2688:
20:
698:
787:
462:. This class has the amalgamation property since any two field extensions of a prime field can be embedded into a common field. However, two arbitrary fields cannot be embedded into a common field when the
305:
In general, the amalgamation property can be considered for a category with a specified choice of the class of morphisms (in place of embeddings). This notion is related to the categorical notion of a
253:
567:
832:
Hodges, Section 1.2 and
Exercise 4 therein. When no relation is present, as in the case of groups, the notion of embedding and of injective morphism are the same, see p. 6.
360:
451:
is the largest. Now any common model with an embedding from these two extensions must be at least of size five so that there are two elements on either side of
296:
276:
318:
The class of sets, where the embeddings are injective functions, and if they are assumed to be inclusions then an amalgam is simply the union of the two sets.
1067:
897:
893:
1742:
1825:
966:
726:
909:
Categorical algebraic properties. A compendium on amalgamation, congruence extension, epimorphisms, residual smallness, and injectivity
43:
that guarantees, under certain conditions, that two structures in the collection can be regarded as substructures of a larger one.
2139:
2297:
1085:
2152:
1475:
901:
584:
2157:
2147:
1884:
1737:
1090:
1081:
2293:
883:
1635:
2390:
2134:
959:
1695:
1388:
1129:
2651:
2353:
2116:
2111:
1936:
1357:
1041:
40:
2646:
2429:
2346:
2059:
1990:
1867:
1109:
206:
2571:
2397:
2083:
1717:
1316:
523:
2449:
2444:
2054:
1793:
1722:
1051:
952:
2378:
1968:
1362:
1330:
1021:
875:
459:
325:
where the embeddings are injective homomorphisms, and (assuming they are inclusions) an amalgam is the
78:
2668:
2617:
2514:
2012:
1973:
1450:
1095:
813:
803:
58:
47:
1124:
2509:
2439:
1978:
1830:
1813:
1536:
1016:
808:
463:
382:
2341:
2318:
2279:
2165:
2106:
1752:
1672:
1516:
1460:
1073:
2631:
2358:
2336:
2303:
2196:
2042:
2027:
2000:
1951:
1835:
1770:
1595:
1561:
1556:
1430:
1261:
1238:
798:
299:
2712:
2561:
2414:
2206:
1924:
1660:
1566:
1425:
1410:
1291:
1266:
2534:
2496:
2373:
2177:
2017:
1941:
1919:
1747:
1705:
1604:
1571:
1435:
1223:
1134:
331:
74:
8:
2663:
2554:
2539:
2519:
2476:
2363:
2313:
2239:
2184:
2121:
1914:
1909:
1857:
1625:
1614:
1286:
1186:
1114:
1105:
1101:
1036:
1031:
24:
298:
has the amalgamation property. The amalgamation property has certain connections to the
2692:
2461:
2424:
2409:
2402:
2385:
2189:
2171:
2037:
1963:
1946:
1899:
1712:
1621:
1455:
1440:
1400:
1352:
1337:
1325:
1281:
1256:
1026:
975:
374:
281:
261:
62:
51:
1645:
2687:
2627:
2434:
2244:
2234:
2126:
2007:
1842:
1818:
1599:
1583:
1488:
1465:
1342:
1311:
1276:
1171:
1006:
912:
879:
102:
2641:
2636:
2529:
2486:
2308:
2269:
2264:
2249:
2075:
2032:
1929:
1727:
1677:
1251:
1213:
928:
309:, in particular, in connection with the strong amalgamation property (see below).
