1411:
Note that we usually do not care about the models of size less than the Löwenheim–Skolem number and often assume that there are none (we will adopt this convention in this article). This is justified since we can always remove all such models from an AEC without influencing its structure above the
2638:
Shelah also has several stronger conjectures: The threshold cardinal for categoricity is the Hanf number of pseudoelementary classes in a language of cardinality LS(K). More specifically when the class is in a countable language and axiomaziable by an
902:
1859:
835:
1077:
2807:
953:
2934:
3106:
3038:
3466:
92:
1148:
3505:
3271:
2846:
1968:
552:
364:
442:
403:
2677:
1758:
1680:
2523:
2180:
1361:
995:
749:
699:
656:
607:
2711:
2274:
2227:
1919:
1296:
1021:
3340:
2556:
2633:
1465:
2437:
2064:
1782:
506:
470:
2991:
1539:
1329:
237:
3222:
3191:
3164:
3133:
2026:
1886:
177:
3310:
3062:
2964:
2580:
1497:
2604:
778:
2300:
147:
2401:
2354:
2327:
1806:
1715:
2491:
1381:
2875:
2761:
2374:
2117:
1988:
1628:
1608:
1579:
1559:
1433:
1405:
1248:
1228:
1208:
1188:
1168:
303:
277:
257:
201:
112:
840:
1811:
3728:
3707:
787:
1026:
3552:
This is due to Will Boney, but combines results of many people, including
Grossberg, Makkai, Shelah, and VanDieren. A proof appears in
3748:
2766:
3665:
909:
2725:
2459:
1084:
2880:
3364:
2443:
2728:), or model-theoretic assumptions (such as amalgamation or tameness). As of 2014, the original conjecture remains open.
3077:
2996:
3425:
58:
3349:
1095:
3471:
3227:
2812:
1924:
3787:
2069:
2642:
1723:
1653:
3642:, Contemporary Mathematics, vol. 302, Providence, RI: American Mathematical Society, pp. 165–204,
2496:
2126:
1687:
1334:
960:
33:
704:
661:
611:
562:
3643:
2682:
2232:
2185:
2736:
The following are some important results about AECs. Except for the last, all results are due to Shelah.
1891:
407:
368:
3782:
1253:
1000:
3319:
2535:
2612:
1438:
2418:
2081:
AECs are very general objects and one usually make some of the assumptions below when studying them:
2035:
511:
312:
1763:
2848:
2412:
475:
446:
3648:
2969:
1502:
1301:
209:
3758:
3620:
3602:
3290:
3200:
3169:
3142:
3111:
2716:
Several approximations have been published (see for example the results section below), assuming
2004:
1864:
21:
155:
3744:
3724:
3703:
3661:
3295:
3047:
2949:
2565:
1470:
41:
3352:, then Shelah's categoricity conjecture holds when we start with categoricity at a successor.
2589:
763:
3653:
3612:
2279:
1785:
1718:
1643:
117:
37:
3675:
2379:
2332:
2305:
1791:
1700:
3671:
2721:
2476:
2408:
1366:
1090:
3289:
is an abstract elementary class with amalgamation that is categorical in a "high-enough"
2454:
Shelah introduced AECs to provide a uniform framework in which to generalize first-order
2415:. These three assumptions allow us to build a universal model-homogeneous monster model
3738:
3716:
3695:
3583:
3571:
2860:
2746:
2359:
2102:
1973:
1613:
1584:
1564:
1544:
1418:
1390:
1233:
1213:
1193:
1173:
1153:
781:
288:
262:
242:
186:
97:
45:
3616:
3776:
3683:
2455:
180:
32:
for short, is a class of models with a partial order similar to the relation of an
3624:
897:{\displaystyle \alpha <\beta <\gamma \implies M_{\alpha }\prec _{K}M_{\beta }}
17:
3632:
3578:, Lecture Notes in Mathematics, vol. 1292, Springer-Verlag, pp. 419–497
2032:, together with elementary substructure, is an AEC with Löwenheim–Skolem number
1998:
755:
306:
3657:
2809:: it is a reduct of a class of models of a first-order theory omitting at most
2717:
2411:, while amalgamation and no maximal models are well-known consequences of the
2407:
Note that in elementary classes, joint embedding holds whenever the theory is
1854:{\displaystyle \langle \operatorname {Mod} (T),\prec _{\mathcal {F}}\rangle }
3743:, University Lecture Series, vol. 50, American Mathematical Society,
2462:, so it is natural to ask whether a similar result holds in AECs. This is
3723:, Studies in Logic (London), vol. 20, College Publications, London,
3702:, Studies in Logic (London), vol. 18, College Publications, London,
3576:
Classification of Non
Elementary Classes II. Abstract Elementary Classes
3587:
3607:
3346:
Shelah's categoricity conjecture for a successor from large cardinals
830:{\displaystyle \{\,M_{\alpha }\mid \alpha <\gamma \,\}\subseteq K}
3763:
1072:{\displaystyle \bigcup _{\alpha <\gamma }M_{\alpha }\prec _{K}N}
2466:. It states that there should be a Hanf number for categoricity:
2802:{\displaystyle \operatorname {PC} _{2^{\operatorname {LS} (K)}}}
3721:
Classification theory for abstract elementary classes. Vol. 2
948:{\displaystyle \bigcup _{\alpha <\gamma }M_{\alpha }\in K}
1842:
1769:
3757:
Boney, Will (2014). "Tameness from large cardinal axioms".
