2872:
2942:
276:, described by Aczel as the highpoint of research of its decade. Boffa's idea was to make foundation fail as badly as it can (or rather, as extensionality permits): Boffa's axiom implies that every
260:
was published, which gave a general method for such proof to be carried out, rekindling some interest in non-well-founded axiomatic systems. The next axiom proposal came in a 1960 congress talk of
193:
372:
It is worth emphasizing that hyperset theory is an extension of classical set theory rather than a replacement: the well-founded sets within a hyperset domain conform to classical set theory.
265:
60:. Although a number of axiomatic systems of non-well-founded sets were proposed afterwards, they did not find much in the way of applications until the book Non-Well-Founded Sets by
225:
Although
Mirimanoff also introduced a notion of isomorphism between possibly non-well-founded sets, he considered neither an axiom of foundation nor of anti-foundation. In 1926,
56:
in a series of papers between 1917 and 1920, in which he formulated the distinction between well-founded and non-well-founded sets; he did not regard well-foundedness as an
442:
358:(apg) and states that two hypersets are equal if and only if they can be pictured by the same apg. Within this framework, it can be shown that the so-called
257:
206:
since it can be proved within ZFC (that is, ZFC without the axiom of regularity) that well-foundedness implies regularity. In variants of ZFC without the
1251:
286:
A more recent approach to non-well-founded set theory, pioneered by M. Forti and F. Honsell in the 1980s, borrows from computer science the concept of a
273:
1125:
Fewer than 1% of that database's theorems are ultimately dependent on this axiom, as can be shown by a command ("show usage") in the
Metamath program.
87:
229:
introduced the first axiom that allowed non-well-founded sets. After
Zermelo adopted Foundation into his own system in 1930 (from previous work of
347:
269:
3399:
1926:
17:
2009:
1150:
997:
452:
856:
Finsler Set Theory: Platonism and
Circularity : Translation of Paul Finsler's Papers on Set Theory with Introductory Comments
664:, CSLI Lecture Notes, vol. 14, Stanford, CA: Stanford University, Center for the Study of Language and Information, pp.
354:
They essentially correspond to four different notions of equality for non-well-founded sets. The first of these, AFA, is based on
2323:
936:
Mirimanoff, D. (1917), "Les antinomies de
Russell et de Burali-Forti et le probleme fondamental de la theorie des ensembles",
2481:
1075:
1057:
906:
863:
818:
1269:
3088:
2908:
2336:
1659:
136:
3416:
2341:
2331:
2068:
1921:
1274:
1265:
2477:
926:
884:
757:
739:
673:
306:
1819:
2574:
2318:
1143:
317:
3394:
1879:
1572:
622:
3274:
1313:
3554:
2835:
2537:
2300:
2295:
2120:
1541:
1225:
1066:
Sangiorgi, Davide (2011), "Origins of bisimulation and coinduction", in
Sangiorgi, Davide; Rutten, Jan (eds.),
3168:
3047:
2830:
2613:
2530:
2243:
2174:
2051:
1293:
506:
244:
Several proofs of the independence of
Foundation from the rest of ZF were published in 1950s particularly by
3411:
2755:
2581:
2267:
1901:
1500:
1086:
Unpublished paper, talk given at the 1960 Stanford
Congress of Logic, Methodology and Philosophy of Science
3404:
3042:
3005:
2633:
2628:
2238:
1977:
1906:
1235:
1136:
233:
1925â1929) interest in non-well-founded sets waned for decades. An early non-well-founded set theory was
309:
anti-foundation axioms are well-known, sometimes abbreviated by the first letter in the following list:
290:. Bisimilar sets are considered indistinguishable and thus equal, which leads to a strengthening of the
2562:
2152:
1546:
1514:
1205:
3059:
3093:
2985:
2973:
2968:
2852:
2801:
2698:
2196:
2157:
1634:
1279:
1308:
3559:
2901:
2693:
2623:
2162:
2014:
1997:
1720:
1200:
430:
355:
3549:
3513:
3431:
3306:
3258:
3072:
2995:
2525:
2502:
2463:
2349:
2290:
1936:
1856:
1700:
1644:
1257:
291:
234:
3465:
3346:
3158:
2978:
2815:
2542:
2520:
2487:
2380:
2226:
2211:
2184:
2135:
2019:
1954:
1779:
1745:
1740:
1614:
1445:
1422:
1100:
420:
516:
3381:
3351:
3295:
3215:
3195:
3173:
2745:
2598:
2390:
2108:
1844:
1750:
1609:
1594:
1475:
1450:
665:
659:
248:(1954), following an announcement of the result in an earlier paper of his from 1941, and by
210:, the possibility of non-well-founded sets with set-like â-chains arises. For example, a set
110:
958:
3455:
3445:
3279:
3210:
3163:
3103:
2990:
2718:
2680:
2557:
2361:
2201:
2125:
2103:
1931:
1889:
1788:
1755:
1619:
1407:
1318:
985:
683:
385:
380:
In published research, non-well-founded sets are also called hypersets, in parallel to the
366:
253:
34:
949:
848:
827:
Finsler, P. (1926), "Ăber die
Grundlagen der Mengenlehre. I: Die Mengen und ihre Axiome",
799:
779:
8:
3450:
3361:
3269:
3264:
3078:
3020:
2958:
2894:
2847:
2738:
2723:
2703:
2660:
2547:
2497:
2423:
2368:
2305:
2098:
2093:
2041:
1809:
1798:
1470:
1370:
1298:
1289:
1285:
1220:
1215:
207:
199:
42:
691:
Ballard, David; HrbĂĄÄek, Karel (1992), "Standard foundations for nonstandard analysis",
3373:
3368:
3153:
3108:
3015:
2876:
2645:
2608:
2593:
2586:
2569:
2373:
2355:
2221:
2147:
2130:
2083:
1896:
1805:
1639:
1624:
1584:
1536:
1521:
1509:
1465:
1440:
1210:
1159:
1030:
716:
708:
1829:
1042:
369:â A â S â F â B. In the Boffa universe, the distinct Quine atoms form a proper class.
