3307:
619:
is non-empty, it must contain all of its finite subsets and a subset of each finite cardinality. One can also prove immediately from the definitions that the intersection of any class of universes is a universe.
296:
is an axiomatic treatment of set theory, used in some automatic proof systems, in which every set belongs to a
Grothendieck universe. The concept of a Grothendieck universe can also be defined in a
772:
251:
196:
987:
1131:
1179:
455:
365:
1425:
1328:
1297:
663:
1359:
490:
429:
915:|; when the axiom of foundation is not assumed, there are counterexamples (we may take for example U to be the set of all finite sets of finite sets etc. of the sets x
112:
542:
516:
391:
339:
1538:
SĂ©minaire de GĂ©omĂ©trie AlgĂ©brique du Bois Marie â 1963â64 â ThĂ©orie des topos et cohomologie Ă©tale des schĂ©mas â (SGA 4) â vol. 1 (Lecture Notes in
Mathematics
1686:
261:
A Grothendieck universe is meant to provide a set in which all of mathematics can be performed. (In fact, uncountable
Grothendieck universes provide
2361:
2444:
1585:
797:) is either zero or strongly inaccessible. Assuming it is non-zero, it is a strong limit cardinal because the power set of any element of
265:
of set theory with the natural â-relation, natural powerset operation etc.). Elements of a
Grothendieck universe are sometimes called
1521:
2758:
2916:
1704:
669:
Other examples are more difficult to construct. Loosely speaking, this is because
Grothendieck universes are equivalent to
2771:
2094:
2776:
2766:
2503:
2356:
1709:
714:
293:
1700:
2912:
1497:
1428:
2254:
213:
155:
3331:
3009:
2753:
1578:
1545:
1062:. The proof of this fact is long, so for details, we again refer to Bourbaki's article, listed in the references.
1260:. This gives another form of the equivalence between Grothendieck universes and strongly inaccessible cardinals:
2314:
2007:
1397:
948:
285:
1748:
3270:
2972:
2735:
2730:
2555:
1976:
1660:
3265:
3048:
2965:
2678:
2609:
2486:
1728:
3190:
3016:
2702:
2336:
1935:
1087:
1490:
From Sets and Types to
Topology and Analysis: Towards Practicable Foundations for Constructive Mathematics
3068:
3063:
2673:
2412:
2341:
1670:
1571:
1427:
cannot be proved from ZFC either. However, strongly inaccessible cardinals are on the lower end of the
2997:
2587:
1981:
1949:
1640:
1147:
3287:
3236:
3133:
2631:
2592:
2069:
1714:
1743:
1482:
696:(C) For each cardinal Îș, there is a strongly inaccessible cardinal λ that is strictly larger than Îș.
3341:
3336:
3128:
3058:
2597:
2449:
2432:
2155:
1635:
903:). By invoking the axiom of foundation, that no set is contained in itself, it can be shown that
434:
344:
2960:
2937:
2898:
2784:
2725:
2371:
2291:
2135:
2079:
1692:
1403:
1306:
1275:
641:
636:
557:
1335:
3250:
2977:
2955:
2922:
2815:
2661:
2646:
2619:
2570:
2454:
2389:
2214:
2180:
2175:
2049:
1880:
1857:
1529:
1453:
1448:
1222:. To show that the universe axiom (U) implies the large cardinal axiom (C), choose a cardinal
460:
399:
270:
3180:
3033:
2825:
2543:
2279:
2044:
2029:
1910:
1885:
670:
289:
85:
945:
has the cardinality of the continuum, but all of its members have finite cardinality and so
521:
495:
370:
318:
3153:
3115:
2992:
2796:
2636:
2560:
2538:
2366:
2324:
2223:
2190:
2054:
1842:
1753:
1458:
1396:
Since the existence of strongly inaccessible cardinals cannot be proved from the axioms of
1549:
8:
3282:
3173:
3158:
3138:
3095:
2982:
2932:
2858:
2803:
2740:
2533:
2528:
2476:
2244:
2233:
1905:
1805:
1733:
1724:
1720:
1655:
1650:
