2222:
2292:
219:
Every well-founded set-like relation can be embedded into a well-founded set-like extensional relation. This implies the following variant of the
Mostowski collapse lemma: every well-founded set-like relation is isomorphic to set-membership on a (non-unique, and not necessarily transitive) class.
601:
2749:
1276:
1359:
500:
290:. In Boffa's set theory, every set-like extensional relation is isomorphic to set-membership on a (non-unique) transitive class. In set theory with
1673:
298:
to set-membership on a unique transitive class, hence every bisimulation-minimal set-like relation is isomorphic to a unique transitive class.
1831:
395:
619:
369:
satisfies the axiom of regularity (it is "internally" well-founded) but it is not well-founded and the collapse lemma does not apply to it.
2438:
2258:
1686:
1009:
2766:
1271:
1691:
1681:
1418:
624:
1169:
615:
1827:
1924:
1668:
493:
291:
2744:
1229:
922:
663:
311:
2185:
1887:
1650:
1645:
1470:
891:
575:
2904:
2518:
2397:
2180:
1963:
1880:
1593:
1524:
1401:
643:
2761:
1251:
2105:
1931:
1617:
850:
2754:
1256:
2392:
2355:
1588:
1327:
585:
486:
314:
is set-like and extensional. If the model is well-founded, then by the
Mostowski collapse lemma it is isomorphic to a
287:
1983:
1978:
1912:
1502:
896:
864:
555:
629:
2443:
2335:
2323:
2318:
2202:
2151:
2048:
1546:
1507:
984:
443:
2043:
658:
2909:
2251:
1973:
1512:
1364:
1347:
1070:
550:
2863:
2781:
2656:
2608:
2422:
2345:
1875:
1852:
1813:
1699:
1640:
1286:
1206:
1050:
994:
607:
416:
145:
2815:
2696:
2508:
2328:
2165:
1892:
1870:
1837:
1730:
1576:
1561:
1534:
1485:
1369:
1304:
1129:
1095:
1090:
964:
795:
772:
438:
321:
Saying that the membership relation of some model of ZF is well-founded is stronger than saying that the
44:
2731:
2645:
2565:
2545:
2523:
2095:
1948:
1740:
1458:
1194:
1100:
959:
944:
825:
800:
345:
is not in the domain of the model, even though all of its members are). More precisely, for no such set
264:
107:
2221:
2805:
2795:
2629:
2560:
2513:
2453:
2340:
2068:
2030:
1907:
1711:
1551:
1475:
1453:
1281:
1239:
1138:
1105:
969:
757:
668:
2800:
2711:
2624:
2619:
2614:
2428:
2370:
2308:
2244:
2197:
2088:
2073:
2053:
2010:
1897:
1847:
1773:
1718:
1655:
1448:
1443:
1391:
1159:
1148:
820:
720:
648:
639:
635:
570:
565:
322:
2723:
2718:
2503:
2458:
2365:
2226:
1995:
1958:
1943:
1936:
1919:
1705:
1571:
1497:
1480:
1433:
1246:
1155:
989:
974:
934:
886:
871:
859:
815:
790:
560:
509:
468:
460:
20:
1723:
1179:
2580:
2417:
2409:
2380:
2350:
2281:
2161:
1968:
1778:
1768:
1660:
1541:
1376:
1352:
1133:
1117:
1022:
999:
876:
845:
810:
705:
540:
404:
391:
36:
2868:
2858:
2843:
2838:
2706:
2360:
2175:
2170:
2063:
2020:
1842:
1803:
1798:
1783:
1609:
1566:
1463:
1261:
1211:
785:
747:
452:
425:
315:
2899:
2737:
2675:
2493:
2313:
2156:
2146:
2100:
2083:
2038:
2000:
1902:
1822:
1629:
1556:
1529:
1517:
1423:
1337:
1311:
1266:
1234:
1035:
837:
780:
730:
695:
653:
387:
75:
2873:
2670:
2651:
2555:
2540:
2497:
2433:
2375:
2141:
2120:
2078:
2058:
1953:
1808:
1406:
1396:
1386:
1381:
1315:
1189:
1065:
954:
949:
927:
528:
408:
286:
The well-foundedness assumption of the
Mostowski lemma can be alleviated or dropped in
176:
2893:
2878:
2848:
2680:
2594:
2589:
2115:
1793:
1300:
1085:
1075:
1045:
1030:
700:
472:
386:, Springer Monographs in Mathematics (third millennium ed.), Berlin, New York:
2828:
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2641:
2570:
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2387:
2291:
2015:
1862:
1763:
1755:
1635:
1583:
1492:
1428:
1411:
1342:
1201:
1060:
762:
545:
307:
295:
268:
180:
2853:
2488:
2125:
2005:
1184:
1174:
1121:
805:
725:
710:
590:
535:
379:
2833:
2701:
2604:
2267:
1055:
910:
881:
687:
32:
2636:
2599:
2550:
2448:
2207:
2110:
1163:
1080:
1040:
1004:
940:
752:
742:
715:
430:
2192:
1990:
1438:
1143:
737:
1788:
580:
464:
2661:
2483:
478:
456:
191:), and the isomorphism is unique. The isomorphism maps each element
424:(1), Institute of Mathematics Polish Academy of Sciences: 143â164,
275:
onto a (non-unique, in general) transitive class. The homomorphism
2533:
2300:
1332:
678:
523:
183:) whose structure under the membership relation is isomorphic to (
2240:
482:
2236:
329:(assuming the consistency of ZF) whose domain has a subset
171:
The
Mostowski collapse lemma states that for every such
255:
can be defined for any well-founded set-like relation
2814:
2777:
2689:
2579:
2467:
2408:
2299:
2274:
2134:
2029:
1861:
1754:
1606:
1299:
1222:
1116:
1020:
909:
836:
771:
686:
677:
599:
516:
441:(1953), "Inner models for set theory, Part III",
451:(2), Association for Symbolic Logic: 145â167,
2252:
494:
318:of ZF and such a transitive model is unique.
8:
325:is true in the model. There exists a model
48:
2259:
2245:
2237:
1320:
915:
683:
501:
487:
479:
429:
40:
409:"An undecidable arithmetical statement"
7:
29:Shepherdson–Mostowski collapse
16:Result in mathematics and set theory
122:-minimal element (i.e. an element
14:
279:is an isomorphism if and only if
199:to the set of images of elements
2290:
2220:
63:is a binary relation on a class
337:-minimal element, but this set
1:
2181:History of mathematical logic
341:is not a "set in the model" (
294:, every set-like relation is
292:Aczel's anti-foundation axiom
288:non-well-founded set theories
2106:Primitive recursive function
156:for every distinct elements
2926:
2750:von NeumannâBernaysâGödel
1170:SchröderâBernstein theorem
897:Monadic predicate calculus
556:Foundations of mathematics
2551:One-to-one correspondence
2288:
2216:
2203:Philosophy of mathematics
2152:Automated theorem proving
1323:
1277:Von NeumannâBernaysâGödel
918:
444:Journal of Symbolic Logic
110:: every nonempty subset
25:Mostowski collapse lemma
1853:Self-verifying theories
1674:Tarski's axiomatization
625:Tarski's undefinability
620:incompleteness theorems
431:10.4064/fm-36-1-143-164
417:Fundamenta Mathematicae
43:, theorem 3) and
2509:Constructible universe
2336:Constructibility (V=L)
2227:Mathematics portal
1838:Proof of impossibility
1486:propositional variable
796:Propositional calculus
265:well-founded recursion
175:there exists a unique
2732:Principia Mathematica