3230:
3067:
3030:
3000:
2931:
2871:
2811:
2618:
2428:
2418:
2310:
2191:
2026:
2002:
1783:
1767:
1672:
1649:
1526:
1495:
1460:
1355:
1190:
1071:
1053:
922:
902:
880:
859:
814:
753:
735:
669:
404:
316:
FA ("Anti-Foundation Axiom") â due to M. Forti and F. Honsell (this is also known as
53:
720:
3518:
3508:
3493:
3488:
3356:
3010:
2825:
2820:
2713:
2670:
2492:
2453:
2448:
2433:
2259:
2216:
2113:
1911:
1861:
1435:
1397:
1034:
1020:
1012:
973:
945:
894:
844:
836:
795:
775:
700:
381:
230:
122:
102:
95:
75:
38:
3387:
3325:
3143:
2963:
2806:
2796:
2750:
2733:
2688:
2650:
2552:
2472:
2279:
2206:
2179:
2167:
2073:
1987:
1961:
1916:
1884:
1685:
1487:
1430:
1380:
1345:
1303:
981:
916:
874:
729:
679:
408:
396:
280:
238:
83:
3523:
3320:
3301:
3205:
3190:
3147:
3083:
3025:
2791:
2770:
2728:
2708:
2603:
2458:
2056:
2046:
2036:
2031:
1965:
1839:
1715:
1604:
1599:
1577:
1178:
277:
37:
that allow sets to be elements of themselves and otherwise violate the rule of
411:. The book is also a good introduction to the topic of non-well-founded sets.
365:
Each of the axioms given above extends the universe of the previous, so that:
3543:
3528:
3330:
3244:
3239:
2765:
2443:
1950:
1735:
1725:
1695:
1680:
1350:
425:
294:. In this context, axioms contradicting the axiom of regularity are known as
249:
3498:
1122:
3478:
3473:
3291:
3220:
3178:
3037:
2941:
2665:
2512:
2413:
2405:
2285:
2233:
2142:
2078:
2061:
1992:
1851:
1710:
1412:
1195:
977:
806:
337:
287:
283:
relation is isomorphic to the elementhood predicate on a transitive class.
245:
226:
106:
3503:
3138:
2775:
2655:
1834:
1824:
1771:
1455:
1375:
1360:
1240:
1185:
993:
392:
91:
61:
264:(never published as a paper), proposing an alternative axiom now called
3483:
3254:
2917:
1705:
1560:
1531:
1337:
1050:
One
Hundred Years of Russell Ìs Paradox: Mathematics, Logic, Philosophy
1025:
1016:
840:
712:
359:
327:
261:
133:, is well-founded if it has no infinite descending membership sequence
3286:
3249:
3200:
3098:
2857:
2760:
1813:
1730:
1690:
1654:
1590:
1402:
1392:
1365:
1128:
629:(Summer 2018 ed.), Metaphysics Research Lab, Stanford University
98:
79:
704:
2842:
2640:
2088:
1793:
1387:
1118:
786:
Boffa, M. (1972), "Forcing et négation de l'axiome de Fondement",
241:, although it is not merely ZF with a replacement for Foundation.
2438:
1230:
1084:
Scott, Dana (1960), "A different kind of model for set theory",
750:
Vicious circles. On the mathematics of non-wellfounded phenomena
3311:
3133:
597:
957:
Nitta, Takashi; Okada, Tomoko; Tzouvaras, Athanassios (2003),
3183:
2950:
2886:
1982:
1328:
1173:
470:
298:, and a set that is not necessarily well-founded is called a
72:
57:
959:"Classification of non-well-founded sets and an application"
71:
The theory of non-well-founded sets has been applied in the
198:
In ZFC, there is no infinite descending â-sequence by the
482:
256:
of 1951, proof which was published in 1957. Then in 1957
46:
811:
The Joy of Sets: Fundamentals of Contemporary Set Theory
752:, CSLI Lecture Notes, vol. 60, CSLI Publications,
202:. In fact, the axiom of regularity is often called the
573:
561:
527:
525:
537:
139:
121:
In 1917, Dmitry Mirimanoff introduced the concept of
766:
Boffa., M. (1968), "Les ensembles extraordinaires",
585:
52:
The study of non-well-founded sets was initiated by
27:
Theory that allows sets to be elements of themselves
956:
603:
549:
522:
362:, formally defined by Q={Q}, exists and is unique.
188:{\displaystyle \cdots \in x_{2}\in x_{1}\in x_{0}.}
621:Moss, Lawrence S. (2018), Zalta, Edward N. (ed.),
458:
187:
1043:"Predicativity, Circularity, and Anti-Foundation"
727:
494:
3541:
1068:Advanced Topics in Bisimulation and Coinduction
768:Bulletin de la Société Mathématique de Belgique
268:. Another axiom proposed in the late 1960s was
690:
488:
2902:
1144:
893:
49:is replaced by axioms implying its negation.