1432:
3311:
3080:
3043:
3028:
3021:
3004:
2808:
2790:
2656:
2582:
2565:
2518:
2331:
2240:
2074:
2059:
2019:
1971:
1956:
1944:
1900:
1875:
1645:
1594:
1533:
1065:
To show that the large cardinal axiom (C) implies the universe axiom (U), choose a set
278:
2264:
3306:
3246:
3053:
2863:
2853:
2745:
2626:
2461:
2437:
2218:
2202:
2107:
2084:
1961:
1930:
1895:
1790:
1625:
1493:
284:
The existence of a nontrivial
Grothendieck universe goes beyond the usual axioms of
3260:
3255:
3148:
3105:
2927:
2888:
2883:
2868:
2694:
2651:
2548:
2346:
2296:
1870:
1832:
1517:
1478:
3241:
3231:
3185:
3168:
3123:
3085:
2987:
2907:
2714:
2641:
2614:
2602:
2508:
2422:
2396:
2351:
2319:
2120:
1922:
1865:
1815:
1780:
1738:
1436:
1431:; thus, most set theories that use large cardinals (such as "ZFC plus there is a
3226:
3205:
3163:
3143:
3038:
2893:
2491:
2481:
2471:
2466:
2400:
2274:
2150:
2039:
2034:
2012:
1613:
64:
3325:
3200:
2878:
2385:
2170:
2160:
2130:
2115:
1785:
1525:
1330:, or a strongly inaccessible cardinal, then there is a Grothendieck universe
3100:
2947:
2848:
2840:
2720:
2668:
2577:
2513:
2496:
2427:
2286:
2145:
1847:
1630:
274:
262:
3210:
3090:
2269:
2259:
2206:
1890:
1810:
1795:
1675:
1620:
20:
2140:
1995:
1966:
1772:
3292:
3195:
2248:
2165:
2125:
2089:
2025:
1837:
1827:
1800:
1563:
138:
623:
3277:
3075:
2523:
2228:
1822:
2873:
1665:
1400:(ZFC), the existence of universes other than the empty set and
2417:
1763:
1608:
548:
It is similarly easy to prove that any
Grothendieck universe
297:
673:. More formally, the following two axioms are equivalent:
1238:
is strongly inaccessible and strictly larger than that of
1230:
1043:
itself corresponds to the empty sequence.) Then the set
628:
615:
In particular, it follows from the last axiom that if
216:
1406:
1338:
1309:
1278:
1181:. By (C), there is a strongly inaccessible cardinal
1150:
1090:
996:
be a strongly inaccessible cardinal. Say that a set
951:
717:
644:
524:
498:
463:
437:
402:
373:
347:
321:
158:
88:
574:All disjoint unions of all families of elements of
1419:
1353:
1322:
1291:
1245:In fact, any Grothendieck universe is of the form
1173:
1125:
989: ; see Bourbaki's article for more details).
981:
766:
657:
536:
510:
484:
449:
423:
385:
359:
333:
308:As an example, we will prove an easy proposition.
245:
190:
106:
1439:") will prove that Grothendieck universes exist.
767:{\displaystyle \mathbf {c} (U)=\sup _{x\in U}|x|}
624:Grothendieck universes and inaccessible cardinals
585:All intersections of all families of elements of
3323:
736:
288:; in particular it would imply the existence of
919:where the index α is any real number, and
246:{\textstyle \bigcup _{\alpha \in I}x_{\alpha }}
1299:, or a strongly inaccessible cardinal. And if
700:To prove this fact, we introduce the function
191:{\displaystyle \{x_{\alpha }\}_{\alpha \in I}}
1579:
1203:be the universe of the previous paragraph.
563:All products of all families of elements of
173:
159:
101:
89:
982:{\displaystyle \mathbf {c} (U)=\aleph _{0}}
813:. To see that it is regular, suppose that
1771:
1586:
1572:
1058:is a Grothendieck universe of cardinality
596:All functions between any two elements of
1477:
1516:
822:is a collection of cardinals indexed by
1435:", "ZFC plus there are infinitely many
681:, there exists a Grothendieck universe
3324:
1593:
1567:
1126:{\displaystyle x_{n+1}=\bigcup x_{n}}
891:has the cardinality of an element of
273:, who used them as a way of avoiding
1492:. Clarendon Press. pp. 78â90.