1638:
The following are examples of abstract elementary classes:
3422:, Corollary 3.5. Note that there is a typo there and that
3685:
Abstract
Elementary Classes: Some Answers, More Questions
2929:{\displaystyle \beth _{(2^{\operatorname {LS} (K)})^{+}}}
3633:"Classification theory for abstract elementary classes"
2089:
if any two model can be embedded inside a common model.
3700:
Classification theory for elementary abstract classes
3588:"Categoricity for abstract classes with amalgamation"
3474:
3428:
3322:
3298:
3230:
3203:
3172:
3145:
3114:
3080:
3050:
2999:
2972:
2952:
2883:
2863:
2815:
2769:
2749:
2685:
2645:
2615:
2592:
2568:
2538:
2499:
2479:
2421:
2382:
2362:
2335:
2308:
2282:
2235:
2188:
2129:
2105:
2038:
2007:
1976:
1927:
1894:
1867:
1814:
1794:
1766:
1726:
1703:
1656:
1616:
1587:
1567:
1547:
1505:
1473:
1441:
1421:
1393:
1369:
1337:
1304:
1256:
1236:
1216:
1196:
1176:
1156:
1098:
1029:
1003:
963:
912:
843:
790:
766:
707:
664:
614:
565:
514:
478:
449:
410:
371:
315:
291:
265:
245:
212:
189:
158:
120:
100:
61:
3499:
3460:
3334:
3304:
3277:Approximations to Shelah's categoricity conjecture
3265:
3216:
3185:
3158:
3127:
3100:
3056:
3032:
2985:
2958:
2928:
2869:
2840:
2801:
2755:
2705:
2679:sentence the threshold number for categoricity is
2671:
2627:
2598:
2574:
2562:has exactly one (up to isomorphism) model of size
2550:
2517:
2485:
2431:
2395:
2368:
2348:
2321:
2294:
2268:
2221:
2174:
2111:
2058:
2020:
1982:
1962:
1913:
1880:
1853:
1800:
1776:
1752:
1709:
1674:
1622:
1602:
1573:
1553:
1533:
1491:
1459:
1427:
1399:
1375:
1355:
1323:
1290:
1242:
1222:
1202:
1182:
1162:
1142:
1071:
1015:
989:
947:
896:
829:
772:
743:
693:
650:
601:
546:
500:
464:
436:
397:
358:
297:
271:
251:
231:
195:
171:
141:
106:
86:
3101:{\displaystyle \operatorname {PC} _{\aleph _{0}}}
3033:{\displaystyle 2^{\lambda }<2^{\lambda ^{+}}}
1888:. This can be generalized to other logics, like
149:, is an AEC if it has the following properties:
3461:{\displaystyle 2^{2^{\operatorname {LS} (K)}}}
8:
1848:
1815:
1263:
1257:
818:
791:
87:{\displaystyle \langle K,\prec _{K}\rangle }
81:
62:
1143:{\displaystyle \mu \geq |L(K)|+\aleph _{0}}
3500:{\displaystyle 2^{\operatorname {LS} (K)}}
3266:{\displaystyle L_{\omega _{1},\omega }(Q)}
2841:{\displaystyle 2^{\operatorname {LS} (K)}}
1990:expresses "there exists uncountably many".
1963:{\displaystyle L_{\omega _{1},\omega }(Q)}
1690:forms an AEC with Löwenheim–Skolem number
863:
859:
3762:
3647:
3606:
3528:
3516:
3479:
3473:
3438:
3433:
3427:
3419:
3407:
3395:
3321:
3297:
3240:
3235:
3229:
3208:
3202:
3177:
3171:
3150:
3144:
3119:
3113:
3090:
3085:
3079:
3049:
3022:
3017:
3004:
2998:
2977:
2971:
2951:
2918:
2896:
2888:
2882:
2862:
2820:
2814:
2779:
2774:
2768:
2748:
2695:
2690:
2684:
2655:
2650:
2644:
2614:
2591:
2567:
2537:
2498:
2478:
2464:Shelah's eventual categoricity conjecture
2423:
2422:
2420:
2387:
2381:
2361:
2340:
2334:
2313:
2307:
2281:
2260:
2250:
2240:
2234:
2213:
2203:
2193:
2187:
2160:
2147:
2134:
2128:
2104:
2048:
2043:
2037:
2012:
2006:
1975:
1937:
1932:
1926:
1899:
1893:
1872:
1866:
1841:
1840:
1813:
1793:
1768:
1767:
1765:
1736:
1731:
1725:
1702:
1655:
1615:
1586:
1566:
1546:
1522:
1504:
1472:
1440:
1420:
1392:
1368:
1336:
1312:
1303:
1277:
1269:
1255:
1235:
1215:
1195:
1175:
1155:
1134:
1122:
1105:
1097:
1060:
1050:
1034:
1028:
1002:
978:
968:
962:
933:
917:
911:
888:
878:
868:
842:
817:
799:
794:
789:
765:
732:
722:
712:
706:
682:
669:
663:
639:
629:
619:
613:
590:
580:
570:
564:
527:
513:
486:
477:
448:
409:
370:
314:
290:
264:
244:
220:
211:
188:
163:
157:
119:
99:
75:
60:
2442:Another assumption that one can make is
1650:is a first-order theory, then the class
1646:is the most basic example of an AEC: If
3376:
3273:can have exactly one uncountable model.
