:
2256:
2255:
2253:
2249:
2241:
2238:
2237:
2236:
2233:
2231:
2228:
2224:
2221:
2220:
2219:
2216:
2214:
2211:
2209:
2206:
2204:
2201:
2199:
2196:
2193:
2189:
2185:
2182:
2178:
2175:
2173:
2170:
2168:
2165:
2164:
2163:
2159:
2156:
2155:
2153:
2151:
2147:
2143:
2135:
2132:
2130:
2127:
2125:
2124:constructible
2122:
2121:
2120:
2117:
2115:
2112:
2110:
2107:
2105:
2102:
2100:
2097:
2095:
2092:
2090:
2087:
2085:
2082:
2080:
2077:
2075:
2072:
2070:
2067:
2065:
2062:
2060:
2057:
2056:
2054:
2052:
2047:
2039:
2036:
2034:
2031:
2029:
2026:
2024:
2021:
2019:
2016:
2014:
2011:
2010:
2008:
2004:
2001:
1999:
1996:
1995:
1994:
1991:
1989:
1986:
1984:
1981:
1979:
1976:
1974:
1970:
1966:
1964:
1961:
1957:
1954:
1953:
1952:
1949:
1948:
1945:
1942:
1940:
1936:
1926:
1923:
1921:
1918:
1916:
1913:
1911:
1908:
1906:
1903:
1901:
1898:
1894:
1891:
1890:
1889:
1886:
1882:
1877:
1876:
1875:
1872:
1871:
1869:
1867:
1863:
1855:
1852:
1850:
1847:
1845:
1842:
1841:
1840:
1837:
1835:
1832:
1830:
1827:
1825:
1822:
1820:
1817:
1815:
1812:
1810:
1807:
1806:
1804:
1802:
1801:Propositional
1798:
1792:
1789:
1787:
1784:
1782:
1779:
1777:
1774:
1772:
1769:
1767:
1764:
1760:
1757:
1756:
1755:
1752:
1750:
1747:
1745:
1742:
1740:
1737:
1735:
1732:
1730:
1729:Logical truth
1727:
1725:
1722:
1721:
1719:
1717:
1713:
1710:
1708:
1704:
1698:
1695:
1693:
1690:
1688:
1685:
1683:
1680:
1678:
1675:
1673:
1669:
1665:
1661:
1659:
1656:
1654:
1651:
1649:
1645:
1642:
1641:
1639:
1637:
1631:
1626:
1620:
1617:
1615:
1612:
1610:
1607:
1605:
1602:
1600:
1597:
1595:
1592:
1590:
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1466:0-486-61630-4
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1415:
1411:
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1392:
1389:
1370:
1364:
1344:
1341:
1338:
1331:that for all
1330:
1314:
1291:
1285:
1275:
1272:
1268:
1265:J. Schlöder,
1262:
1259:
1252:
1248:
1245:
1243:
1240:
1238:
1235:
1233:
1230:
1228:
1225:
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1220:
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1216:
1212:
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1114:
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1103:
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1085:
1066:
1063:
1060:
1057:
1052:
1048:
1036:
1032:
1028:
1024:
1023:
1022:
1020:
1015:
1013:
1008:
1000:
998:
996:
992:
988:
984:
980:
976:
972:
969:
964:
959:
952:
944:
925:
919:
911:
907:
903:
897:
891:
884:
881:
877:
873:
866:
862:
858:
855:
849:
845:
842:
841:
840:
838:
834:
827:
820:
813:
809:
805:
799:
795:
771:
752:
746:
740:
737:
731:
725:
718:
717:
716:
714:
710:
707:
703:
699:
695:
691:
687:
683:
679:
676:
659:
656:
653:
650:
645:
641:
629:
625:
621:
617:
612:
610:
603:
601:
599:
598:limit ordinal
578:
575:
572:
549:
543:
536:follows from
520:
514:
494:
487:
486:limit ordinal
483:
480:
465:
462:
459:
436:
430:
407:
401:
394:follows from
378:
375:
372:
366:
346:
343:
340:
333:
329:
326:
308:
302:
294:
291:
290:
289:
286:
272:
249:
243:
223:
220:
217:
194:
188:
168:
160:
141:
135:
123:
121:
119:
115:
111:
107:
103:
99:
92:
88:
84:
68:
46:
42:
32:
19:
3864:Georg Cantor
3859:Paul Bernays
3790:MorseâKelley
3765:
3698:
3697:Subset
3644:hereditarily
3606:Venn diagram
3600:
3564:ordered pair
3479:Intersection
3423:Axiom schema
3246:
3044:Ultraproduct
2891:Model theory
2856:Independence
2792:Formal proof
2784:Proof theory
2767:
2740:
2697:real numbers
2669:second-order
2580:Substitution
2457:Metalanguage
2398:conservative
2371:Axiom schema
2315:Constructive
2285:MorseâKelley
2251:Set theories
2230:Aleph number
2223:inaccessible
2129:Grothendieck
2013:intersection
1900:Higher-order
1888:Second-order
1834:Truth tables
1791:Venn diagram
1574:Formal proof
1496:
1486:Lezama, Mark
1451:
1429:
1425:
1421:
1417:
1413:
1409:
1404:
1391:
1274:
1261:
1247:Zorn's lemma
1199:
1195:
1191:
1187:
1181:
1177:
1173:
1171:
1168:
1162:
1158:
1154:
1150:
1139:
1131:
1127:
1120:
1112:
1108:
1101:
1100:. Then let
1094:
1087:
1031:real numbers
1016:
1004:
994:
990:
986:
979:proper class
974:
970:
965:
957:
950:
948:
942:
879:
878:)), for all
875:
871:
864:
860:
856:
847:
843:
836:
832:
825:
818:
811:
807:
806:
803:
797:
793:
769:
712:
708:
697:
693:
689:
685:
681:
680:
677:
623:
620:vector space
613:
608:
607:
594:
481:
327:
292:
287:
127:
97:
96:
90:
86:
82:
3889:Thomas Jech
3732:Alternative
3711:Uncountable
3665:Ultrafilter
3524:Cardinality
3428:replacement
3369:Determinacy
3154:Type theory
3102:undecidable
3034:Truth value
2921:equivalence
2600:non-logical
2213:Enumeration
2203:Isomorphism
2150:cardinality
2134:Von Neumann
2099:Ultrafilter
2064:Uncountable
1998:equivalence
1915:Quantifiers
1905:Fixed-point
1874:First-order
1754:Consistency
1739:Proposition
1716:Traditional
1687:Lindström's
1677:Compactness
1619:Type theory
1564:Cardinality
1232:â-induction
997:is a set.)
482:Limit case:
295:Prove that
3929:Categories
3884:Kurt Gödel
3869:Paul Cohen
3706:Transitive
3474:Identities
3458:Complement
3445:Operations
3406:Regularity
3374:projective
3337:Adjunction
3296:Set theory
2965:elementary
2658:arithmetic
2526:Quantifier
2504:functional
2376:Expression
2094:Transitive
2038:identities
2023:complement
1956:hereditary
1939:Set theory
1440:References
1420:such that
1176:value for
1027:well-order
1019:Vitali set
1012:Borel sets
993:such that
839:such that
293:Zero case:
91:limit case
3945:Recursion
3817:Paradoxes
3737:Axiomatic
3716:Universal
3692:Singleton
3687:Recursive
3630:Countable
3625:Amorphous
3484:Power set
3401:Power set
3352:dependent
3347:countable
3236:Supertask
3139:Recursion
3097:decidable
2931:saturated
2909:of models
2832:deductive
2827:axiomatic
2747:Hilbert's
2734:Euclidean
2715:canonical
2638:axiomatic
2570:Signature
2499:Predicate
2388:Extension
2310:Ackermann
2235:Operation
2114:Universal
2104:Recursive
2079:Singleton
2074:Inhabited