853:
788:Acad. Roy. Belgique, MĂ©m. Cl. Sci., Coll. 8â
991:
876:Cantorian set theory and limitation of size
809:(1993), "§7. Non-Well-Founded Set Theory",
747:
731:The Liar: An Essay on Truth and Circularity
448:
2909:
2895:
1336:
1151:
1137:
935:
543:
407:. The book's proposals contributed to the
1065:
1024:
476:
748:Barwise, Jon; Moss, Lawrence S. (1996),
41:. In non-well-founded set theories, the
1040:
872:
826:
728:Barwise, Jon; Etchemendy, John (1987),
627:The Stanford Encyclopedia of Philosophy
512:
464:
391:The hypersets were extensively used by
14:
3542:
1158:
805:
765:
2890:
1132:
1083:
785:
657:
616:
614:
612:
591:
579:
567:
555:
531:
1098:
914:
854:Finsler, Paul; Booth, David (1996).
620:
500:
1105:Stanford Encyclopedia of Philosophy
899:Nonstandard Analysis, Axiomatically
604:Nitta, Okada & Tzouvaras (2003)
24:
1092:
998:"Issues in commonsense set theory"
609:
252:who gave a different proof in his
25:
3571:
1112:
65:
2940:
2870:
639:
375:
336:AFA ("Finslerâs AFA") â due to
109:), and in a different setting,
82:processes in computer science (
2916:
1070:, Cambridge University Press,
1005:Artificial Intelligence Review
13:
1:
2831:History of mathematical logic
651:
346:AFA ("Boffaâs AFA") â due to
326:AFA ("Scottâs AFA") â due to
318:Aczel's anti-foundation axiom
31:Non-well-founded set theories
18:Non-well-founded set theories
2756:Primitive recursive function
1101:"Non-wellfounded Set Theory"
966:Mathematical Logic Quarterly
623:"Non-wellfounded Set Theory"
489:Ballard & HrbĂĄÄek (1992)
7:
1048:, in Link, Godehard (ed.),
938:L'Enseignement Mathématique
879:, Oxford University Press,
734:, Oxford University Press,
414:
105:), philosophy (work on the
10:
3576:
3400:von NeumannâBernaysâGödel
1820:SchröderâBernstein theorem
1547:Monadic predicate calculus
1206:Foundations of mathematics
1099:Moss, Lawrence S. (2018).
897:; Reeken, Michael (2004),
813:(2nd ed.), Springer,
116:
3464:
3427:
3339:
3229:
3201:One-to-one correspondence
3117:
3058:
2949:
2938:
2924:
2866:
2853:Philosophy of mathematics
2802:Automated theorem proving
2784:
2679:
2511:
2404:
2256:
1973:
1949:
1927:Von NeumannâBernaysâGödel
1872:
1766:
1670:
1568:
1559:
1486:
1421:
1327:
1249:
1166:
873:Hallett, Michael (1986),
693:Journal of Symbolic Logic
449:Pakkan & Akman (1994)
356:accessible pointed graphs
436:
431:Turtles all the way down
2503:Self-verifying theories
2324:Tarski's axiomatization
1275:Tarski's undefinability
1270:incompleteness theorems
292:axiom of extensionality
235:Willard Van Orman Quine
3159:Constructible universe
2986:Constructibility (V=L)
2877:Mathematics portal
2488:Proof of impossibility
2136:propositional variable
1446:Propositional calculus
978:10.1002/malq.200310018
921:, Dover Publications,
915:Levy, Azriel (2012) ,
421:Alternative set theory
296:anti-foundation axioms
189:
3555:Systems of set theory
3382:Principia Mathematica
3216:Transfinite induction
3075:(i.e. set difference)
2746:Kolmogorov complexity
2699:Computably enumerable
2599:Model complete theory
2391:Principia Mathematica
1451:Propositional formula
1280:BanachâTarski paradox
1052:, Walter de Gruyter,
661:Non-Well-Founded Sets
658:Aczel, Peter (1988),
479:, pp. 17â19, 26.
222:is non-well-founded.
190:
111:non-standard analysis
3456:Burali-Forti paradox
3211:Set-builder notation
3164:Continuum hypothesis
3104:Symmetric difference
2694:ChurchâTuring thesis
2681:Computability theory
1890:continuum hypothesis
1408:Square of opposition
1266:Gödel's completeness
1123:axiom of Regularity.
1041:Rathjen, M. (2004),
645:Hypersets (ucsd.edu)
386:nonstandard analysis
254:Habilitationsschrift
137:
35:axiomatic set theory
3417:TarskiâGrothendieck
2848:Mathematical object
2739:P versus NP problem
2704:Computable function
2498:Reverse mathematics
2424:Logical consequence
2301:primitive recursive
2296:elementary function
2069:Free/bound variable
1922:TarskiâGrothendieck
1441:Logical connectives
1371:Logical equivalence
1221:Logical consequence
399:in their 1987 book
208:axiom of regularity
200:axiom of regularity
78:of non-terminating
3006:Limitation of size
2646:Transfer principle
2609:Semantics of logic
2594:Categorical theory
2570:Non-standard model
2084:Logical connective
1211:Information theory
1160:Mathematical logic
1017:10.1007/BF00849061
841:10.1007/BF01283862
185:
3537:
3536:
3446:Russell's paradox
3395:ZermeloâFraenkel
3296:Dedekind-infinite
3169:Diagonal argument
3068:Cartesian product
2932:Set (mathematics)
2884:
2883:
2816:Abstract category
2619:Theories of truth
2429:Rule of inference
2419:Natural deduction
2400:
2399:
1945:
1944:
1650:Cartesian product
1555:
1554:
1461:Many-valued logic
1436:Boolean functions
1319:Russell's paradox
1294:diagonal argument
1191:First-order logic
1077:978-1-107-00497-9
1059:978-3-11-019968-0
908:978-3-540-22243-9
895:Kanovei, Vladimir
865:978-3-7643-5400-8
852:; translation in
820:978-0-387-94094-6
582:, pp. 108â9.
570:, pp. 107â8.