2713:. This conjecture dates back to 1976.
2672:{\displaystyle L_{\omega _{1},\omega }}
2449:
1861:is an AEC with Löwenheim–Skolem number
1753:{\displaystyle L_{\omega _{1},\omega }}
1675:{\displaystyle \operatorname {Mod} (T)}
114:a class of structures in some language
3540:
3383:
2936:has models of arbitrarily large sizes.
2720:assumptions (such as the existence of
2518:{\displaystyle \operatorname {LS} (K)}
2175:{\displaystyle M_{0},M_{1},M_{2}\in K}
1356:{\displaystyle \operatorname {LS} (K)}
990:{\displaystyle M_{\alpha }\prec _{K}N}
44:model theory. They were introduced by
3553:
2458:. Classification theory started with
2439:, exactly as in the elementary case.
744:{\displaystyle M_{1}\prec _{K}M_{2}.}
694:{\displaystyle M_{1}\subseteq M_{2},}
651:{\displaystyle M_{2}\prec _{K}M_{3},}
602:{\displaystyle M_{1}\prec _{K}M_{3},}
7:
3044:has amalgamation for models of size
2706:{\displaystyle \beth _{\omega _{1}}}
2269:{\displaystyle M_{0}\prec _{K}M_{2}}
2222:{\displaystyle M_{0}\prec _{K}M_{1}}
2096:if any model has a proper extension.
2424:
1914:{\displaystyle L_{\kappa ,\omega }}
1630:is clear from context, we omit it.
437:{\displaystyle g\colon N\simeq N',}
398:{\displaystyle f\colon M\simeq M',}
3316:is categorical in all high-enough
3283:Downward transfer from a successor
3205:
3174:
3147:
3116:
3087:
2070:Zilber's pseudo-exponential fields
2045:
2009:
1869:
1291:{\displaystyle \|N\|\leq |A|+\mu }
1131:
1016:{\displaystyle \alpha <\gamma }
14:
3682:Baldwin, John T. (July 7, 2006),
3335:{\displaystyle \mu \leq \lambda }
3108:AEC with Löwenheim–Skolem number
2551:{\displaystyle \lambda \geq \mu }
3595:Annals of Pure and Applied Logic
3224:. In particular, no sentence of
2726:generalized continuum hypothesis
2628:{\displaystyle \theta \geq \mu }
2450:Shelah's categoricity conjecture
1460:{\displaystyle f:M\rightarrow N}
3574:(1987), John T. Baldwin (ed.),
2432:{\displaystyle {\mathfrak {C}}}
2059:{\displaystyle 2^{\aleph _{0}}}
1170:is a subset of the universe of
547:{\displaystyle M'\prec _{K}N'.}
359:{\displaystyle M,N,M',N'\in K,}
3492:
3486:
3451:
3445:
3365:Tame abstract elementary class
3260:
3254:
2940:Amalgamation from categoricity
2915:
2909:
2903:
2889:
2833:
2827:
2792:
2786:
2512:
2506:
1957:
1951:
1830:
1824:
1777:{\displaystyle {\mathcal {F}}}
1669:
1663:
1597:
1591:
1515:
1509:
1451:
1350:
1344:
1278:
1270:
1123:
1119:
1113:
1106:
860:
136:
130:
1:
3617:10.1016/s0168-0072(98)00016-5
2741:Shelah's Presentation Theorem
2460:Morley's categoricity theorem
501:{\displaystyle M\prec _{K}N,}
465:{\displaystyle f\subseteq g,}
2986:{\displaystyle \lambda ^{+}}
1534:{\displaystyle f\prec _{K}N}
1324:{\displaystyle N\prec _{K}M}
232:{\displaystyle M\prec _{K}N}
3217:{\displaystyle \aleph _{2}}
3186:{\displaystyle \aleph _{1}}
3159:{\displaystyle \aleph _{0}}
3128:{\displaystyle \aleph _{0}}
3068:Existence from categoricity
2473:there should be a cardinal
2021:{\displaystyle \aleph _{1}}
1997:is a first-order countable
1881:{\displaystyle \aleph _{0}}
3804:
3350:strongly compact cardinals
3348:: If there are class-many
2877:which has a model of size
172:{\displaystyle \prec _{K}}
3737:Baldwin, John T. (2009),
2946:is an AEC categorical in
2855:Hanf number for existence
1412:Löwenheim–Skolem number.