2622:
2612:
2566:
2549:
2504:
2466:
2368:
2288:
2095:
2022:
1995:
1983:
1889:
1803:
1777:
1732:
1700:
1501:
1303:
1246:
1196:
1161:
1119:
425:
does not have the amalgamation property. The counterexample for this starts with
70:
2607:
2586:
2544:
2524:
2419:
2274:
1872:
1862:
1852:
1847:
1781:
1655:
1531:
1420:
1415:
1393:
994:
933:
867:
414:
has the joint embedding property because all three models can be embedded into
326:
2706:
2581:
2259:
1766:
1551:
1541:
1511:
1496:
1166:
2481:
2328:
2229:
2221:
2101:
2049:
1958:
1894:
1877:
1808:
1667:
1526:
1228:
1011:
370:
363:
32:
2591:
2471:
1650:
1640:
1587:
1271:
1191:
1176:
1056:
1001:
66:
19:
1521:
1376:
1347:
1153:
322:
160:
of structures has the amalgamation property if for every amalgam with
2673:
2576:
1629:
1546:
1506:
1470:
1406:
1218:
1208:
1181:
944:
904:(Department of Mathematics and Computer Science, Chapman University).
782:{\displaystyle h\lbrack X\rbrack =\lbrace h(x)\mid x\in X\rbrace .\,}
2658:
2456:
1904:
1609:
1203:
381:
A similar but different notion to the amalgamation property is the
306:
2254:
1046:
919:
Macpherson, Donald (2011), "A survey of homogeneous structures",
50:, which characterises classes of finite structures that arise as
389:(or simply the set) containing three models with linear orders,
1798:
1144:
989:
141:
is an injective morphism which induces an isomorphism from
278:
has the amalgamation property if the class of models of
61:
of the amalgamation property appears in many areas of
729:
587:
526:
334:
284:
264:
209:
693:{\displaystyle f'\cap g'=(f'\circ f)=(g'\circ g)\,}
902:online database of classes of algebraic structures
781:
692:
561:
385:. To see the difference, first consider the class
354:
290:
270:
247:
2704:
911:, Studia Sci. Math. Hungar 18 (1), 79-141, 1983
377:from an amalgamation class of finite structure.
469:
69:as an incestual accessibility relation, and in
960:
172: â Ă, there exist both a structure
772:
745:
739:
733:
907:E.W. Kiss, L. Mårki, P. Pröhle, W. Tholen,
1152:
967:
953:
918:
850:
932:
778:
689:
558:
244:
18:
447:is the smallest and the other in which
2705:
974:
866:
841:Kiss, Mårki, Pröhle, Tholen, Section 6
248:{\displaystyle f'\circ f=g'\circ g.\,}
93:can be formally defined as a 5-tuple (
46:This property plays a crucial role in
948:
562:{\displaystyle f'\circ f=g'\circ g\,}
436:and extends in two different ways to
54:of countable homogeneous structures.
373:. This is due to the fact that any
13:
14:
2724:
486:(DAP), if for every amalgam with
2686:
39:is a property of collections of
860:
101:are structures having the same
844:
835:
826:
757:
751:
686:
680:
677:
660:
654:
648:
645:
628:
622:
616:
602:
596:
484:disjoint amalgamation property
1:
2647:History of mathematical logic
819:
494:there exist both a structure
84:
31:In the mathematical field of
27:of the amalgamation property.
2572:Primitive recursive function
898:strong amalgamation property
480:strong amalgamation property
470:Strong amalgamation property
432:containing a single element
7:
792:
460:algebraically closed fields
312:
10:
2729:
1636:SchröderâBernstein theorem
1363:Monadic predicate calculus
1022:Foundations of mathematics
934:10.1016/j.disc.2011.01.024
876:Cambridge University Press
458:Now consider the class of
410:of size three. This class
2682:
2669:Philosophy of mathematics
2618:Automated theorem proving
2600:
2495:
2327:
2220:
2072:
1789:
1765:
1743:Von NeumannâBernaysâGödel
1688:
1582:
1486:
1384:
1375:
1302:
1237:
1143:
1065:
982:
804:Pushout (category theory)
809:Joint embedding property
383:joint embedding property
2319:Self-verifying theories
2140:Tarski's axiomatization
1091:Tarski's undefinability
1086:incompleteness theorems
482:(SAP), also called the
16:Concept in model theory
2693:Mathematics portal
2304:Proof of impossibility
1952:propositional variable
1262:Propositional calculus
872:A shorter model theory
799:Span (category theory)
783:
694:
563:
478:of structures has the
466:of the fields differ.