544:Mirimanoff (1917)
382:hyperreal numbers
274:superuniversality
54:Dmitry Mirimanoff
16:(Redirected from
3567:
3519:Bertrand Russell
3509:John von Neumann
3494:Abraham Fraenkel
3489:Richard Dedekind
3451:Suslin's problem
3362:Cantor's theorem
3079:De Morgan's laws
2944:
2911:
2904:
2897:
2888:
2887:
2875:
2874:
2826:History of logic
2821:Category of sets
2714:Decision problem
2493:Ordinal analysis
2434:Sequent calculus
2332:Boolean algebras
2272:
2271:
2246:
2217:logical/constant
1971:
1970:
1957:
1880:ZermeloâFraenkel
1631:Set operations:
1566:
1565:
1503:
1334:
1333:
1314:LöwenheimâSkolem
1201:Formal semantics
1153:
1146:
1139:
1130:
1129:
1108:
1088:
1080:
1062:
1047:
1037:
1028:
1002:
988:
963:
953:
932:
918:Basic set theory
911:
890:
869:
851:
823:
802:
782:
762:
744:
724:
687:
646:
643:
637:
636:
635:
634:
618:
607:
601:
595:
589:
583:
577:
571:
565:
559:
553:
547:
541:
535:
529:
520:
510:
504:
498:
492:
486:
480:
477:Sangiorgi (2011)
474:
468:
462:
456:
446:
258:Rieger's theorem
204:foundation axiom
194:
192:
191:
186:
181:
180:
168:
167:
155:
154:
123:well-foundedness
103:situation theory
96:natural language
43:foundation axiom
39:well-foundedness
33:are variants of
21:
3575:
3574:
3570:
3569:
3568:
3566:
3565:
3564:
3560:Wellfoundedness
3540:
3539:
3538:
3533:
3460:
3439:
3423:
3388:New Foundations
3335:
3225:
3144:Cardinal number
3127:
3113:
3054:
2945:
2936:
2920:
2915:
2885:
2880:
2869:
2862:
2807:Category theory
2797:Algebraic logic
2780:
2751:Lambda calculus
2689:Church encoding
2675:
2651:Truth predicate
2507:
2473:Complete theory
2396:
2265:
2261:
2257:
2252:
2244:
1964: and
1960:
1955:
1941:
1917:New Foundations
1885:axiom of choice
1868:
1830:Gödel numbering
1770: and
1762:
1666:
1551:
1501:
1482:
1431:Boolean algebra
1417:
1381:Equiconsistency
1346:Classical logic
1323:
1304:Halting problem
1292: and
1268: and
1256: and
1255:
1250:Theorems (
1245:
1162:
1157:
1115:
1095:
1093:Further reading
1078:
1060:
1045:
1000:
992:Pakkan, M. J.;
961:
929:
909:
887:
866:
821:
760:
742:
705:10.2307/2275304
676:
654:
649:
644:
640:
632:
630:
619:
610:
602:
598:
590:
586:
578:
574:
566:
562:
554:
550:
542:
538:
530:
523:
511:
507:
499:
495:
487:
483:
475:
471:
463:
459:
447:
443:
439:
417:
409:theory of truth
397:John Etchemendy
378:
239:New Foundations
176:
172:
163:
159:
150:
146:
138:
135:
134:
132:
119:
88:final semantics
84:process algebra
66:hyperset theory
28:
23:
22:
15:
12:
11:
5:
3573:
3563:
3562:
3557:
3552:
3550:Self-reference
3535:
3534:
3532:
3531:
3526:
3524:Thoralf Skolem
3521:
3516:
3511:
3506:
3501:
3496:
3491:
3486:
3481:
3476:
3470:
3468:
3462:
3461:
3459:
3458:
3453:
3448:
3442:
3440:
3438:
3437:
3434:
3428:
3425:
3424:
3422:
3421:
3420:
3419:
3414:
3409:
3408:
3407:
3392:
3391:
3390:
3378:
3377:
3376:
3365:
3364:
3359:
3354:
3349:
3343:
3341:
3337:
3336:
3334:
3333:
3328:
3323:
3318:
3309:
3304:
3299:
3289:
3284:
3283:
3282:
3277:
3272:
3262:
3252:
3247:
3242:
3236:
3234:
3227:
3226:
3224:
3223:
3218:
3213:
3208:
3206:Ordinal number
3203:
3198:
3193:
3188:
3187:
3186:
3181:
3171:
3166:
3161:
3156:
3151:
3141:
3136:
3130:
3128:
3126:
3125:
3122:
3118:
3115:
3114:
3112:
3111:
3106:
3101:
3096:
3091:
3086:
3084:Disjoint union
3081:
3076:
3070:
3064:
3062:
3056:
3055:
3053:
3052:
3051:
3050:
3045:
3034:
3033:
3031:Martin's axiom
3028:
3023:
3018:
3013:
3008:
3003:
2998:
2996:Extensionality
2993:
2988:
2983:
2982:
2981:
2976:
2971:
2961:
2955:
2953:
2947:
2946:
2939:
2937:
2935:
2934:
2928:
2926:
2922:
2921:
2914:
2913:
2906:
2899:
2891:
2882:
2881:
2867:
2864:
2863:
2861:
2860:
2855:
2850:
2845:
2840:
2839:
2838:
2828:
2823:
2818:
2809:
2804:
2799:
2794:
2792:Abstract logic
2788:
2786:
2782:
2781:
2779:
2778:
2773:
2771:Turing machine
2768:
2763:
2758:
2753:
2748:
2743:
2742:
2741:
2736:
2731:
2726:
2721:
2711:
2709:Computable set
2706:
2701:
2696:
2691:
2685:
2683:
2677:
2676:
2674:
2673:
2668:
2663:
2658:
2653:
2648:
2643:
2638:
2637:
2636:
2631:
2626:
2616:
2611:
2606:
2604:Satisfiability
2601:
2596:
2591:
2590:
2589:
2579:
2578:
2577:
2567:
2566:
2565:
2560:
2555:
2550:
2545:
2535:
2534:
2533:
2528:
2521:Interpretation
2517:
2515:
2509:
2508:
2506:
2505:
2500:
2495:
2490:
2485:
2475:
2470:
2469:
2468:
2467:
2466:
2456:
2451:
2441:
2436:
2431:
2426:
2421:
2416:
2410:
2408:
2402:
2401:
2398:
2397:
2395:
2394:
2386:
2385:
2384:
2383:
2378:
2377:
2376:
2371:
2366:
2346:
2345:
2344:
2342:minimal axioms
2339:
2328:
2327:
2326:
2315:
2314:
2313:
2308:
2303:
2298:
2293:
2288:
2275:
2273:
2254:
2253:
2251:
2250:
2249:
2248:
2236:
2231:
2230:
2229:
2224:
2219:
2214:
2204:
2199:
2194:
2189:
2188:
2187:
2182:
2172:
2171:
2170:
2165:
2160:
2155:
2145:
2140:
2139:
2138:
2133:
2128:
2118:
2117:
2116:
2111:
2106:
2101:
2096:
2091:
2081:
2076:
2071:
2066:
2065:
2064:
2059:
2054:
2049:
2039:
2034:
2032:Formation rule
2029:
2024:
2023:
2022:
2017:
2007:
2006:
2005:
1995:
1990:
1985:
1980:
1974:
1968:
1951:Formal systems
1947:
1946:
1943:
1942:
1940:
1939:
1934:
1929:
1924:
1919:
1914:
1909:
1904:
1899:
1894:
1893:
1892:
1887:
1876:
1874:
1870:
1869:
1867:
1866:
1865:
1864:
1854:
1849:
1848:
1847:
1840:Large cardinal
1837:
1832:
1827:
1822:
1817:
1803:
1802:
1801:
1796:
1791:
1776:
1774:
1764:
1763:
1761:
1760:
1759:
1758:
1753:
1748:
1738:
1733:
1728:
1723:
1718:
1713:
1708:
1703:
1698:
1693:
1688:
1683:
1677:
1675:
1668:
1667:
1665:
1664:
1663:
1662:
1657:
1652:
1647:
1642:
1637:
1629:
1628:
1627:
1622:
1612:
1607:
1605:Extensionality
1602:
1600:Ordinal number
1597:
1587:
1582:
1581:
1580:
1569:
1563:
1557:
1556:
1553:
1552:
1550:
1549:
1544:
1539:
1534:
1529:
1524:
1519:
1518:
1517:
1507:
1506:
1505:
1492:
1490:
1484:
1483:
1481:
1480:
1479:
1478:
1473:
1468:
1458:
1453:
1448:
1443:
1438:
1433:
1427:
1425:
1419:
1418:
1416:
1415:
1410:
1405:
1400:
1395:
1390:
1385:
1384:
1383:
1373:
1368:
1363:
1358:
1353:
1348:
1342:
1340:
1331:
1325:
1324:
1322:
1321:
1316:
1311:
1306:
1301:
1296:
1284:Cantor's
1282:
1277:
1272:
1262:
1260:
1247:
1246:
1244:
1243:
1238:
1233:
1228:
1223:
1218:
1213:
1208:
1203:
1198:
1193:
1188:
1183:
1182:
1181:
1170:
1168:
1164:
1163:
1156:
1155:
1148:
1141:
1133:
1127:
1126:
1114:
1113:External links
1111:
1110:
1109:
1094:
1091:
1090:
1089:
1081:
1076:
1063:
1058:
1038:
1011:(4): 279â308,
989:
972:(2): 187â200,
954:
933:
927:
912:
907:
891:
885:
870:
864:
824:
819:
803:
783:
763:
758:
745:
740:
725:
699:(2): 741â748,
688:
674:
653:
650:
648:
647:
638:
608:
596:
594:, p. 110.
584:
572:
560:
558:, p. 107.
548:
536:
534:, p. 105.