26:abstract elementary class
3631:Grossberg, Rami (2002),
3305:{\displaystyle \lambda }
3057:{\displaystyle \lambda }
2959:{\displaystyle \lambda }
2575:{\displaystyle \lambda }
1492:{\displaystyle M,N\in K}
1230:whose universe contains
2599:{\displaystyle \theta }
1688:elementary substructure
1561:is an isomorphism from
1385:Löwenheim–Skolem number
773:{\displaystyle \gamma }
34:elementary substructure
3658:10.1090/conm/302/05080
3501:
3468:should be replaced by
3462:
3336:
3306:
3267:
3218:
3187:
3160:
3129:
3102:
3058:
3034:
2987:
2960:
2930:
2871:
2842:
2803:
2757:
2707:
2673:
2629:
2600:
2576:
2552:
2519:
2487:
2433:
2397:
2370:
2350:
2323:
2296:
2295:{\displaystyle N\in K}
2270:
2223:
2176:
2113:
2060:
2022:
1984:
1964:
1915:
1882:
1855:
1802:
1778:
1754:
1711:
1676:
1624:
1604:
1575:
1555:
1535:
1493:
1461:
1429:
1401:
1377:
1363:denote the least such
1357:
1325:
1292:
1244:
1224:
1204:
1184:
1164:
1144:
1073:
1017:
991:
949:
898:
831:
774:
745:
695:
652:
603:
548:
502:
466:
438:
399:
360:
299:
273:
253:
233:
197:
173:
143:
142:{\displaystyle L=L(K)}
108:
88:
20:, a discipline within
3502:
3463:
3337:
3307:
3268:
3219:
3188:
3161:
3130:
3103:
3059:
3035:
2988:
2961:
2931:
2872:
2843:
2804:
2758:
2724:or variations of the
2708:
2674:
2630:
2601:
2577:
2553:
2520:
2488:
2456:classification theory
2434:
2398:
2396:{\displaystyle M_{0}}
2371:
2351:
2349:{\displaystyle M_{2}}
2324:
2322:{\displaystyle M_{1}}
2297:
2271:
2224:
2177:
2114:
2061:
2028:-saturated models of
2023:
1985:
1965:
1916:
1883:
1856:
1803:
1801:{\displaystyle \phi }
1779:
1755:
1717:is a sentence in the
1712:
1710:{\displaystyle \phi }
1677:
1625:
1605:
1576:
1556:
1536:
1494:
1462:
1430:
1402:
1378:
1358:
1326:
1293:
1245:
1225:
1205:
1185:
1165:
1145:
1074:
1018:
992:
950:
899:
832:
775:
746:
696:
653:
604:
549:
503:
467:
439:
400:
361:
300:
274:
259:is a substructure of
254:
234:
198:
174:
144:
109:
89:
3472:
3426:
3320:
3296:
3228:
3201:
3197:has a model of size
3170:
3143:
3112:
3078:
3048:
2997:
2970:
2950:
2881:
2861:
2813:
2767:
2747:
2683:
2643:
2613:
2590:
2566:
2536:
2497:
2486:{\displaystyle \mu }
2477:
2419:
2380:
2360:
2333:
2306:
2280:
2233:
2186:
2127:
2103:
2036:
2005:
1974:
1925:
1892:
1865:
1812:
1792:
1764:
1724:
1701:
1654:
1614:
1585:
1565:
1545:
1503:
1471:
1439:
1435:-embedding is a map
1419:
1391:
1376:{\displaystyle \mu }
1367:
1335:
1302:
1254:
1234:
1214:
1194:
1174:
1154:
1096:
1027:
1001:
961:
910:
841:
788:
764:
705:
662:
612:
563:
512:
476:
447:
408:
369:
313:
289:
263:
243:
210:
187:
156:
118:
98:
59:
2413:compactness theorem
2001:theory, the set of
3497:
3458:
3332:
3302:
3263:
3214:
3183:
3156:
3139:is categorical in
3125:
3098:
3054:
3030:
2983:
2956:
2926:
2867:
2838:
2799:
2753:
2703:
2669:
2625:
2596:
2586:is categorical in
2572:
2548:
2529:is categorical in
2515:
2493:depending only on
2483:
2429:
2393:
2366:
2346:
2319:
2302:and embeddings of
2292:
2266:
2219:
2172:
2123:if for any triple
2109:
2077:Common assumptions
2056:
2018:
1980:
1960:
1911:
1878:
1851:
1798:
1774:
1750:
1707:
1672:
1620:
1600:
1571:
1551:
1531:
1489:
1457:
1425:
1397:
1373:
1353:
1321:
1288:
1240:
1220:
1200:
1180:
1160:
1140:
1069:
1045:
1013:
987:
945:
928:
894:
827:
770:
741:
691:
648:
599:
544:
498:
462:
434:
395:
356:
295:
269:
249:
229:
193:
169:
139:
104:
84:
22:mathematical logic
3730:978-1-904987-72-7
3709:978-1-904987-71-0
3640:Logic and algebra
2870:{\displaystyle K}
2756:{\displaystyle K}
2369:{\displaystyle N}
2112:{\displaystyle K}
1983:{\displaystyle Q}
1623:{\displaystyle K}
1603:{\displaystyle f}
1574:{\displaystyle M}
1554:{\displaystyle f}
1428:{\displaystyle K}
1400:{\displaystyle K}
1243:{\displaystyle A}
1223:{\displaystyle K}
1203:{\displaystyle N}
1183:{\displaystyle M}
1163:{\displaystyle A}
1089:: There exists a
1030:
913:
837:is a chain (i.e.