2566:Transfinite induction
2425:(i.e. set difference)
2096:Kolmogorov complexity
2049:Computably enumerable
1949:Model complete theory
1741:Principia Mathematica
801:Propositional formula
630:BanachâTarski paradox
96:} is a set for every
37:Andrzej Mostowski
2905:Lemmas in set theory
2806:Burali-Forti paradox
2561:Set-builder notation
2514:Continuum hypothesis
2454:Symmetric difference
2044:ChurchâTuring thesis
2031:Computability theory
1240:continuum hypothesis
758:Square of opposition
616:Gödel's completeness
45:John Shepherdson
27:, also known as the
2767:TarskiâGrothendieck
2198:Mathematical object
2089:P versus NP problem
2054:Computable function
1848:Reverse mathematics
1774:Logical consequence
1651:primitive recursive
1646:elementary function
1419:Free/bound variable
1272:TarskiâGrothendieck
791:Logical connectives
721:Logical equivalence
571:Logical consequence
323:axiom of regularity
2356:Limitation of size
1996:Transfer principle
1959:Semantics of logic
1944:Categorical theory
1920:Non-standard model
1434:Logical connective
561:Information theory
510:Mathematical logic
405:Mostowski, Andrzej
31:, is a theorem of
21:mathematical logic
2887:
2886:
2796:Russell's paradox
2745:ZermeloâFraenkel
2646:Dedekind-infinite
2519:Diagonal argument
2418:Cartesian product
2282:Set (mathematics)
2234:
2233:
2166:Abstract category
1969:Theories of truth
1779:Rule of inference
1769:Natural deduction
1750:
1749:
1295:
1294:
1000:Cartesian product
905:
904:
811:Many-valued logic
786:Boolean functions
669:Russell's paradox
644:diagonal argument
541:First-order logic
439:Shepherdson, John
397:978-3-540-44085-7
2917:
2869:Bertrand Russell
2859:John von Neumann
2844:Abraham Fraenkel
2839:Richard Dedekind
2801:Suslin's problem
2712:Cantor's theorem
2429:De Morgan's laws
2294:
2261:
2254:
2247:
2238:
2225:
2224:
2176:History of logic
2171:Category of sets
2064:Decision problem
1843:Ordinal analysis
1784:Sequent calculus
1682:Boolean algebras
1622:
1621:
1596:
1567:logical/constant
1321:
1307:
1230:ZermeloâFraenkel
981:Set operations:
916:
853:
684:
664:LöwenheimâSkolem
551:Formal semantics
503:
496:
489:
480:
475:
434:
433:
413:
400:
316:transitive model
283:is extensional.
267:. It provides a
211:(Jech 2003:69).
179:class (possibly
2925:
2924:
2920:
2919:
2918:
2916:
2915:
2914:
2910:Wellfoundedness
2890:
2889:
2888:
2883:
2810:
2789:
2773:
2738:New Foundations
2685:
2575:
2494:Cardinal number
2477:
2463:
2404:
2295:
2286:
2270:
2265:
2235:
2230:
2219:
2212:
2157:Category theory
2147:Algebraic logic
2130:
2101:Lambda calculus
2039:Church encoding
2025:
2001:Truth predicate
1857:
1823:Complete theory
1746:
1615:
1611:
1607:
1602:
1594:
1314: and
1310:
1305:
1291:
1267:New Foundations
1235:axiom of choice
1218:
1180:Gödel numbering
1120: and
1112:
1016:
901:
851:
832:
781:Boolean algebra
767:
731:Equiconsistency
696:Classical logic
673:
654:Halting problem
642: and
618: and
606: and
605:
600:Theorems (
595:
512:
507:
457:10.2307/2268947
437:
411:
403:
398:
388:Springer-Verlag
378:
375:
304:
217:
215:Generalizations