13:
1548:. pp. 185â217. Archived from
1311:
1280:
970:
269:. The idea of universes is due to
14:
3353:
1174:{\displaystyle \bigcup _{n}x_{n}}
866:can be replaced by an element of
3305:
1133:be the union of the elements of
953:
719:
607:whose cardinal is an element of
1544:(in French). Berlin; New York:
845:). Then, by the definition of
671:strongly inaccessible cardinals
290:strongly inaccessible cardinals
31:with the following properties:
1471:
1348:
1342:
1264:For any Grothendieck universe
963:
957:
760:
752:
729:
723:
473:
467:
418:
412:
294:TarskiâGrothendieck set theory
1:
3266:History of mathematical logic
1510:
1054:of all sets strictly of type
781:| we mean the cardinality of
303:
3191:Primitive recursive function
870:. The union of elements of
450:{\displaystyle y\subseteq x}
360:{\displaystyle y\subseteq x}
7:
1442:
1420:{\displaystyle V_{\omega }}
1398:ZermeloâFraenkel set theory
1323:{\displaystyle \aleph _{0}}
1292:{\displaystyle \aleph _{0}}
826:, where the cardinality of
658:{\displaystyle V_{\omega }}
286:ZermeloâFraenkel set theory
198:is a family of elements of
10:
3358:
2255:SchröderâBernstein theorem
1982:Monadic predicate calculus
1641:Foundations of mathematics
1354:{\displaystyle u(\kappa )}
1207:is strictly of type Îș, so
3301:
3288:Philosophy of mathematics
3237:Automated theorem proving
3219:
3114:
2946:
2839:
2691:
2408:
2384:
2362:Von NeumannâBernaysâGödel
2307:
2201:
2105:
2003:
1994:
1921:
1856:
1762:
1684:
1601:
874:indexed by an element of
785:. Then for any universe
589:indexed by an element of
578:indexed by an element of
567:indexed by an element of
485:{\displaystyle P(x)\in U}
424:{\displaystyle y\in P(x)}
1464:
637:hereditarily finite sets
560:of each of its elements,
145:, is also an element of
3332:Set-theoretic universes
2938:Self-verifying theories
2759:Tarski's axiomatization
1710:Tarski's undefinability
1705:incompleteness theorems
1429:list of large cardinals
107:{\displaystyle \{x,y\}}
3312:Mathematics portal
2923:Proof of impossibility
2571:propositional variable
1881:Propositional calculus
1530:Alexandre Grothendieck
1483:"Universes in Toposes"
1454:Universe (mathematics)
1449:Constructible universe
1421:
1355:
1324:
1293:
1234:. The cardinality of
1175:
1127:
983:
768:
659:
538:
537:{\displaystyle y\in U}
512:
511:{\displaystyle x\in U}
486:
451:
425:
387:
386:{\displaystyle y\in U}
361:
335:
334:{\displaystyle x\in U}
271:Alexander Grothendieck
247:
192:
108:
55:is also an element of
3181:Kolmogorov complexity
3134:Computably enumerable
3034:Model complete theory
2826:Principia Mathematica
1886:Propositional formula
1715:BanachâTarski paradox
1422:
1356:
1325:
1294:
1176:
1128:
984:
895:, hence is less than
805:and every element of
769:
660:
539:
513:
487:
452:
426:
388:
362:
336:
248:
193:
109:
78:are both elements of
25:Grothendieck universe
16:Set-theoretic concept
3129:ChurchâTuring thesis
3116:Computability theory
2325:continuum hypothesis
1843:Square of opposition
1701:Gödel's completeness
1459:Von Neumann universe
1404:
1336:
1307:
1276:
1148:
1088:
1004:if for any sequence
1000:is strictly of type
949:
882:, so the sum of the
715:
642:
522:
496:
461:
435:
400:
371:
345:
319:
214:
156:
86:
3283:Mathematical object
3174:P versus NP problem
3139:Computable function
2933:Reverse mathematics
2859:Logical consequence
2736:primitive recursive
2731:elementary function
2504:Free/bound variable
2357:TarskiâGrothendieck
1876:Logical connectives
1806:Logical equivalence