298:{\displaystyle K}
272:{\displaystyle N}
252:{\displaystyle M}
196:{\displaystyle K}
107:{\displaystyle K}
3795:
3768:
3766:
3753:
3733:
3712:
3691:
3690:
3678:
3651:
3637:
3627:
3610:
3592:
3579:
3557:
3550:
3544:
3538:
3532:
3526:
3520:
3514:
3508:
3506:
3504:
3503:
3498:
3496:
3495:
3467:
3465:
3464:
3459:
3457:
3456:
3455:
3454:
3417:
3411:
3405:
3399:
3393:
3387:
3381:
3341:
3339:
3338:
3333:
3311:
3309:
3308:
3303:
3272:
3270:
3269:
3264:
3253:
3252:
3245:
3244:
3223:
3221:
3220:
3215:
3213:
3212:
3192:
3190:
3189:
3184:
3182:
3181:
3165:
3163:
3162:
3157:
3155:
3154:
3134:
3132:
3131:
3126:
3124:
3123:
3107:
3105:
3104:
3099:
3097:
3096:
3095:
3094:
3063:
3061:
3060:
3055:
3039:
3037:
3036:
3031:
3029:
3028:
3027:
3026:
3009:
3008:
2992:
2990:
2989:
2984:
2982:
2981:
2965:
2963:
2962:
2957:
2935:
2933:
2932:
2927:
2925:
2924:
2923:
2922:
2913:
2912:
2876:
2874:
2873:
2868:
2847:
2845:
2844:
2839:
2837:
2836:
2808:
2806:
2805:
2800:
2798:
2797:
2796:
2795:
2762:
2760:
2759:
2754:
2712:
2710:
2709:
2704:
2702:
2701:
2700:
2699:
2678:
2676:
2675:
2670:
2668:
2667:
2660:
2659:
2634:
2632:
2631:
2626:
2605:
2603:
2602:
2597:
2581:
2579:
2578:
2573:
2557:
2555:
2554:
2549:
2524:
2522:
2521:
2516:
2492:
2490:
2489:
2484:
2438:
2436:
2435:
2430:
2428:
2427:
2402:
2400:
2399:
2394:
2392:
2391:
2375:
2373:
2372:
2367:
2355:
2353:
2352:
2347:
2345:
2344:
2328:
2326:
2325:
2320:
2318:
2317:
2301:
2299:
2298:
2293:
2275:
2273:
2272:
2267:
2265:
2264:
2255:
2254:
2245:
2244:
2228:
2226:
2225:
2220:
2218:
2217:
2208:
2207:
2198:
2197:
2181:
2179:
2178:
2173:
2165:
2164:
2152:
2151:
2139:
2138:
2118:
2116:
2115:
2110:
2094:no maximal model
2065:
2063:
2062:
2057:
2055:
2054:
2053:
2052:
2027:
2025:
2024:
2019:
2017:
2016:
1989:
1987:
1986:
1981:
1969:
1967:
1966:
1961:
1950:
1949:
1942:
1941:
1920:
1918:
1917:
1912:
1910:
1909:
1887:
1885:
1884:
1879:
1877:
1876:
1860:
1858:
1857:
1852:
1847:
1846:
1845:
1807:
1805:
1804:
1799:
1783:
1781:
1780:
1775:
1773:
1772:
1759:
1757:
1756:
1751:
1749:
1748:
1741:
1740:
1719:infinitary logic
1716:
1714:
1713:
1708:
1681:
1679:
1678:
1673:
1644:Elementary class
1629:
1627:
1626:
1621:
1609:
1607:
1606:
1601:
1580:
1578:
1577:
1572:
1560:
1558:
1557:
1552:
1540:
1538:
1537:
1532:
1527:
1526:
1498:
1496:
1495:
1490:
1466:
1464:
1463:
1458:
1434:
1432:
1431:
1426:
1406:
1404:
1403:
1398:
1383:and call it the
1382:
1380:
1379:
1374:
1362:
1360:
1359:
1354:
1330:
1328:
1327:
1322:
1317:
1316:
1297:
1295:
1294:
1289:
1281:
1273:
1249:
1247:
1246:
1241:
1229:
1227:
1226:
1221:
1209:
1207:
1206:
1201:
1190:, then there is
1189:
1187:
1186:
1181:
1169:
1167:
1166:
1161:
1149:
1147:
1146:
1141:
1139:
1138:
1126:
1109:
1085:Löwenheim–Skolem
1078:
1076:
1075:
1070:
1065:
1064:
1055:
1054:
1044:
1022:
1020:
1019:
1014:
996:
994:
993:
988:
983:
982:
973:
972:
954:
952:
951:
946:
938:
937:
927:
903:
901:
900:
895:
893:
892:
883:
882:
873:
872:
836:
834:
833:
828:
804:
803:
779:
777:
776:
771:
750:
748:
747:
742:
737:
736:
727:
726:
717:
716:
700:
698:
697:
692:
687:
686:
674:
673:
657:
655:
654:
649:
644:
643:
634:
633:
624:
623:
608:
606:
605:
600:
595:
594:
585:
584:
575:
574:
553:
551:
550:
545:
540:
532:
531:
522:
507:
505:
504:
499:
491:
490:
471:
469:
468:
463:
443:
441:
440:
435:
430:
404:
402:
401:
396:
391:
365:
363:
362:
357:
346:
335:
305:is closed under
304:
302:
301:
296:
278:
276:
275:
270:
258:
256:
255:
250:
238:
236:
235:
230:
225:
224:
202:
200:
199:
194:
178:
176:
175:
170:
168:
167:
148:
146:
145:
140:
113:
111:
110:
105:
93:
91:
90:
85:
80:
79:
38:elementary class
3803:
3802:
3798:
3797:
3796:
3794:
3793:
3792:
3788:Category theory
3773:
3772:
3771:
3756:
3751:
3736:
3731:
3717:Shelah, Saharon
3715:
3710:
3696:Shelah, Saharon
3694:
3688:
3681:
3668:
3635:
3630:
3590:
3584:Shelah, Saharon
3582:
3572:Shelah, Saharon
3570:
3566:
3561:
3560:
3551:
3547:
3539:
3535:
3527:
3523:
3515:
3511:
3475:
3470:
3469:
3434:
3429:
3424:
3423:
3418:
3414:
3406:
3402:
3394:
3390:
3382:
3378:
3373:
3361:
3318:
3317:
3294:
3293:
3236:
3231:
3226:
3225:
3204:
3199:
3198:
3173:
3168:
3167:
3146:
3141:
3140:
3115:
3110:
3109:
3086:
3081:
3076:
3075:
3046:
3045:
3018:
3013:
3000:
2995:
2994:
2973:
2968:
2967:
2948:
2947:
2914:
2892:
2884:
2879:
2878:
2859:
2858:
2816:
2811:
2810:
2775:
2770:
2765:
2764:
2745:
2744:
2734:
2722:large cardinals
2691:
2686:
2681:
2680:
2651:
2646:
2641:
2640:
2611:
2610:
2588:
2587:
2564:
2563:
2534:
2533:
2495:
2494:
2475:
2474:
2452:
2417:
2416:
2383:
2378:
2377:
2358:
2357:
2336:
2331:
2330:
2309:
2304:
2303:
2278:
2277:
2256:
2246:
2236:
2231:
2230:
2209:
2199:
2189:
2184:
2183:
2156:
2143:
2130:
2125:
2124:
2101:
2100:
2087:joint embedding
2079:
2044:
2039:
2034:
2033:
2008:
2003:
2002:
1972:
1971:
1933:
1928:
1923:
1922:
1895:
1890:
1889:
1868:
1863:
1862:
1836:
1810:
1809:
1790:
1789:
1784:is a countable
1762:
1761:
1732:
1727:
1722:
1721:
1699:
1698:
1652:
1651:
1636:
1612:
1611:
1583:
1582:
1563:
1562:
1543:
1542:
1518:
1501:
1500:
1469:
1468:
1437:
1436:
1417:
1416:
1389:
1388:
1365:
1364:
1333:
1332:
1308:
1300:
1299:
1252:
1251:
1232:
1231:
1212:
1211:
1192:
1191:
1172:
1171:
1152:
1151:
1150:, such that if
1130:
1094:
1093:
1056:
1046:
1025:
1024:
999:
998:
974:
964:
959:
958:
929:
908:
907:
884:
874:
864:
839:
838:
795:
786:
785:
762:
761:
728:
718:
708:
703:
702:
678:
665:
660:
659:
635:
625:
615:
610:
609:
586:
576:
566:
561:
560:
533:
523:
515:
510:
509:
482:
474:
473:
445:
444:
423:
406:
405:
384:
367:
366:
339:
328:
311:
310:
287:
286:
261:
260:
241:
240:
216:
208:
207:
185:
184:
159:
154:
153:
116:
115:
96:
95:
71:
57:
56:
54:
12:
11:
5:
3801:
3799:
3791:
3790:
3785:
3775:
3774:
3770:
3769:
3754:
3750:978-0821848937
3749:
3734:
3729:
3713:
3708:
3692:
3679:
3666:
3628:
3601:(1): 261–294,
3580:
3567:
3565:
3562:
3559:
3558:
3556:, Theorem 7.5.
3545:
3533:
3531:, Theorem 5.1.
3529:Grossberg 2002
3521:
3519:, Theorem 4.3.
3517:Grossberg 2002
3509:
3494:
3491:
3488:
3485:
3482:
3478:
3453:
3450:
3447:
3444:
3441:
3437:
3432:
3420:Grossberg 2002
3412:
3410:, Theorem 3.4.
3408:Grossberg 2002
3400:
3396:Grossberg 2002
3388:
3375:
3374:
3372:
3369:
3368:
3367:
3360:
3357:
3356:
3355:
3354:
3353:
3343:
3331:
3328:
3325:
3301:
3274:
3262:
3259:
3256:
3251:
3248:
3243:
3239:
3234:
3211:
3207:
3180:
3176:
3153:
3149:
3122:
3118:
3093:
3089:
3084:
3065:
3053:
3025:
3021:
3016:
3012:
3007:
3003:
2980:
2976:
2955:
2937:
2921:
2917:
2911:
2908:
2905:
2902:
2899:
2895:
2891:
2887:
2866:
2852:
2835:
2832:
2829:
2826:
2823:
2819:
2794:
2791:
2788:
2785:
2782:
2778:
2773:
2752:
2733:
2730:
2698:
2694:
2689:
2666:
2663:
2658:
2654:
2649:
2624:
2621:
2618:
2595:
2571:
2547:
2544:
2541:
2514:
2511:
2508:
2505:
2502:
2482:
2469:For every AEC
2451:
2448:
2426:
2405:
2404:
2390:
2386:
2365:
2343:
2339:
2316:
2312:
2291:
2288:
2285:
2263:
2259:
2253:
2249:
2243:
2239:
2216:
2212:
2206:
2202:
2196:
2192:
2171:
2168:
2163:
2159:
2155:
2150:
2146:
2142:
2137:
2133:
2108:
2097:
2090:
2078:
2075:
2074:
2073:
2067:
2051:
2047:
2042:
2015:
2011:
1991:
1979:
1959:
1956:
1953:
1948:
1945:
1940:
1936:
1931:
1908:
1905:
1902:
1898:
1875:
1871:
1850:
1844:
1839:
1835:
1832:
1829:
1826:
1823:
1820:
1817:
1797:
1771:
1747:
1744:
1739:
1735:
1730:
1706:
1695:
1686:together with
1671:
1668:
1665:
1662:
1659:
1635:
1632:
1619:
1599:
1596:
1593:
1590:
1570:
1550:
1530:
1525:
1521:
1517:
1514:
1511:
1508:
1488:
1485:
1482:
1479:
1476:
1456:
1453:
1450:
1447:
1444:
1424:
1409:
1408:
1396:
1372:
1352:
1349:
1346:
1343:
1340:
1320:
1315:
1311:
1307:
1287:
1284:
1280:
1276:
1272:
1268:
1265:
1262:
1259:
1239:
1219:
1199:
1179:
1159:
1137:
1133:
1129:
1125:
1121:
1118:
1115:
1112:
1108:
1104:
1101:
1081:
1080:
1079:
1068:
1063:
1059:
1053:
1049:
1043:
1040:
1037:
1033:
1012:
1009:
1006:
986:
981:
977:
971:
967:
955:
944:
941:
936:
932:
926:
923:
920:
916:
891:
887:
881:
877:
871:
867:
862:
858:
855:
852:
849:
846:
826:
823:
820:
816:
813:
810:
807:
802:
798:
793:
769:
754:Tarski–Vaught
751:
740:
735:
731:
725:
721:
715:
711:
690:
685:
681:
677:
672:
668:
647:
642:
638:
632:
628:
622:
618:
598:
593:
589:
583:
579:
573:
569:
554:
543:
539:
536:
530:
526:
521:
518:
497:
494:
489:
485:
481:
461:
458:
455:
452:
433:
429:
426:
422:
419:
416:
413:
394:
390:
387:
383:
380:
377:
374:
355:
352:
349:
345:
342:
338:
334:
331:
327:
324:
321:
318:
294:
280:
268:
248:
228:
223:
219:
215:
204:
192:
166:
162:
138:
135:
132:
129:
126:
123:
103:
83:
78:
74:
70:
67:
64:
53:
50:
46:Saharon Shelah
13:
10:
9:
6:
4:
3:
2:
3800:
3789:
3786:
3784:
3781:
3780:
3778:
3765:
3760:
3755:
3752:
3746:
3742:
3741:
3735:
3732:
3726:
3722:
3718:
3714:
3711:
3705:
3701:
3697:
3693:
3687:
3686:
3680:
3677:
3673:
3669:
3667:9780821829844
3663:
3659:
3655:
3650:
3649:10.1.1.6.9630
3645:
3641:
3634:
3629:
3626:
3622:
3618:
3614:
3609:
3604:
3600:
3596:
3589:
3585:
3581:
3577:
3573:
3569:
3568:
3563:
3555:
3549:
3546:
3542:
3537:
3534:
3530:
3525:
3522:
3518:
3513:
3510:
3489:
3483:
3480:
3476:
3448:
3442:
3439:
3435:
3430:
3421:
3416:
3413:
3409:
3404:
3401:
3397:
3392:
3389:
3385:
3380:
3377:
3370:
3366:
3363:
3362:
3358:
3351:
3347:
3344:
3329:
3326:
3323:
3315:
3299:
3292:
3288:
3284:
3281:
3280:
3278:
3275:
3257:
3249:
3246:
3241:
3237:
3232:
3209:
3196:
3178:
3151:
3138:
3120:
3091:
3082:
3073:
3069:
3066:
3051:
3043:
3023:
3019:
3014:
3010:
3005:
3001:
2978:
2974:
2953:
2945:
2941:
2938:
2919:
2906:
2900:
2897:
2893:
2885:
2864:
2856:
2853:
2850:
2830:
2824:
2821:
2817:
2789:
2783:
2780:
2776:
2771:
2750:
2742:
2739:
2738:
2737:
2731:
2729:
2727:
2723:
2719:
2718:set-theoretic
2714:
2696:
2692:
2687:
2664:
2661:
2656:
2652:
2647:
2636:
2622:
2619:
2616:
2609:
2593:
2585:
2569:
2561:
2545:
2542:
2539:
2532:
2528:
2525:such that if
2509:
2503:
2500:
2480:
2472:
2467:
2465:
2461:
2457:
2447:
2445:
2440:
2414:
2410:
2388:
2384:
2363:
2341:
2337:
2314:
2310:
2289:
2286:
2283:
2261:
2257:
2251:
2247:
2241:
2237:
2214:
2210:
2204:
2200:
2194:
2190:
2169:
2166:
2161:
2157:
2153:
2148:
2144:
2140:
2135:
2131:
2122:
2106:
2098:
2095:
2091:
2088:
2084:
2083:
2082:
2076:
2071:
2068:
2049:
2040:
2031:
2013:
2000:
1996:
1992:
1977:
1954:
1946:
1943:
1938:
1934:
1929:
1906:
1903:
1900:
1896:
1873:
1837:
1833:
1827:
1821:
1818:
1795:
1787:
1745:
1742:
1737:
1733:
1728:
1720:
1704:
1696:
1693:
1689:
1685:
1682:of models of
1666:
1660:
1657:
1649:
1645:
1641:
1640:
1639:
1633:
1631:
1617:
1594:
1588:
1568:
1548:
1528:
1523:
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1004:
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905:
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865:
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850:
847:
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821:
814:
811:
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800:
796:
783:
767:
759:
757:
752:
738:
733:
729:
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688:
683:
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675:
670:
666:
645:
640:
636:
630:
626:
620:
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587:
581:
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558:
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541:
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182:
181:partial order
164:
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76:
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68:
65:
51:
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35:
31:
27:
23:
19:
3783:Model theory
3740:Categoricity
3739:
3720:
3699:
3684:
3639:
3608:math/9809197
3598:
3594:
3575:
3548:
3536:
3524:
3512:
3415:
3403:
3398:, Section 1.