521:
513:Hallett (1986)
505:
493:
481:
469:
465:Rathjen (2004)
457:
440:
438:
435:
434:
433:
428:
423:
416:
413:
405:liar's paradox
377:
374:
352:
351:
341:
331:
321:
305:Four mutually
196:
195:
184:
179:
175:
171:
166:
162:
158:
153:
149:
145:
142:
130:
118:
115:
26:
9:
6:
4:
3:
2:
3572:
3561:
3558:
3556:
3553:
3551:
3548:
3547:
3545:
3530:
3529:Ernst Zermelo
3527:
3525:
3522:
3520:
3517:
3515:
3514:Willard Quine
3512:
3510:
3507:
3505:
3502:
3500:
3497:
3495:
3492:
3490:
3487:
3485:
3482:
3480:
3477:
3475:
3472:
3471:
3469:
3467:
3466:Set theorists
3463:
3457:
3454:
3452:
3449:
3447:
3444:
3443:
3441:
3435:
3433:
3430:
3429:
3426:
3418:
3415:
3413:
3412:KripkeâPlatek
3410:
3406:
3403:
3402:
3401:
3398:
3397:
3396:
3393:
3389:
3386:
3385:
3384:
3383:
3379:
3375:
3372:
3371:
3370:
3367:
3366:
3363:
3360:
3358:
3355:
3353:
3350:
3348:
3345:
3344:
3342:
3338:
3332:
3329:
3327:
3324:
3322:
3319:
3317:
3315:
3310:
3308:
3305:
3303:
3300:
3297:
3293:
3290:
3288:
3285:
3281:
3278:
3276:
3273:
3271:
3268:
3267:
3266:
3263:
3260:
3256:
3253:
3251:
3248:
3246:
3243:
3241:
3238:
3237:
3235:
3232:
3228:
3222:
3219:
3217:
3214:
3212:
3209:
3207:
3204:
3202:
3199:
3197:
3194:
3192:
3189:
3185:
3182:
3180:
3177:
3176:
3175:
3172:
3170:
3167:
3165:
3162:
3160:
3157:
3155:
3152:
3149:
3145:
3142:
3140:
3137:
3135:
3132:
3131:
3129:
3123:
3120:
3119:
3116:
3110:
3107:
3105:
3102:
3100:
3097:
3095:
3092:
3090:
3087:
3085:
3082:
3080:
3077:
3074:
3071:
3069:
3066:
3065:
3063:
3061:
3057:
3049:
3048:specification
3046:
3044:
3041:
3040:
3039:
3036:
3035:
3032:
3029:
3027:
3024:
3022:
3019:
3017:
3014:
3012:
3009:
3007:
3004:
3002:
2999:
2997:
2994:
2992:
2989:
2987:
2984:
2980:
2977:
2975:
2972:
2970:
2967:
2966:
2965:
2962:
2960:
2957:
2956:
2954:
2952:
2948:
2943:
2933:
2930:
2929:
2927:
2923:
2919:
2912:
2907:
2905:
2900:
2898:
2893:
2892:
2889:
2879:
2878:
2873:
2865:
2859:
2856:
2854:
2851:
2849:
2846:
2844:
2841:
2837:
2834:
2833:
2832:
2829:
2827:
2824:
2822:
2819:
2817:
2813:
2810:
2808:
2805:
2803:
2800:
2798:
2795:
2793:
2790:
2789:
2787:
2783:
2777:
2774:
2772:
2769:
2767:
2766:Recursive set
2764:
2762:
2759:
2757:
2754:
2752:
2749:
2747:
2744:
2740:
2737:
2735:
2732:
2730:
2727:
2725:
2722:
2720:
2717:
2716:
2715:
2712:
2710:
2707:
2705:
2702:
2700:
2697:
2695:
2692:
2690:
2687:
2686:
2684:
2682:
2678:
2672:
2669:
2667:
2664:
2662:
2659:
2657:
2654:
2652:
2649:
2647:
2644:
2642:
2639:
2635:
2632:
2630:
2627:
2625:
2622:
2621:
2620:
2617:
2615:
2612:
2610:
2607:
2605:
2602:
2600:
2597:
2595:
2592:
2588:
2585:
2584:
2583:
2580:
2576:
2575:of arithmetic
2573:
2572:
2571:
2568:
2564:
2561:
2559:
2556:
2554:
2551:
2549:
2546:
2544:
2541:
2540:
2539:
2536:
2532:
2529:
2527:
2524:
2523:
2522:
2519:
2518:
2516:
2514:
2510:
2504:
2501:
2499:
2496:
2494:
2491:
2489:
2486:
2483:
2482:from ZFC
2479:
2476:
2474:
2471:
2465:
2462:
2461:
2460:
2457:
2455:
2452:
2450:
2447:
2446:
2445:
2442:
2440:
2437:
2435:
2432:
2430:
2427:
2425:
2422:
2420:
2417:
2415:
2412:
2411:
2409:
2407:
2403:
2393:
2392:
2388:
2387:
2382:
2381:non-Euclidean
2379:
2375:
2372:
2370:
2367:
2365:
2364:
2360:
2359:
2357:
2354:
2353:
2351:
2347:
2343:
2340:
2338:
2335:
2334:
2333:
2329:
2325:
2322:
2321:
2320:
2316:
2312:
2309:
2307:
2304:
2302:
2299:
2297:
2294:
2292:
2289:
2287:
2284:
2283:
2281:
2277:
2276:
2274:
2269:
2263:
2258:Example
2255:
2247:
2242:
2241:
2240:
2237:
2235:
2232:
2228:
2225:
2223:
2220:
2218:
2215:
2213:
2210:
2209:
2208:
2205:
2203:
2200:
2198:
2195:
2193:
2190:
2186:
2183:
2181:
2178:
2177:
2176:
2173:
2169:
2166:
2164:
2161:
2159:
2156:
2154:
2151:
2150:
2149:
2146:
2144:
2141:
2137:
2134:
2132:
2129:
2127:
2124:
2123:
2122:
2119:
2115:
2112:
2110:
2107:
2105:
2102:
2100:
2097:
2095:
2092:
2090:
2087:
2086:
2085:
2082:
2080:
2077:
2075:
2072:
2070:
2067:
2063:
2060:
2058:
2055:
2053:
2050:
2048:
2045:
2044:
2043:
2040:
2038:
2035:
2033:
2030:
2028:
2025:
2021:
2018:
2016:
2015:by definition
2013:
2012:
2011:
2008:
2004:
2001:
2000:
1999:
1996:
1994:
1991:
1989:
1986:
1984:
1981:
1979:
1976:
1975:
1972:
1969:
1967:
1963:
1958:
1952:
1948:
1938:
1935:
1933:
1930:
1928:
1925:
1923:
1920:
1918:
1915:
1913:
1910:
1908:
1905:
1903:
1902:KripkeâPlatek
1900:
1898:
1895:
1891:
1888:
1886:
1883:
1882:
1881:
1878:
1877:
1875:
1871:
1863:
1860:
1859:
1858:
1855:
1853:
1850:
1846:
1843:
1842:
1841:
1838:
1836:
1833:
1831:
1828:
1826:
1823:
1821:
1818:
1815:
1811:
1807:
1804:
1800:
1797:
1795:
1792:
1790:
1787:
1786:
1785:
1781:
1778:
1777:
1775:
1773:
1769:
1765:
1757:
1754:
1752:
1749:
1747:
1746:constructible
1744:
1743:
1742:
1739:
1737:
1734:
1732:
1729:
1727:
1724:
1722:
1719:
1717:
1714:
1712:
1709:
1707:
1704:
1702:
1699:
1697:
1694:
1692:
1689:
1687:
1684:
1682:
1679:
1678:
1676:
1674:
1669:
1661:
1658:
1656:
1653:
1651:
1648:
1646:
1643:
1641:
1638:
1636:
1633:
1632:
1630:
1626:
1623:
1621:
1618:
1617:
1616:
1613:
1611:
1608:
1606:
1603:
1601:
1598:
1596:
1592:
1588:
1586:
1583:
1579:
1576:
1575:
1574:
1571:
1570:
1567:
1564:
1562:
1558:
1548:
1545:
1543:
1540:
1538:
1535:
1533:
1530:
1528:
1525:
1523:
1520:
1516:
1513:
1512:
1511:
1508:
1504:
1499:
1498:
1497:
1494:
1493:
1491:
1489:
1485:
1477:
1474:
1472:
1469:
1467:
1464:
1463:
1462:
1459:
1457:
1454:
1452:
1449:
1447:
1444:
1442:
1439:
1437:
1434:
1432:
1429:
1428:
1426:
1424:
1423:Propositional
1420:
1414:
1411:
1409:
1406:
1404:
1401:
1399:
1396:
1394:
1391:
1389:
1386:
1382:
1379:
1378:
1377:
1374:
1372:
1369:
1367:
1364:
1362:
1359:
1357:
1354:
1352:
1351:Logical truth
1349:
1347:
1344:
1343:
1341:
1339:
1335:
1332:
1330:
1326:
1320:
1317:
1315:
1312:
1310:
1307:
1305:
1302:
1300:
1297:
1295:
1291:
1287:
1283:
1281:
1278:
1276:
1273:
1271:
1267:
1264:
1263:
1261:
1259:
1253:
1248:
1242:
1239:
1237:
1234:
1232:
1229:
1227:
1224:
1222:
1219:
1217:
1214:
1212:
1209:
1207:
1204:
1202:
1199:
1197:
1194:
1192:
1189:
1187:
1184:
1180:
1177:
1176:
1175:
1172:
1171:
1169:
1165:
1161:
1154:
1149:
1147:
1142:
1140:
1135:
1134:
1131:
1124:
1120:
1117:
1116:
1106:
1102:
1097:
1096:
1087:
1082:
1079:
1073:
1069:
1064:
1061:
1055:
1051:
1044:
1039:
1036:
1032:
1027:
1022:
1018:
1014:
1010:
1006:
999:
995:
990:
987:
983:
979:
975:
971:
967:
960:
955:
951:
947:
943:
939:
934:
930:
928:9780486150734
924:
920:
919:
913:
910:
904:
900:
896:
892:
888:
886:9780198532835
882:
878:
877:
871:
867:
861:
857:
850:
846:
842:
838:
834:
830:
825:
822:
816:
812:
808:
807:Devlin, Keith
804:
801:
797:
793:
789:
784:
781:
777:
773:
769:
764:
761:
759:1-57586-009-0
755:
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746:
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741:9780195059441
737:
733:
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685:
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675:0-937073-22-9
671:
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581:
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569:
564:
557:
552:
545:
540:
533:
528:
526:
518:
514:
509:
503:, p. 68.
502:
497:
490:
485:
478:
473:
466:
461:
454:
450:
445:
441:
432:
429:
427:
426:Universal set
424:
422:
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412:
410:
406:
402:
398:
394:
389:
387:
383:
373:
370:
368:
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361:
357:
349:
348:Maurice Boffa
345:
342:
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332:
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319:
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312:
311:
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308:
303:
301:
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293:
289:
284:
282:
279:
275:
271:
270:Maurice Boffa
267:
263:
259:
255:
251:
250:Ernst Specker
247:
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223:
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182:
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128:
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108:
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100:
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93:
89:
85:
81:
80:computational
77:
74:
69:
67:
63:
59:
55:
50:
48:
44:
40:
36:
32:
19:
3479:Georg Cantor
3474:Paul Bernays
3405:MorseâKelley
3380:
3313:
3312:Subset
3259:hereditarily
3221:Venn diagram
3179:ordered pair
3094:Intersection
3038:Axiom schema
2868:
2666:Ultraproduct
2513:Model theory
2478:Independence
2414:Formal proof
2406:Proof theory
2389:
2362:
2319:real numbers
2291:second-order
2202:Substitution
2079:Metalanguage
2020:conservative
1993:Axiom schema
1937:Constructive
1907:MorseâKelley
1873:Set theories
1852:Aleph number
1845:inaccessible
1751:Grothendieck
1635:intersection
1522:Higher-order
1510:Second-order
1456:Truth tables
1413:Venn diagram
1196:Formal proof
1121:page on the
1104:
1085:
1067:
1049:
1008:
1004:
969:
965:
941:
937:
917:
901:, Springer,
898:
875:
858:. Springer.