1656:Logical consequence
1433:measurable cardinal
3081:Transfer principle
3044:Semantics of logic
3029:Categorical theory
3005:Non-standard model
2519:Logical connective
1646:Information theory
1595:Mathematical logic
1534:Jean-Louis Verdier
1417:
1351:
1320:
1289:
1272:| is either zero,
1171:
1160:
1123:
979:
764:
750:
655:
632:The empty set, and
534:
508:
482:
447:
421:
383:
357:
331:
279:algebraic geometry
243:
232:
188:
104:
3319:
3318:
3251:Abstract category
3054:Theories of truth
2864:Rule of inference
2854:Natural deduction
2835:
2834:
2380:
2379:
2085:Cartesian product
1990:
1989:
1896:Many-valued logic
1871:Boolean functions
1754:Russell's paradox
1729:diagonal argument
1626:First-order logic
1518:Bourbaki, Nicolas
1479:Streicher, Thomas
1151:
878:is an element of
801:is an element of
735:
677:(U) For each set
253:is an element of
217:
210:, then the union
206:is an element of
125:is an element of
114:is an element of
47:is an element of
39:is an element of
3349:
3310:
3309:
3261:History of logic
3256:Category of sets
3149:Decision problem
2928:Ordinal analysis
2869:Sequent calculus
2767:Boolean algebras
2707:
2706:
2681:
2652:logical/constant
2406:
2405:
2392:
2315:ZermeloâFraenkel
2066:Set operations:
2001:
2000:
1938:
1769:
1768:
1749:LöwenheimâSkolem
1636:Formal semantics
1588:
1581:
1574:
1565:
1564:
1560:
1558:
1557:
1504:
1503:
1487:
1475:
1437:Woodin cardinals
1426:
1424:
1423:
1418:
1416:
1415:
1391:
1376:
1363:. Furthermore,
1362:
1360:
1358:
1357:
1352:
1329:
1327:
1326:
1321:
1319:
1318:
1302:
1298:
1296:
1295:
1290:
1288:
1287:
1259:
1255:
1241:
1229:
1225:
1221:
1202:
1191:
1184:
1180:
1178:
1177:
1172:
1170:
1169:
1159:
1132:
1130:
1129:
1124:
1122:
1121:
1106:
1105:
1061:
1057:
1053:
1038:
1003:
995:
988:
986:
985:
980:
978:
977:
956:
856:
829:
825:
773:
771:
770:
765:
763:
755:
749:
722:
664:
662:
661:
656:
654:
653:
543:
541:
540:
535:
517:
515:
514:
509:
491:
489:
488:
483:
456:
454:
453:
448:
430:
428:
427:
422:
392:
390:
389:
384:
366:
364:
363:
358:
340:
338:
337:
332:
252:
250:
249:
244:
242:
241:
231:
205:
197:
195:
194:
189:
187:
186:
171:
170:
113:
111:
110:
105:
3357:
3356:
3352:
3351:
3350:
3348:
3347:
3346:
3342:Large cardinals
3337:Category theory
3322:
3321:
3320:
3315:
3304:
3297:
3242:Category theory
3232:Algebraic logic
3215:
3186:Lambda calculus
3124:Church encoding
3110:
3086:Truth predicate
2942:
2908:Complete theory
2831:
2700:
2696:
2692:
2687:
2679:
2399: and
2395:
2390:
2376:
2352:New Foundations
2320:axiom of choice
2303:
2265:Gödel numbering
2205: and
2197:
2101:
1986:
1936:
1917:
1866:Boolean algebra
1852:
1816:Equiconsistency
1781:Classical logic
1758:
1739:Halting problem
1727: and
1703: and
1691: and
1690:
1685:Theorems (
1680:
1597:
1592:
1555:
1553:
1546:Springer-Verlag
1513:
1508:
1507:
1500:
1485:
1476:
1472:
1467:
1445:
1411:
1407:
1405:
1402:
1401:
1378:
1364:
1337:
1334:
1333:
1331:
1314:
1310:
1308:
1305:
1304:
1300:
1283:
1279:
1277:
1274:
1273:
1257:
1246:
1239:
1227:
1223:
1208:
1193:
1186:
1182:
1165:
1161:
1155:
1149:
1146:
1145:
1138:
1117:
1113:
1095:
1091:
1089:
1086:
1085:
1080:, and for each
1075:
1059:
1055:
1044:
1033:
1020:
1013:
1005:
1001:
993:
973:
969:
952:
950:
947:
946:
936:
927:
918:
890:
865:
854:
835:
827:
823:
821:
809:is a subset of
759:
751:
739:
718:
716:
713:
712:
649:
645:
643:
640:
639:
635:The set of all
626:
603:All subsets of
523:
520:
519:
497:
494:
493:
462:
459:
458:
436:
433:
432:
401:
398:
397:
372:
369:
368:
346:
343:
342:
320:
317:
316:
306:
237:
233:
221:
215:
212:
211:
203:
176:
172:
166:
162:
157:
154:
153:
87:
84:
83:
17:
12:
11:
5:
3355:
3345:
3344:
3339:
3334:
3317:
3316:
3302:
3299:
3298:
3296:
3295:
3290:
3285:
3280:
3275:
3274:
3273:
3263:
3258:
3253:
3244:
3239:
3234:
3229:
3227:Abstract logic
3223:
3221:
3217:
3216:
3214:
3213:
3208:
3206:Turing machine
3203:
3198:
3193:
3188:
3183:
3178:
3177:
3176:
3171:
3166:
3161:
3156:
3146:
3144:Computable set
3141:
3136:
3131:
3126:
3120:
3118:
3112:
3111:
3109:
3108:
3103:
3098:
3093:
3088:
3083:
3078:
3073:
3072:
3071:
3066:
3061:
3051:
3046:
3041:
3039:Satisfiability
3036:
3031:
3026:
3025:
3024:
3014:
3013:
3012:
3002:
3001:
3000:
2995:
2990:
2985:
2980:
2970:
2969:
2968:
2963:
2956:Interpretation
2952:
2950:
2944:
2943:
2941:
2940:
2935:
2930:
2925:
2920:
2910:
2905:
2904:
2903:
2902:
2901:
2891:
2886:
2876:
2871:
2866:
2861:
2856:
2851:
2845:
2843:
2837:
2836:
2833:
2832:
2830:
2829:
2821:
2820:
2819:
2818:
2813:
2812:
2811:
2806:
2801:
2781:
2780:
2779:
2777:minimal axioms
2774:
2763:
2762:
2761:
2750:
2749:
2748:
2743:
2738:
2733:
2728:
2723:
2710:
2708:
2689:
2688:
2686:
2685:
2684:
2683:
2671:
2666:
2665:
2664:
2659:
2654:
2649:
2639:
2634:
2629:
2624:
2623:
2622:
2617:
2607:
2606:
2605:
2600:
2595:
2590:
2580:
2575:
2574:
2573:
2568:
2563:
2553:
2552:
2551:
2546:
2541:
2536:
2531:
2526:
2516:
2511:
2506:
2501:
2500:
2499:
2494:
2489:
2484:
2474:
2469:
2467:Formation rule
2464:
2459:
2458:
2457:
2452:
2442:
2441:
2440:
2430:
2425:
2420:
2415:
2409:
2403:
2386:Formal systems
2382:
2381:
2378:
2377:
2375:
2374:
2369:
2364:
2359:
2354:
2349:
2344:
2339:
2334:
2329:
2328:
2327:
2322:
2311:
2309:
2305:
2304:
2302:
2301:
2300:
2299:
2289:
2284:
2283:
2282:
2275:Large cardinal
2272:
2267:
2262:
2257:
2252:
2238:
2237:
2236:
2231:
2226:
2211:
2209:
2199:
2198:
2196:
2195:
2194:
2193:
2188:
2183:
2173:
2168:
2163:
2158:
2153:
2148:
2143:
2138:
2133:
2128:
2123:
2118:
2112:
2110:
2103:
2102:
2100:
2099:
2098:
2097:
2092:
2087:
2082:
2077:
2072:
2064:
2063:
2062:
2057:
2047:
2042:
2040:Extensionality
2037:
2035:Ordinal number
2032:
2022:
2017:
2016:
2015:
2004:
1998:
1992:
1991:
1988:
1987:
1985:
1984:
1979:
1974:
1969:
1964:
1959:
1954:
1953:
1952:
1942:
1941:
1940:
1927:
1925:
1919:
1918:
1916:
1915:
1914:
1913:
1908:
1903:
1893:
1888:
1883:
1878:
1873:
1868:
1862:
1860:
1854:
1853:
1851:
1850:
1845:
1840:
1835:
1830:
1825:
1820:
1819:
1818:
1808:
1803:
1798:
1793:
1788:
1783:
1777:
1775:
1766:
1760:
1759:
1757:
1756:
1751:
1746:
1741:
1736:
1731:
1719:Cantor's
1717:
1712:
1707:
1697:
1695:
1682:
1681:
1679:
1678:
1673:
1668:
1663:
1658:
1653:
1648:
1643:
1638:
1633:
1628:
1623:
1618:
1617:
1616:
1605:
1603:
1599:
1598:
1591:
1590:
1583:
1576:
1568:
1562:
1561:
1512:
1509:
1506:
1505:
1498:
1469:
1468:
1466:
1463:
1462:
1461:
1456:
1451:
1444:
1441:
1414:
1410:
1394:
1393:
1350:
1347:
1344:
1341:
1317:
1313:
1286:
1282:
1168:
1164:
1158:
1154:
1136:
1120:
1116:
1112:
1109:
1104:
1101:
1098:
1094:
1073:
1029:
1018:
1009:
976:
972:
968:
965:
962:
959:
955:
932:
923:
916:
886:
861:
833:
817:
775:
774:
762:
758:
754:
748:
745:
742:
738:
734:
731:
728:
725:
721:
698:
697:
694:
667:
666:
652:
648:
633:
625:
622:
613:
612:
601:
594:
583:
572:
561:
546:
545:
533:
530:
527:
507:
504:
501:
481:
478:
475:
472:
469:
466:
446:
443:
440:
420:
417:
414:
411:
408:
405:
394:
382:
379:
376:
356:
353:
350:
330:
327:
324:
305:
302:
275:proper classes
259:
258:
240:
236:
230:
227:
224:
220:
185:
182:
179:
175:
169:
165:
161:
150:
119:
103:
100:
97:
94:
91:
68:
65:transitive set
15:
9:
6:
4:
3:
2:
3354:
3343:
3340:
3338:
3335:
3333:
3330:
3329:
3327:
3314:
3313:
3308:
3300:
3294:
3291:
3289:
3286:
3284:
3281:
3279:
3276:
3272:
3269:
3268:
3267:
3264:
3262:
3259:
3257:
3254:
3252:
3248:
3245:
3243:
3240:
3238:
3235:
3233:
3230:
3228:
3225:
3224:
3222:
3218:
3212:
3209:
3207:
3204:
3202:
3201:Recursive set
3199:
3197:
3194:
3192:
3189:
3187:
3184:
3182:
3179:
3175:
3172:
3170:
3167:
3165:
3162:
3160:
3157:
3155:
3152:
3151:
3150:
3147:
3145:
3142:
3140:
3137:
3135:
3132:
3130:
3127:
3125:
3122:
3121:
3119:
3117:
3113:
3107:
3104:
3102:
3099:
3097:
3094:
3092:
3089:
3087:
3084:
3082:
3079:
3077:
3074:
3070:
3067:
3065:
3062:
3060:
3057:
3056:
3055:
3052:
3050:
3047:
3045:
3042:
3040:
3037:
3035:
3032:
3030:
3027:
3023:
3020:
3019:
3018:
3015:
3011:
3010:of arithmetic
3008:
3007:
3006:
3003:
2999:
2996:
2994:
2991:
2989:
2986:
2984:
2981:
2979:
2976:
2975:
2974:
2971:
2967:
2964:
2962:
2959:
2958:
2957:
2954:
2953:
2951:
2949:
2945:
2939:
2936:
2934:
2931:
2929:
2926:
2924:
2921:
2918:
2917:from ZFC
2914:
2911:
2909:
2906:
2900:
2897:
2896:
2895:
2892:
2890:
2887:
2885:
2882:
2881:
2880:
2877:
2875:
2872:
2870:
2867:
2865:
2862:
2860:
2857:
2855:
2852:
2850:
2847:
2846:
2844:
2842:
2838:
2828:
2827:
2823:
2822:
2817:
2816:non-Euclidean
2814:
2810:
2807:
2805:
2802:
2800:
2799:
2795:
2794:
2792:
2789:
2788:
2786:
2782:
2778:
2775:
2773:
2770:
2769:
2768:
2764:
2760:
2757:
2756:
2755:
2751:
2747:
2744:
2742:
2739:
2737:
2734:
2732:
2729:
2727:
2724:
2722:
2719:
2718:
2716:
2712:
2711:
2709:
2704:
2698:
2693:Example
2690:
2682:
2677:
2676:
2675:
2672:
2670:
2667:
2663:
2660:
2658:
2655:
2653:
2650:
2648:
2645:
2644:
2643:
2640:
2638:
2635:
2633:
2630:
2628:
2625:
2621:
2618:
2616:
2613:
2612:
2611:
2608:
2604:
2601:
2599:
2596:
2594:
2591:
2589:
2586:
2585:
2584:
2581:
2579:
2576:
2572:
2569:
2567:
2564:
2562:
2559:
2558:
2557:
2554:
2550:
2547:
2545:
2542:
2540:
2537:
2535:
2532:
2530:
2527:
2525:
2522:
2521:
2520:
2517:
2515:
2512:
2510:
2507:
2505:
2502:
2498:
2495:
2493:
2490:
2488:
2485:
2483:
2480:
2479:
2478:
2475:
2473:
2470:
2468:
2465:
2463:
2460:
2456:
2453:
2451:
2450:by definition
2448:
2447:
2446:
2443:
2439:
2436:
2435:
2434:
2431:
2429:
2426:
2424:
2421:
2419:
2416:
2414:
2411:
2410:
2407:
2404:
2402:
2398:
2393:
2387:
2383:
2373:
2370:
2368:
2365:
2363:
2360:
2358:
2355:
2353:
2350:
2348:
2345:
2343:
2340:
2338:
2337:KripkeâPlatek
2335:
2333:
2330:
2326:
2323:
2321:
2318:
2317:
2316:
2313:
2312:
2310:
2306:
2298:
2295:
2294:
2293:
2290:
2288:
2285:
2281:
2278:
2277:
2276:
2273:
2271:
2268:
2266:
2263:
2261:
2258:
2256:
2253:
2250:
2246:
2242:
2239:
2235:
2232:
2230:
2227:
2225:
2222:
2221:
2220:
2216:
2213:
2212:
2210:
2208:
2204:
2200:
2192:
2189:
2187:
2184:
2182:
2181:constructible
2179:
2178:
2177:
2174:
2172:
2169:
2167:
2164:
2162:
2159:
2157:
2154:
2152:
2149:
2147:
2144:
2142:
2139:
2137:
2134:
2132:
2129:
2127:
2124:
2122:
2119:
2117:
2114:
2113:
2111:
2109:
2104:
2096:
2093:
2091:
2088:
2086:
2083:
2081:
2078:
2076:
2073:
2071:
2068:
2067:
2065:
2061:
2058:
2056:
2053:
2052:
2051:
2048:
2046:
2043:
2041:
2038:
2036:
2033:
2031:
2027:
2023:
2021:
2018:
2014:
2011:
2010:
2009:
2006:
2005:
2002:
1999:
1997:
1993:
1983:
1980:
1978:
1975:
1973:
1970:
1968:
1965:
1963:
1960:
1958:
1955:
1951:
1948:
1947:
1946:
1943:
1939:
1934:
1933:
1932:
1929:
1928:
1926:
1924:
1920:
1912:
1909:
1907:
1904:
1902:
1899:
1898:
1897:
1894:
1892:
1889:
1887:
1884:
1882:
1879:
1877:
1874:
1872:
1869:
1867:
1864:
1863:
1861:
1859:
1858:Propositional
1855:
1849:
1846:
1844:
1841:
1839:
1836:
1834:
1831:
1829:
1826:
1824:
1821:
1817:
1814:
1813:
1812:
1809:
1807:
1804:
1802:
1799:
1797:
1794:
1792:
1789:
1787:
1786:Logical truth
1784:
1782:
1779:
1778:
1776:
1774:
1770:
1767:
1765:
1761:
1755:
1752:
1750:
1747:
1745:
1742:
1740:
1737:
1735:
1732:
1730:
1726:
1722:
1718:
1716:
1713:
1711:
1708:
1706:
1702:
1699:
1698:
1696:
1694:
1688:
1683:
1677:
1674:
1672:
1669:
1667:
1664:
1662:
1659:
1657:
1654:
1652:
1649:
1647:
1644:
1642:
1639:
1637:
1634:
1632:
1629:
1627:
1624:
1622:
1619:
1615:
1612:
1611:
1610:
1607:
1606:
1604:
1600:
1596:
1589:
1584:
1582:
1577:
1575:
1570:
1569:
1566:
1552:on 2016-08-09
1551:
1547:
1543:
1541:
1535:
1531:
1527:
1526:Michael Artin
1523:
1519:
1515:
1514:
1501:
1499:9780198566519
1495:
1491:
1484:
1480:
1474:
1470:
1460:
1457:
1455:
1452:
1450:
1447:
1446:
1440:
1438:
1434:
1430:
1412:
1408:
1399:
1390:
1386:
1382:
1375:
1371:
1367:
1345:
1339:
1315:
1284:
1271:
1267:
1263:
1262:
1261:
1253:
1249:
1243:
1237:
1233:
1219:
1215:
1211:
1206:
1200:
1196:
1190:
1166:
1162:
1156:
1152:
1143:
1139:
1118:
1114:
1110:
1107:
1102:
1099:
1096:
1092:
1083:
1079:
1072:
1068:
1063:
1051:
1047:
1042:
1037:
1032:
1028:
1024:
1017:
1012:
1008:
999:
990:
974:
966:
960:
944:
940:
935:
931:
926:
922:
914:
910:
906:
902:
898:
894:
889:
885:
881:
877:
873:
869:
864:
860:
852:
848:
844:
840:
837:is less than
836:
820:
816:
812:
808:
804:
800:
796:
792:
788:
784:
780:
756:
746:
743:
740:
732:
726:
711:
710:
709:
707:
703:
695:
692:
688:
684:
680:
676:
675:
674:
672:
650:
646:
638:
634:
631:
630:
629:
621:
618:
610:
606:
602:
599:
595:
592:
588:
584:
581:
577:
573:
570:
566:
562:
559:
555:
554:
553:
551:
531:
528:
525:
505:
502:
499:
479:
476:
470:
464:
444:
441:
438:
415:
409:
406:
403:
395:
380:
377:
374:
354:
351:
348:
328:
325:
322:
314:
311:
310:
309:
301:
299:
295:
291:
287:
282:
280:
276:
272:
268:
264:
256:
238:
234:
228:
225:
222:
218:
209:
201:
183:
180:
177:
167:
163:
151:
148:
144:
140:
136:
132:
128:
124:
120:
117:
98:
95:
92:
81:
77:
73:
69:
66:
62:
58:
54:
50:
46:
42:
38:
34:
33:
32:
30:
26:
22:
3303:
3101:Ultraproduct
2948:Model theory
2913:Independence
2849:Formal proof
2841:Proof theory
2824:
2797:
2754:real numbers
2726:second-order
2637:Substitution
2514:Metalanguage
2455:conservative
2428:Axiom schema
2372:Constructive
2342:MorseâKelley
2308:Set theories
2287:Aleph number
2280:inaccessible
2186:Grothendieck
2185:
2070:intersection
1957:Higher-order
1945:Second-order
1891:Truth tables
1848:Venn diagram
1631:Formal proof
1554:. Retrieved
1550:the original
1539:
1537:
1489:
1473:
1395:
1388:
1384:
1380:
1373:
1369:
1365:
1269:
1265:
1251:
1247:
1244:
1235:
1231:
1217:
1213:
1209:
1204:
1198:
1194:
1188:
1141:
1134:
1081:
1077:
1070:
1066:
1064:
1049:
1045:
1040:
1035:
1030:
1026:
1022:
1015:
1010:
1006:
997:
991:
942:
938:
933:
929:
924:
920:
912:
908:
904:
900:
896:
892:
887:
883:
879:
875:
871:
867:
862:
858:
850:
846:
842:
838:
831:
830:and of each
818:
814:
810:
806:
802:
798:
794:
790:
786:
782:
778:
776:
708:). Define:
705:
701:
699:
690:
686:
682:
678:
668:
627:
616:
614:
608:
604:
597:
590:
586:
579:
575:
568:
564:
549:
547:
312:
307:
283:
266:
260:
254:
207:
199:
146:
142:
134:
130:
126:
122:
115:
79:
75:
71:
60:
56:
52:
48:
44:
40:
36:
28:
24:
18:
3211:Type theory
3159:undecidable
3091:Truth value
2978:equivalence
2657:non-logical
2270:Enumeration
2260:Isomorphism
2207:cardinality
2191:Von Neumann
2156:Ultrafilter
2121:Uncountable
2055:equivalence
1972:Quantifiers
1962:Fixed-point
1931:First-order
1811:Consistency
1796:Proposition
1773:Traditional
1744:Lindström's
1734:Compactness
1676:Type theory
1621:Cardinality
937:} for each
313:Proposition
21:mathematics
3326:Categories
3022:elementary
2715:arithmetic
2583:Quantifier
2561:functional
2433:Expression
2151:Transitive
2095:identities
2080:complement
2013:hereditary
1996:Set theory
1556:2010-12-06
1511:References
1185:such that
911:) equals |
777:where by |
685:such that
558:singletons
552:contains:
304:Properties
267:small sets
3293:Supertask
3196:Recursion
3154:decidable
2988:saturated
2966:of models
2889:deductive
2884:axiomatic
2804:Hilbert's
2791:Euclidean
2772:canonical
2695:axiomatic
2627:Signature
2556:Predicate
2445:Extension
2367:Ackermann
2292:Operation
2171:Universal
2161:Recursive
2136:Singleton
2131:Inhabited
2116:Countable
2106:Types of
2090:power set
2060:partition
1977:Predicate
1923:Predicate
1838:Syllogism
1828:Soundness
1801:Inference
1791:Tautology
1693:paradoxes
1522:"Univers"
1413:ω
1346:κ
1312:ℵ
1303:is zero,
1281:ℵ
1256:for some
1187:|y| <
1153:⋃
1111:⋃
971:ℵ
857:and each
744:∈
651:ω
529:∈
503:∈
477:∈
442:⊆
407:∈
378:∈
352:⊆
326:∈
239:α
226:∈
223:α
219:⋃
202:, and if
181:∈
178:α
168:α
139:power set
27:is a set
3278:Logicism
3271:timeline
3247:Concrete
3106:Validity
3076:T-schema
3069:Kripke's
3064:Tarski's
3059:semantic
3049:Strength
2998:submodel
2993:spectrum
2961:function
2809:Tarski's
2798:Elements
2785:geometry
2741:Robinson
2662:variable
2647:function
2620:spectrum
2610:Sentence
2566:variable
2509:Language
2462:Relation
2423:Automata
2413:Alphabet
2397:language
2251:-jection
2229:codomain
2215:Function
2176:Universe
2146:Infinite
2050:Relation
1833:Validity
1823:Argument
1721:theorem,
1536:(eds.).
1520:(1972).
1481:(2006).
1443:See also
1014:â ... â
492:because
431:because
3220:Related
3017:Diagram
2915: (
2894:Hilbert
2879:Systems
2874:Theorem
2752:of the
2697:systems
2477:Formula
2472:Grammar
2388: (
2332:General
2045:Forcing
2030:Element
1950:Monadic
1725:paradox
1666:Theorem
1602:General
1361:
1332:
1192:. Let
1140:. Let
1069:. Let
1034:| <
941:. Then
396:Proof.
367:, then
137:), the
129:, then
82:, then
51:, then
43:and if
2983:finite
2746:Skolem
2699:
2674:Theory
2642:Symbol
2632:String
2615:atomic
2492:ground
2487:closed
2482:atomic
2438:ground
2401:syntax
2297:binary
2224:domain
2141:Finite
1906:finite
1764:Logics
1723:
1671:Theory
1496:
1377:, and
1084:, let
939:α
934:α
925:α
917:α
263:models
2973:Model
2721:Peano
2578:Proof
2418:Arity
2347:Naive
2234:image
2166:Fuzzy
2126:Empty
2075:union
2020:Class
1661:Model
1651:Lemma
1609:Axiom
1524:. In
1486:(PDF)
1465:Notes
1387:)| =
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600:, and
518:, so
315:. If
298:topos
63:is a
3096:Type
2899:list
2703:list
2680:list
2669:Term
2603:rank
2497:open
2391:list
2203:Maps
2108:sets
1967:Free
1937:list
1687:list
1614:list
1494:ISBN
1039:. (
992:Let
556:All
341:and
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2765:of
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2219:Map
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2008:Set
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1268:, |
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