3391:
3379:
3345:
3313:
3286:
3282:
3276:
3194:
3136:
3071:
3067:
3041:
2943:
2939:
2854:
2740:
2735:
2715:
2637:
2607:
2583:
2559:
2530:
2526:
2470:
2468:
2463:
2453:
2441:
2406:
2121:amalgamation
2120:
2093:
2086:
2080:
2072:form an AEC.
2029:
1994:
1691:
1683:
1647:
1637:
1414:
1410:
1384:
1083:
753:
556:
307:isomorphisms
283:Isomorphisms
282:
55:
29:
25:
18:model theory
15:
3764:1303.0550v4
3541:Shelah 1999
3384:Shelah 1987
2276:, there is
2092:An AEC has
2085:An AEC has
1999:superstable
1788:containing
42:first-order
3777:Categories
3564:References
3554:Boney 2014
2857:: Any AEC
2743:: Any AEC
2403:pointwise.
1499:such that
1250:such that
997:, for all
309:, and if
52:Definition
3644:CiteSeerX
3484:
3443:
3330:λ
3327:≤
3324:μ
3300:λ
3291:successor
3250:ω
3238:ω
3206:ℵ
3175:ℵ
3148:ℵ
3117:ℵ
3088:ℵ
3052:λ
3020:λ
3006:λ
2975:λ
2954:λ
2901:
2886:ℶ
2825:
2784:
2693:ω
2688:ℶ
2665:ω
2653:ω
2623:μ
2620:≥
2617:θ
2594:θ
2570:λ
2546:μ
2543:≥
2540:λ
2504:
2481:μ
2376:that fix
2287:∈
2248:≺
2201:≺
2167:∈
2046:ℵ
2010:ℵ
1947:ω
1935:ω
1907:ω
1901:κ
1870:ℵ
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1838:≺
1822:
1816:⟨
1796:ϕ
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1705:ϕ
1661:
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1484:∈
1452:→
1371:μ
1342:
1331:. We let
1310:≺
1286:μ
1267:≤
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1132:ℵ
1103:≥
1100:μ
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1036:α
1032:⋃
1011:γ
1005:α
976:≺
970:α
940:∈
935:α
925:γ
919:α
915:⋃
904:), then:
890:β
876:≺
870:α
861:⟹
857:γ
851:β
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822:⊆
815:γ
809:α
806:∣
801:α
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720:≺
676:⊆
627:≺
578:≺
557:Coherence
525:≺
484:≺
454:⊆
421:≃
415::
382:≃
376::
348:∈
218:≺
161:≺
82:⟩
73:≺
63:⟨
3719:(2009),
3698:(2009),
3625:27872122
3586:(1999),
3359:See also
2582:), then
2444:tameness
2409:complete
1970:, where
1786:fragment
1634:Examples
1091:cardinal
538:′
520:′
428:′
389:′
344:′
333:′
3676:1928390
3312:, then
3193:, then
3040:, then
2732:Results
2356:inside
2099:An AEC
1808:, then
1023:, then
782:ordinal
3747:
3727:
3706:
3674:
3664:
3646:
3623:
2558:(i.e.
1760:, and
780:is an
758:axioms
94:, for
36:of an
3759:arXiv
3689:(PDF)
3636:(PDF)
3621:S2CID
3603:arXiv
3591:(PDF)
3371:Notes
3285:: If
3074:is a
3070:: If
2942:: If
2849:types
2182:with
1921:, or
1610:. If
1581:onto
1087:axiom
760:: If
756:chain
701:then
559:: If
508:then
239:then
179:is a
28:, or
24:, an
3745:ISBN
3725:ISBN
3704:ISBN
3662:ISBN
3166:and
3135:and
3011:<
2993:and
2966:and
2606:for
2531:some
2329:and
2119:has
1541:and
1467:for
1298:and
1039:<
1008:<
922:<
854:<
848:<
812:<
784:and
658:and
472:and
3654:doi
3613:doi
2763:is
2608:all
1993:If
1819:Mod
1697:If
1692:|T|
1658:Mod
1642:An
1387:of
1210:in
957:If
206:If
183:on
40:in
30:AEC
16:In
3779::
3672:MR
3670:,
3660:,
3652:,
3638:,
3619:,
3611:,
3599:98
3597:,
3593:,
3481:LS
3440:LS
3279::
3083:PC
2898:LS
2822:LS
2781:LS
2772:PC
2635:.
2501:LS
2446:.
2229:,
1415:A
1339:LS
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48:.
3767:.
3761::
3656::
3615::
3605::
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3507:.
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2050:0
2041:2
2030:T
2014:1
1995:T
1978:Q
1958:)
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1952:(
1944:,
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1592:[
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1345:(
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1114:(
1111:L
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819:}
797:M
792:{
739:.
734:2
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714:1
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641:3
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542:.
535:N
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493:N
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480:M
460:,
457:g
451:f
432:,
425:N
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412:g
393:,
386:M
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373:f
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351:K
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330:M
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323:N
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317:M
293:K
279:.
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203:.
191:K
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134:K
131:(
128:L
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102:K
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