855:
832:
828:
810:
791:
790:, SĂ©rie II,
787:
771:
767:
749:
730:
696:
692:
660:
641:
631:, retrieved
626:
599:
592:Aczel (1988)
587:
580:Aczel (1988)
575:
568:Aczel (1988)
563:
556:Aczel (1988)
551:
539:
532:Aczel (1988)
508:
496:
484:
472:
460:
453:section link
444:
400:
390:
379:
376:Applications
371:
364:
353:
343:
338:Paul Finsler
333:
323:
313:
304:
299:
295:
288:bisimulation
285:
272:'s axiom of
246:Paul Bernays
243:
227:Paul Finsler
224:
219:
215:
211:
203:
197:
120:
107:Liar Paradox
70:
51:
30:
29:
3504:Thomas Jech
3347:Alternative
3326:Uncountable
3280:Ultrafilter
3139:Cardinality
3043:replacement
2991:Determinacy
2776:Type theory
2724:undecidable
2656:Truth value
2543:equivalence
2222:non-logical
1835:Enumeration
1825:Isomorphism
1772:cardinality
1756:Von Neumann
1721:Ultrafilter
1686:Uncountable
1620:equivalence
1537:Quantifiers
1527:Fixed-point
1496:First-order
1376:Consistency
1361:Proposition
1338:Traditional
1309:Lindström's
1299:Compactness
1241:Type theory
1186:Cardinality
1026:11693/25955
835:: 683â713,
501:Levy (2012)
393:Jon Barwise
307:independent
278:extensional
231:von Neumann
92:linguistics
64:introduces
62:Peter Aczel
3544:Categories
3499:Kurt Gödel
3484:Paul Cohen
3321:Transitive
3089:Identities
3073:Complement
3060:Operations
3021:Regularity
2959:Adjunction
2918:Set theory
2587:elementary
2280:arithmetic
2148:Quantifier
2126:functional
1998:Expression
1716:Transitive
1660:identities
1645:complement
1578:hereditary
1561:Set theory
950:46.0306.01
849:52.0192.01
800:0286.02068
780:0179.01602
652:References
633:2024-05-30
515:, p.
360:Quine atom
328:Dana Scott
262:Dana Scott
214:such that
125:of a set:
3432:Paradoxes
3352:Axiomatic
3331:Universal
3307:Singleton
3302:Recursive
3245:Countable
3240:Amorphous
3099:Power set
3016:Power set
2974:dependent
2969:countable
2858:Supertask
2761:Recursion
2719:decidable
2553:saturated
2531:of models
2454:deductive
2449:axiomatic
2369:Hilbert's
2356:Euclidean
2337:canonical
2260:axiomatic
2192:Signature
2121:Predicate
2010:Extension
1932:Ackermann
1857:Operation
1736:Universal
1726:Recursive
1701:Singleton
1696:Inhabited
1681:Countable
1671:Types of
1655:power set
1625:partition
1542:Predicate
1488:Predicate
1403:Syllogism
1393:Soundness
1366:Inference
1356:Tautology
1258:paradoxes
994:Akman, V.
944:: 37â52,
403:, on the
170:∈
157:∈
144:∈
141:⋯
99:semantics
76:modelling
68:in 1988.
3436:Problems
3340:Theories
3316:Superset
3292:Infinite
3121:Concepts
3001:Infinity
2925:Overview
2843:Logicism
2836:timeline
2812:Concrete
2671:Validity
2641:T-schema
2634:Kripke's
2629:Tarski's
2624:semantic
2614:Strength
2563:submodel
2558:spectrum
2526:function
2374:Tarski's
2363:Elements
2350:geometry
2306:Robinson
2227:variable
2212:function
2185:spectrum
2175:Sentence
2131:variable
2074:Language
2027:Relation
1988:Automata
1978:Alphabet
1962:language
1816:-jection
1794:codomain
1780:Function
1741:Universe
1711:Infinite
1615:Relation
1398:Validity
1388:Argument
1286:theorem,
1119:Metamath
996:(1994),
829:Math. Z.
774:: 3â15,
721:39158351
415:See also
401:The Liar
300:hyperset
281:set-like
129:A set, x
3374:General
3369:Zermelo
3275:subbase
3257: (
3196:Forcing
3174:Element
3146: (
3124:Methods
3011:Pairing
2785:Related
2582:Diagram
2480: (
2459:Hilbert
2444:Systems
2439:Theorem
2317:of the
2262:systems
2042:Formula
2037:Grammar
1953: (
1897:General
1610:Forcing
1595:Element
1515:Monadic
1290:paradox
1231:Theorem
1167:General
1035:6323872
986:1961461
713:2275304
684:0940014
117:Details
73:logical
3265:Filter
3255:Finite
3191:Family
3134:Almost
2979:global
2964:Choice
2951:Axioms
2548:finite
2311:Skolem
2264:
2239:Theory
2207:Symbol
2197:String
2180:atomic
2057:ground
2052:closed
2047:atomic
2003:ground
1966:syntax
1862:binary
1789:domain
1706:Finite
1471:finite
1329:Logics
1288:
1236:Theory
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925:
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862:
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682:
672:
666:xx+137
3357:Naive
3287:Fuzzy
3250:Empty
3233:types
3184:tuple
3154:Class
3148:large
3109:Union
3026:Union
2538:Model
2286:Peano
2143:Proof
1983:Arity
1912:Naive
1799:image
1731:Fuzzy
1691:Empty
1640:union
1585:Class
1226:Model
1216:Lemma
1174:Axiom
1046:(PDF)
1031:S2CID
1001:(PDF)
962:(PDF)
794:(7),
717:S2CID
709:JSTOR
437:Notes
58:axiom
3270:base
2661:Type
2464:list
2268:list
2245:list
2234:Term
2168:rank
2062:open
1956:list
1768:Maps
1673:sets
1532:Free
1502:list
1252:list
1179:list
1072:ISBN
1054:ISBN
923:ISBN
903:ISBN
881:ISBN
860:ISBN
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754:ISBN
736:ISBN
670:ISBN
395:and
266:SAFA
94:and
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2348:of
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2278:of
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1784:Map
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1573:Set
1021:hdl
1013:doi
974:doi
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845:JFM
837:doi
796:Zbl
776:Zbl
701:doi
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384:of
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