2059:Countable
2049:Types of
2033:power set
2003:partition
1920:Predicate
1866:Predicate
1781:Syllogism
1771:Soundness
1744:Inference
1734:Tautology
1636:paradoxes
1498:MathWorld
1371:β
1339:β
1315:β
1207:countable
1192:condition
1145:is not a
1070:⟩
1067:β
1061:α
1058:∣
1053:α
1045:⟨
926:λ
923:↾
898:λ
780:↾
753:α
750:↾
732:α
660:α
654:β
651:∣
646:β
579:λ
573:β
550:β
521:λ
495:λ
466:α
460:β
437:β
408:α
373:α
341:α
250:α
224:α
218:β
195:β
169:α
142:α
83:base case
69:ω
47:ω
43:ω
3821:Problems
3725:Theories
3701:Superset
3677:Infinite
3506:Concepts
3386:Infinity
3303:Overview
3221:Logicism
3214:timeline
3190:Concrete
3049:Validity
3019:T-schema
3012:Kripke's
3007:Tarski's
3002:semantic
2992:Strength
2941:submodel
2936:spectrum
2904:function
2752:Tarski's
2741:Elements
2728:geometry
2684:Robinson
2605:variable
2590:function
2563:spectrum
2553:Sentence
2509:variable
2452:Language
2405:Relation
2366:Automata
2356:Alphabet
2340:language
2194:-jection
2172:codomain
2158:Function
2119:Universe
2089:Infinite
1993:Relation
1776:Validity
1766:Argument
1664:theorem,
1397:function
1386:is true.
1221:See also
1119:, where
835:: Ord â
772:, where
711:: Ord â
624:α > 0
565:for all
452:for all
324:is true.
159:property
3759:General
3754:Zermelo
3660:subbase
3642: (
3581:Forcing
3559:Element
3531: (
3509:Methods
3396:Pairing
3163:Related
2960:Diagram
2858: (
2837:Hilbert
2822:Systems
2817:Theorem
2695:of the
2640:systems
2420:Formula
2415:Grammar
2331: (
2275:General
1988:Forcing
1973:Element
1893:Monadic
1668:paradox
1609:Theorem
1545:General
1086:. Let
1025:First,
863:+ 1) =
700:is the
696:(where
236:, then
3650:Filter
3640:Finite
3576:Family
3519:Almost
3357:global
3342:Choice
3329:Axioms
2926:finite
2689:Skolem
2642:
2617:Theory
2585:Symbol
2575:String
2558:atomic
2435:ground
2430:closed
2425:atomic
2381:ground
2344:syntax
2240:binary
2167:domain
2084:Finite
1849:finite
1707:Logics
1666:
1614:Theory
1488:&
1463:
1428:
1424:
1174:unique
1107:equal
1093:equal
882:â Ord,
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3742:Naive
3672:Fuzzy
3635:Empty
3618:types
3569:tuple
3539:Class
3533:large
3494:Union
3411:Union
2916:Model
2664:Peano
2521:Proof
2361:Arity
2290:Naive
2177:image
2109:Fuzzy
2069:Empty
2018:union
1963:Class
1604:Model
1594:Lemma
1552:Axiom
1253:Notes
1194:that
702:class
616:basis
157:be a
3655:base
3039:Type
2842:list
2646:list
2623:list
2612:Term
2546:rank
2440:open
2334:list
2146:Maps
2051:sets
1910:Free
1880:list
1630:list
1557:list
1461:ISBN
1342:<
1064:<
1029:the
945:â 0.
657:<
628:span
576:<
463:<
221:<
128:Let
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2726:of
2708:of
2656:of
2188:Sur
2162:Map
1969:Ur-
1951:Set
995:yRx
973:. (
118:ZFC
112:or
104:to
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3112:NP
2736::
2730::
2660::
2337:),
2192:Bi
2184:In
1495:.
1484:;
1459:,
1455:,
1357:,
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