57:
17:
12:
11:
5:
2923:
2921:
2913:
2912:
2907:
2902:
2892:
2891:
2885:
2884:
2882:
2881:
2876:
2874:Thoralf Skolem
2871:
2866:
2861:
2856:
2851:
2846:
2841:
2836:
2831:
2826:
2820:
2818:
2812:
2811:
2809:
2808:
2803:
2798:
2792:
2790:
2788:
2787:
2784:
2778:
2775:
2774:
2772:
2771:
2770:
2769:
2764:
2759:
2758:
2757:
2742:
2741:
2740:
2728:
2727:
2726:
2715:
2714:
2709:
2704:
2699:
2693:
2691:
2687:
2686:
2684:
2683:
2678:
2673:
2668:
2659:
2654:
2649:
2639:
2634:
2633:
2632:
2627:
2622:
2612:
2602:
2597:
2592:
2586:
2584:
2577:
2576:
2574:
2573:
2568:
2563:
2558:
2556:Ordinal number
2553:
2548:
2543:
2538:
2537:
2536:
2531:
2521:
2516:
2511:
2506:
2501:
2491:
2486:
2480:
2478:
2476:
2475:
2472:
2468:
2465:
2464:
2462:
2461:
2456:
2451:
2446:
2441:
2436:
2434:Disjoint union
2431:
2426:
2420:
2414:
2412:
2406:
2405:
2403:
2402:
2401:
2400:
2395:
2384:
2383:
2381:Martin's axiom
2378:
2373:
2368:
2363:
2358:
2353:
2348:
2346:Extensionality
2343:
2338:
2333:
2332:
2331:
2326:
2321:
2311:
2305:
2303:
2297:
2296:
2289:
2287:
2285:
2284:
2278:
2276:
2272:
2271:
2266:
2264:
2263:
2256:
2249:
2241:
2232:
2231:
2217:
2214:
2213:
2211:
2210:
2205:
2200:
2195:
2190:
2189:
2188:
2178:
2173:
2168:
2159:
2154:
2149:
2144:
2142:Abstract logic
2138:
2136:
2132:
2131:
2129:
2128:
2123:
2121:Turing machine
2118:
2113:
2108:
2103:
2098:
2093:
2092:
2091:
2086:
2081:
2076:
2071:
2061:
2059:Computable set
2056:
2051:
2046:
2041:
2035:
2033:
2027:
2026:
2024:
2023:
2018:
2013:
2008:
2003:
1998:
1993:
1988:
1987:
1986:
1981:
1976:
1966:
1961:
1956:
1954:Satisfiability
1951:
1946:
1941:
1940:
1939:
1929:
1928:
1927:
1917:
1916:
1915:
1910:
1905:
1900:
1895:
1885:
1884:
1883:
1878:
1871:Interpretation
1867:
1865:
1859:
1858:
1856:
1855:
1850:
1845:
1840:
1835:
1825:
1820:
1819:
1818:
1817:
1816:
1806:
1801:
1791:
1786:
1781:
1776:
1771:
1766:
1760:
1758:
1752:
1751:
1748:
1747:
1745:
1744:
1736:
1735:
1734:
1733:
1728:
1727:
1726:
1721:
1716:
1696:
1695:
1694:
1692:minimal axioms
1689:
1678:
1677:
1676:
1665:
1664:
1663:
1658:
1653:
1648:
1643:
1638:
1625:
1623:
1604:
1603:
1601:
1600:
1599:
1598:
1586:
1581:
1580:
1579:
1574:
1569:
1564:
1554:
1549:
1544:
1539:
1538:
1537:
1532:
1522:
1521:
1520:
1515:
1510:
1505:
1495:
1490:
1489:
1488:
1483:
1478:
1468:
1467:
1466:
1461:
1456:
1451:
1446:
1441:
1431:
1426:
1421:
1416:
1415:
1414:
1409:
1404:
1399:
1389:
1384:
1382:Formation rule
1379:
1374:
1373:
1372:
1367:
1357:
1356:
1355:
1345:
1340:
1335:
1330:
1324:
1318:
1301:Formal systems
1297:
1296:
1293:
1292:
1290:
1289:
1284:
1279:
1274:
1269:
1264:
1259:
1254:
1249:
1244:
1243:
1242:
1237:
1226:
1224:
1220:
1219:
1217:
1216:
1215:
1214:
1204:
1199:
1198:
1197:
1190:Large cardinal
1187:
1182:
1177:
1172:
1167:
1153:
1152:
1151:
1146:
1141:
1126:
1124:
1114:
1113:
1111:
1110:
1109:
1108:
1103:
1098:
1088:
1083:
1078:
1073:
1068:
1063:
1058:
1053:
1048:
1043:
1038:
1033:
1027:
1025:
1018:
1017:
1015:
1014:
1013:
1012:
1007:
1002:
997:
992:
987:
979:
978:
977:
972:
962:
957:
955:Extensionality
952:
950:Ordinal number
947:
937:
932:
931:
930:
919:
913:
907:
906:
903:
902:
900:
899:
894:
889:
884:
879:
874:
869:
868:
867:
857:
856:
855:
842:
840:
834:
833:
831:
830:
829:
828:
823:
818:
808:
803:
798:
793:
788:
783:
777:
775:
769:
768:
766:
765:
760:
755:
750:
745:
740:
735:
734:
733:
723:
718:
713:
708:
703:
698:
692:
690:
681:
675:
674:
672:
671:
666:
661:
656:
651:
646:
634:Cantor's
632:
627:
622:
612:
610:
597:
596:
594:
593:
588:
583:
578:
573:
568:
563:
558:
553:
548:
543:
538:
533:
532:
531:
520:
518:
514:
513:
508:
506:
505:
498:
491:
483:
477:
476:
435:
401:
396:
374:
371:
303:
300:
216:
213:
169:
168:
139:
101:
56:
53:
35:introduced by
15:
13:
10:
9:
6:
4:
3:
2:
2922:
2911:
2908:
2906:
2903:
2901:
2898:
2897:
2895:
2880:
2879:Ernst Zermelo
2877:
2875:
2872:
2870:
2867:
2865:
2864:Willard Quine
2862:
2860:
2857:
2855:
2852:
2850:
2847:
2845:
2842:
2840:
2837:
2835:
2832:
2830:
2827:
2825:
2822:
2821:
2819:
2817:
2816:Set theorists
2813:
2807:
2804:
2802:
2799:
2797:
2794:
2793:
2791:
2785:
2783:
2780:
2779:
2776:
2768:
2765:
2763:
2762:KripkeâPlatek
2760:
2756:
2753:
2752:
2751:
2748:
2747:
2746:
2743:
2739:
2736:
2735:
2734:
2733:
2729:
2725:
2722:
2721:
2720:
2717:
2716:
2713:
2710:
2708:
2705:
2703:
2700:
2698:
2695:
2694:
2692:
2688:
2682:
2679:
2677:
2674:
2672:
2669:
2667:
2665:
2660:
2658:
2655:
2653:
2650:
2647:
2643:
2640:
2638:
2635:
2631:
2628:
2626:
2623:
2621:
2618:
2617:
2616:
2613:
2610:
2606:
2603:
2601:
2598:
2596:
2593:
2591:
2588:
2587:
2585:
2582:
2578:
2572:
2569:
2567:
2564:
2562:
2559:
2557:
2554:
2552:
2549:
2547:
2544:
2542:
2539:
2535:
2532:
2530:
2527:
2526:
2525:
2522:
2520:
2517:
2515:
2512:
2510:
2507:
2505:
2502:
2499:
2495:
2492:
2490:
2487:
2485:
2482:
2481:
2479:
2473:
2470:
2469:
2466:
2460:
2457:
2455:
2452:
2450:
2447:
2445:
2442:
2440:
2437:
2435:
2432:
2430:
2427:
2424:
2421:
2419:
2416:
2415:
2413:
2411:
2407:
2399:
2398:specification
2396:
2394:
2391:
2390:
2389:
2386:
2385:
2382:
2379:
2377:
2374:
2372:
2369:
2367:
2364:
2362:
2359:
2357:
2354:
2352:
2349:
2347:
2344:
2342:
2339:
2337:
2334:
2330:
2327:
2325:
2322:
2320:
2317:
2316:
2315:
2312:
2310:
2307:
2306:
2304:
2302:
2298:
2293:
2283:
2280:
2279:
2277:
2273:
2269:
2262:
2257:
2255:
2250:
2248:
2243:
2242:
2239:
2229:
2228:
2223:
2215:
2209:
2206:
2204:
2201:
2199:
2196:
2194:
2191:
2187:
2184:
2183:
2182:
2179:
2177:
2174:
2172:
2169:
2167:
2163:
2160:
2158:
2155:
2153:
2150:
2148:
2145:
2143:
2140:
2139:
2137:
2133:
2127:
2124:
2122:
2119:
2117:
2116:Recursive set
2114:
2112:
2109:
2107:
2104:
2102:
2099:
2097:
2094:
2090:
2087:
2085:
2082:
2080:
2077:
2075:
2072:
2070:
2067:
2066:
2065:
2062:
2060:
2057:
2055:
2052:
2050:
2047:
2045:
2042:
2040:
2037:
2036:
2034:
2032:
2028:
2022:
2019:
2017:
2014:
2012:
2009:
2007:
2004:
2002:
1999:
1997:
1994:
1992:
1989:
1985:
1982:
1980:
1977:
1975:
1972:
1971:
1970:
1967:
1965:
1962:
1960:
1957:
1955:
1952:
1950:
1947:
1945:
1942:
1938:
1935:
1934:
1933:
1930:
1926:
1925:of arithmetic
1923:
1922:
1921:
1918:
1914:
1911:
1909:
1906:
1904:
1901:
1899:
1896:
1894:
1891:
1890:
1889:
1886:
1882:
1879:
1877:
1874:
1873:
1872:
1869:
1868:
1866:
1864:
1860:
1854:
1851:
1849:
1846:
1844:
1841:
1839:
1836:
1833:
1832:from ZFC
1829:
1826:
1824:
1821:
1815:
1812:
1811:
1810:
1807:
1805:
1802:
1800:
1797:
1796:
1795:
1792:
1790:
1787:
1785:
1782:
1780:
1777:
1775:
1772:
1770:
1767:
1765:
1762:
1761:
1759:
1757:
1753:
1743:
1742:
1738:
1737:
1732:
1731:non-Euclidean
1729:
1725:
1722:
1720:
1717:
1715:
1714:
1710:
1709:
1707:
1704:
1703:
1701:
1697:
1693:
1690:
1688:
1685:
1684:
1683:
1679:
1675:
1672:
1671:
1670:
1666:
1662:
1659:
1657:
1654:
1652:
1649:
1647:
1644:
1642:
1639:
1637:
1634:
1633:
1631:
1627:
1626:
1624:
1619:
1613:
1608:Example
1605:
1597:
1592:
1591:
1590:
1587:
1585:
1582:
1578:
1575:
1573:
1570:
1568:
1565:
1563:
1560:
1559:
1558:
1555:
1553:
1550:
1548:
1545:
1543:
1540:
1536:
1533:
1531:
1528:
1527:
1526:
1523:
1519:
1516:
1514:
1511:
1509:
1506:
1504:
1501:
1500:
1499:
1496:
1494:
1491:
1487:
1484:
1482:
1479:
1477:
1474:
1473:
1472:
1469:
1465:
1462:
1460:
1457:
1455:
1452:
1450:
1447:
1445:
1442:
1440:
1437:
1436:
1435:
1432:
1430:
1427:
1425:
1422:
1420:
1417:
1413:
1410:
1408:
1405:
1403:
1400:
1398:
1395:
1394:
1393:
1390:
1388:
1385:
1383:
1380:
1378:
1375:
1371:
1368:
1366:
1365:by definition
1363:
1362:
1361:
1358:
1354:
1351:
1350:
1349:
1346:
1344:
1341:
1339:
1336:
1334:
1331:
1329:
1326:
1325:
1322:
1319:
1317:
1313:
1308:
1302:
1298:
1288:
1285:
1283:
1280:
1278:
1275:
1273:
1270:
1268:
1265:
1263:
1260:
1258:
1255:
1253:
1252:KripkeâPlatek
1250:
1248:
1245:
1241:
1238:
1236:
1233:
1232:
1231:
1228:
1227:
1225:
1221:
1213:
1210:
1209:
1208:
1205:
1203:
1200:
1196:
1193:
1192:
1191:
1188:
1186:
1183:
1181:
1178:
1176:
1173:
1171:
1168:
1165:
1161:
1157:
1154:
1150:
1147:
1145:
1142:
1140:
1137:
1136:
1135:
1131:
1128:
1127:
1125:
1123:
1119:
1115:
1107:
1104:
1102:
1099:
1097:
1096:constructible
1094:
1093:
1092:
1089:
1087:
1084:
1082:
1079:
1077:
1074:
1072:
1069:
1067:
1064:
1062:
1059:
1057:
1054:
1052:
1049:
1047:
1044:
1042:
1039:
1037:
1034:
1032:
1029:
1028:
1026:
1024:
1019:
1011:
1008:
1006:
1003:
1001:
998:
996:
993:
991:
988:
986:
983:
982:
980:
976:
973:
971:
968:
967:
966:
963:
961:
958:
956:
953:
951:
948:
946:
942:
938:
936:
933:
929:
926:
925:
924:
921:
920:
917:
914:
912:
908:
898:
895:
893:
890:
888:
885:
883:
880:
878:
875:
873:
870:
866:
863:
862:
861:
858:
854:
849:
848:
847:
844:
843:
841:
839:
835:
827:
824:
822:
819:
817:
814:
813:
812:
809:
807:
804:
802:
799:
797:
794:
792:
789:
787:
784:
782:
779:
778:
776:
774:
773:Propositional
770:
764:
761:
759:
756:
754:
751:
749:
746:
744:
741:
739:
736:
732:
729:
728:
727:
724:
722:
719:
717:
714:
712:
709:
707:
704:
702:
701:Logical truth
699:
697:
694:
693:
691:
689:
685:
682:
680:
676:
670:
667:
665:
662:
660:
657:
655:
652:
650:
647:
645:
641:
637:
633:
631:
628:
626:
623:
621:
617:
614:
613:
611:
609:
603:
598:
592:
589:
587:
584:
582:
579:
577:
574:
572:
569:
567:
564:
562:
559:
557:
554:
552:
549:
547:
544:
542:
539:
537:
534:
530:
527:
526:
525:
522:
521:
519:
515:
511:
504:
499:
497:
492:
490:
485:
484:
481:
474:
470:
466:
462:
458:
454:
450:
446:
445:
440:
436:
432:
427:
423:
419:
418:
410:
406:
402:
399:
393:
389:
385:
381:
377:
376:
372:
370:
368:
364:
360:
356:
352:
349:there exists
348:
344:
340:
336:
332:
328:
324:
319:
317:
313:
309:
301:
299:
297:
293:
289:
284:
282:
278:
274:
270:
266:
262:
258:
254:
250:
246:
242:
238:
234:
230:
226:
221:
214:
212:
210:
206:
202:
198:
194:
190:
186:
182:
178:
174:
167:
163:
159:
155:
151:
147:
143:
140:
137:
133:
129:
125:
121:
117:
113:
109:
105:
102:
99:
95:
92:
89:
85:
81:
77:
73:
70:
69:
68:
66:
62:
59:Suppose that
54:
52:
50:
46:
42:
38:
34:
30:
26:
22:
2829:Georg Cantor
2824:Paul Bernays
2755:MorseâKelley
2730:
2663:
2662:Subset
2609:hereditarily
2571:Venn diagram
2529:ordered pair
2444:Intersection
2388:Axiom schema
2218:
2016:Ultraproduct
1863:Model theory
1828:Independence
1764:Formal proof
1756:Proof theory
1739:
1712:
1669:real numbers
1641:second-order
1552:Substitution
1429:Metalanguage
1370:conservative
1343:Axiom schema
1287:Constructive
1257:MorseâKelley
1223:Set theories
1202:Aleph number
1195:inaccessible
1101:Grothendieck
985:intersection
872:Higher-order
860:Second-order
806:Truth tables
763:Venn diagram
546:Formal proof
448:
442:
421:
415:
383:
380:Jech, Thomas
366:
362:
358:
354:
350:
346:
342:
338:
334:
330:
326:
320:
305:
285:
280:
276:
272:
269:homomorphism
260:
256:
252:
248:
244:
240:
236:
232:
228:
224:
222:
218:
208:
204:
200:
196:
192:
188:
184:
172:
170:
165:
161:
157:
153:
149:
141:
135:
131:
127:
123:
119:
118:contains an
115:
111:
108:well-founded
103:
97:
93:
90:
87:
83:
79:
71:
64:
60:
58:
28:
24:
18:
2854:Thomas Jech
2697:Alternative
2676:Uncountable
2630:Ultrafilter
2489:Cardinality
2393:replacement
2341:Determinacy
2126:Type theory
2074:undecidable
2006:Truth value
1893:equivalence
1572:non-logical
1185:Enumeration
1175:Isomorphism
1122:cardinality
1106:Von Neumann
1071:Ultrafilter
1036:Uncountable
970:equivalence
887:Quantifiers
877:Fixed-point
846:First-order
726:Consistency
711:Proposition
688:Traditional
659:Lindström's
649:Compactness
591:Type theory
536:Cardinality
302:Application
146:extensional
2894:Categories
2849:Kurt Gödel
2834:Paul Cohen
2671:Transitive
2439:Identities
2423:Complement
2410:Operations
2371:Regularity
2309:Adjunction
2268:Set theory
1937:elementary
1630:arithmetic
1498:Quantifier
1476:functional
1348:Expression
1066:Transitive
1010:identities
995:complement
928:hereditary
911:Set theory
384:Set Theory
373:References
357:such that
306:Every set
247:} for all
227:such that
223:A mapping
207:such that
177:transitive
138:is empty),
130:such that
67:such that
33:set theory
2782:Paradoxes
2702:Axiomatic
2681:Universal
2657:Singleton
2652:Recursive
2595:Countable
2590:Amorphous
2449:Power set
2366:Power set
2324:dependent
2319:countable
2208:Supertask
2111:Recursion
2069:decidable
1903:saturated
1881:of models
1804:deductive
1799:axiomatic
1719:Hilbert's
1706:Euclidean
1687:canonical
1610:axiomatic
1542:Signature
1471:Predicate
1360:Extension
1282:Ackermann
1207:Operation
1086:Universal
1076:Recursive
1051:Singleton
1046:Inhabited
1031:Countable
1021:Types of
1005:power set
975:partition
892:Predicate
838:Predicate
753:Syllogism
743:Soundness
716:Inference
706:Tautology
608:paradoxes
296:bisimilar
243:) :
55:Statement
2786:Problems
2690:Theories
2666:Superset
2642:Infinite
2471:Concepts
2351:Infinity
2275:Overview
2193:Logicism
2186:timeline
2162:Concrete
2021:Validity
1991:T-schema
1984:Kripke's
1979:Tarski's
1974:semantic
1964:Strength
1913:submodel
1908:spectrum
1876:function
1724:Tarski's
1713:Elements
1700:geometry
1656:Robinson
1577:variable
1562:function
1535:spectrum
1525:Sentence
1481:variable
1424:Language
1377:Relation
1338:Automata
1328:Alphabet
1312:language
1166:-jection
1144:codomain
1130:Function
1091:Universe
1061:Infinite
965:Relation
748:Validity
738:Argument
636:theorem,
473:35526998
407:(1949),
382:(2003),
333:with no
86: :
76:set-like
2724:General
2719:Zermelo
2625:subbase
2607: (
2546:Forcing
2524:Element
2496: (
2474:Methods
2361:Pairing
2135:Related
1932:Diagram
1830: (
1809:Hilbert
1794:Systems
1789:Theorem
1667:of the
1612:systems
1392:Formula
1387:Grammar
1303: (
1247:General
960:Forcing
945:Element
865:Monadic
640:paradox
581:Theorem
517:General
465:2268947
47: (
39: (
2900:Lemmas
2615:Filter
2605:Finite
2541:Family
2484:Almost
2329:global
2314:Choice
2301:Axioms
1898:finite
1661:Skolem
1614:
1589:Theory
1557:Symbol
1547:String
1530:atomic
1407:ground
1402:closed
1397:atomic
1353:ground
1316:syntax
1212:binary
1139:domain
1056:Finite
821:finite
679:Logics
638:
586:Theory
471:
463:
394:
181:proper
23:, the
2707:Naive
2637:Fuzzy
2600:Empty
2583:types
2534:tuple
2504:Class
2498:large
2459:Union
2376:Union
1888:Model
1636:Peano
1493:Proof
1333:Arity
1262:Naive
1149:image
1081:Fuzzy
1041:Empty
990:union
935:Class
576:Model
566:Lemma
524:Axiom
469:S2CID
461:JSTOR
412:(PDF)
365:. So
308:model
245:y R x
235:) = {
209:y R x
2620:base
2011:Type
1814:list
1618:list
1595:list
1584:Term
1518:rank
1412:open
1306:list
1118:Maps
1023:sets
882:Free
852:list
602:list
529:list
392:ISBN
160:and
49:1953
41:1949
2581:Set
1698:of
1680:of
1628:of
1160:Sur
1134:Map
941:Ur-
923:Set
453:doi
426:doi
353:in
310:of
271:of
263:by
259:on
251:in
203:of
195:of
164:of
144:is
114:of
106:is
82:= {
74:is
51:).
19:In
2896::
2084:NP
1708::
1702::
1632::
1309:),
1164:Bi
1156:In
467:,
459:,
449:18
447:,
422:36
420:,
414:,
390:,
361:=
312:ZF
187:,
152:â
148::
134:â©
126:â
78::
2664:·
2648:)
2644:(
2611:)
2500:)
2260:e
2253:t
2246:v
2164:/
2079:P
1834:)
1620:)
1616:(
1513:â
1508:!
1503:â
1464:=
1459:â
1454:â
1449:â§
1444:âš
1439:ÂŹ
1162:/
1158:/
1132:/
943:)
939:(
826:â
816:3
604:)
502:e
495:t
488:v
455::
428::
367:M
363:R
359:A
355:M
351:x
347:A
343:A
339:A
335:R
331:A
327:M
281:R
277:F
273:R
261:X
257:R
253:X
249:x
241:y
239:(
237:F
233:x
231:(
229:F
225:F
205:X
201:y
197:X
193:x
189:R
185:X
173:R
166:X
162:y
158:x
154:R
150:R
142:R
136:S
132:R
128:S
124:x
120:R
116:X
112:S
104:R
100:,
98:x
94:x
91:R
88:y
84:y
80:R
72:R
65:X
61:R
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