356:
300:quantifier elimination
292:
272:
249:
79:ChurchâRosser property
65:. Examples include in
28:
2562:Kolmogorov complexity
2515:Computably enumerable
2415:Model complete theory
2207:Principia Mathematica
1267:Propositional formula
1096:BanachâTarski paradox
894:amalgamation property
784:
695:
564:
375:homogeneous structure
357:
355:{\displaystyle B*C/A}
293:
273:
258:A first-order theory
250:
37:amalgamation property
22:
2510:ChurchâTuring thesis
2497:Computability theory
1706:continuum hypothesis
1224:Square of opposition
1082:Gödel's completeness
921:Discrete Mathematics
727:
585:
524:
369:The class of finite
332:
282:
262:
207:
145:to the substructure
2664:Mathematical object
2555:P versus NP problem
2520:Computable function
2314:Reverse mathematics
2240:Logical consequence
2117:primitive recursive
2112:elementary function
1885:Free/bound variable
1738:TarskiâGrothendieck
1257:Logical connectives
1187:Logical equivalence
1037:Logical consequence
913:whole journal issue
25:commutative diagram
2462:Transfer principle
2425:Semantics of logic
2410:Categorical theory
2386:Non-standard model
1900:Logical connective
1027:Information theory
976:Mathematical logic
779:
706:where for any set
690:
559:
352:
288:
268:
245:
63:mathematical logic
29:
2700:
2699:
2632:Abstract category
2435:Theories of truth
2245:Rule of inference
2235:Natural deduction
2216:
2215:
1761:
1760:
1466:Cartesian product
1371:
1370:
1277:Many-valued logic
1252:Boolean functions
1135:Russell's paradox
1110:diagonal argument
1007:First-order logic
851:Macpherson (2011)
814:Fraïssé's theorem
403:of size two, and
362:, where * is the
291:{\displaystyle T}
271:{\displaystyle T}
48:Fraïssé's theorem
2720:
2691:
2690:
2642:History of logic
2637:Category of sets
2530:Decision problem
2309:Ordinal analysis
2250:Sequent calculus
2148:Boolean algebras
2088:
2087:
2062:
2033:logical/constant
1787:
1786:
1773:
1696:ZermeloâFraenkel
1447:Set operations:
1382:
1381:
1319:
1150:
1149:
1130:LöwenheimâSkolem
1017:Formal semantics
969:
962:
955:
946:
945:
937:
936:
927:(2): 1599â1634,
889:
854:
848:
842:
839:
833:
830:
788:
786:
785:
780:
699:
697:
696:
691:
670:
638:
615:
595:
568:
566:
565:
560:
551:
534:
371:linear orderings
361:
359:
358:
353:
348:
297:
295:
294:
289:
277:
275:
274:
269:
254:
252:
251:
246:
234:
217:
2728:
2727:
2723:
2722:
2721:
2719:
2718:
2717:
2703:
2702:
2701:
2696:
2685:
2678:
2623:Category theory
2613:Algebraic logic
2596:
2567:Lambda calculus
2505:Church encoding
2491:
2467:Truth predicate
2323:
2289:Complete theory
2212:
2081:
2077:
2073:
2068:
2060:
1780: and
1776:
1771:
1757:
1733:New Foundations
1701:axiom of choice
1684:
1646:Gödel numbering
1586: and
1578:
1482:
1367:
1317:
1298:
1247:Boolean algebra
1233:
1197:Equiconsistency
1162:Classical logic
1139:
1120:Halting problem
1108: and
1084: and
1072: and
1071:
1066:Theorems (
1061:
978:
973:
942:
886:
868:Hodges, Wilfrid
863:
858:
857:
849:
845:
840:
836:
831:
827:
822:
795:
728:
725:
724:
663:
631:
608:
588:
586:
583:
582:
544:
527:
525:
522:
521:
502:and embeddings
472:
443:, one in which
442:
431:
420:
409:
402:
395:
344:
333:
330:
329:
315:
283:
280:
279:
263:
260:
259:
227:
210:
208:
205:
204:
180:and embeddings
87:
73:as a manner of
71:lambda calculus
17:
12:
11:
5:
2726:
2716:
2715:
2698:
2697:
2683:
2680:
2679:
2677:
2676:
2671:
2666:
2661:
2656:
2655:
2654:
2644:
2639:
2634:
2625:
2620:
2615:
2610:
2608:Abstract logic
2604:
2602:
2598:
2597:
2595:
2594:
2589:
2587:Turing machine
2584:
2579:
2574:
2569:
2564:
2559:
2558:
2557:
2552:
2547:
2542:
2537:
2527:
2525:Computable set
2522:
2517:
2512:
2507:
2501:
2499:
2493:
2492:
2490:
2489:
2484:
2479:
2474:
2469:
2464:
2459:
2454:
2453:
2452:
2447:
2442:
2432:
2427:
2422:
2420:Satisfiability
2417:
2412:
2407:
2406:
2405:
2395:
2394:
2393:
2383:
2382:
2381:
2376:
2371:
2366:
2361:
2351:
2350:
2349:
2344:
2337:Interpretation
2333:
2331:
2325:
2324:
2322:
2321:
2316:
2311:
2306:
2301:
2291:
2286:
2285:
2284:
2283:
2282:
2272:
2267:
2257:
2252:
2247:
2242:
2237:
2232:
2226:
2224:
2218:
2217:
2214:
2213:
2211:
2210:
2202:
2201:
2200:
2199:
2194:
2193:
2192:
2187:
2182:
2162:
2161:
2160:
2158:minimal axioms
2155:
2144:
2143:
2142:
2131:
2130:
2129:
2124:
2119:
2114:
2109:
2104:
2091:
2089:
2070:
2069:
2067:
2066:
2065:
2064:
2052:
2047:
2046:
2045:
2040:
2035:
2030:
2020:
2015:
2010:
2005:
2004:
2003:
1998:
1988:
1987:
1986:
1981:
1976:
1971:
1961:
1956:
1955:
1954:
1949:
1944:
1934:
1933:
1932:
1927:
1922:
1917:
1912:
1907:
1897:
1892:
1887:
1882:
1881:
1880:
1875:
1870:
1865:
1855:
1850:
1848:Formation rule
1845:
1840:
1839:
1838:
1833:
1823:
1822:
1821:
1811:
1806:
1801:
1796:
1790:
1784:
1767:Formal systems
1763:
1762:
1759:
1758:
1756:
1755:
1750:
1745:
1740:
1735:
1730:
1725:
1720:
1715:
1710:
1709:
1708:
1703:
1692:
1690:
1686:
1685:
1683:
1682:
1681:
1680:
1670:
1665:
1664:
1663:
1656:Large cardinal
1653:
1648:
1643:
1638:
1633:
1619:
1618:
1617:
1612:
1607:
1592:
1590:
1580:
1579:
1577:
1576:
1575:
1574:
1569:
1564:
1554:
1549:
1544:
1539:
1534:
1529:
1524:
1519:
1514:
1509:
1504:
1499:
1493:
1491:
1484:
1483:
1481:
1480:
1479:
1478:
1473:
1468:
1463:
1458:
1453:
1445:
1444:
1443:
1438:
1428:
1423:
1421:Extensionality
1418:
1416:Ordinal number
1413:
1403:
1398:
1397:
1396:
1385:
1379:
1373:
1372:
1369:
1368:
1366:
1365:
1360:
1355:
1350:
1345:
1340:
1335:
1334:
1333:
1323:
1322:
1321:
1308:
1306:
1300:
1299:
1297:
1296:
1295:
1294:
1289:
1284:
1274:
1269:
1264:
1259:
1254:
1249:
1243:
1241:
1235:
1234:
1232:
1231:
1226:
1221:
1216:
1211:
1206:
1201:
1200:
1199:
1189:
1184:
1179:
1174:
1169:
1164:
1158:
1156:
1147:
1141:
1140:
1138:
1137:
1132:
1127:
1122:
1117:
1112:
1100:Cantor's
1098:
1093:
1088:
1078:
1076:
1063:
1062:
1060:
1059:
1054:
1049:
1044:
1039:
1034:
1029:
1024:
1019:
1014:
1009:
1004:
999:
998:
997:
986:
984:
980:
979:
972:
971:
964:
957:
949:
940:
939:
916:
905:
890:
884:
862:
859:
856:
855:
843:
834:
824:
823:
821:
818:
817:
816:
811:
806:
801:
794:
791:
790:
789:
777:
774:
771:
768:
765:
762:
759:
756:
753:
750:
747:
744:
741:
738:
735:
732:
721:
720:
719:
718:
701:
700:
688:
685:
682:
679:
676:
673:
669:
666:
662:
659:
656:
653:
650:
647:
644:
641:
637:
634:
630:
627:
624:
621:
618:
614:
611:
607:
604:
601:
598:
594:
591:
579:
578:
577:
576:
570:
569:
557:
554:
550:
547:
543:
540:
537:
533:
530:
471:
468:
464:characteristic
440:
429:
418:
407:
400:
393:
379:
378:
367:
351:
347:
343:
340:
337:
327:quotient group
319:
314:
311:
287:
267:
256:
255:
243:
240:
237:
233:
230:
226:
223:
220:
216:
213:
125:. Recall that
86:
83:
15:
9:
6:
4:
3:
2:
2725:
2714:
2711:
2710:
2708:
2695:
2694:
2689:
2681:
2675:
2672:
2670:
2667:
2665:
2662:
2660:
2657:
2653:
2650:
2649:
2648:
2645:
2643:
2640:
2638:
2635:
2633:
2629:
2626:
2624:
2621:
2619:
2616:
2614:
2611:
2609:
2606:
2605:
2603:
2599:
2593:
2590:
2588:
2585:
2583:
2582:Recursive set
2580:
2578:
2575:
2573:
2570:
2568:
2565:
2563:
2560:
2556:
2553:
2551:
2548:
2546:
2543:
2541:
2538:
2536:
2533:
2532:
2531:
2528:
2526:
2523:
2521:
2518:
2516:
2513:
2511:
2508:
2506:
2503:
2502:
2500:
2498:
2494:
2488:
2485:
2483:
2480:
2478:
2475:
2473:
2470:
2468:
2465:
2463:
2460:
2458:
2455:
2451:
2448:
2446:
2443:
2441:
2438:
2437:
2436:
2433:
2431:
2428:
2426:
2423:
2421:
2418:
2416:
2413:
2411:
2408:
2404:
2401:
2400:
2399:
2396:
2392:
2391:of arithmetic
2389:
2388:
2387:
2384:
2380:
2377:
2375:
2372:
2370:
2367:
2365:
2362:
2360:
2357:
2356:
2355:
2352:
2348:
2345:
2343:
2340:
2339:
2338:
2335:
2334:
2332:
2330:
2326:
2320:
2317:
2315:
2312:
2310:
2307:
2305:
2302:
2299:
2298:from ZFC
2295:
2292:
2290:
2287:
2281:
2278:
2277:
2276:
2273:
2271:
2268:
2266:
2263:
2262:
2261:
2258:
2256:
2253:
2251:
2248:
2246:
2243:
2241:
2238:
2236:
2233:
2231:
2228:
2227:
2225:
2223:
2219:
2209:
2208:
2204:
2203:
2198:
2197:non-Euclidean
2195:
2191:
2188:
2186:
2183:
2181:
2180:
2176:
2175:
2173:
2170:
2169:
2167:
2163:
2159:
2156:
2154:
2151:
2150:
2149:
2145:
2141:
2138:
2137:
2136:
2132:
2128:
2125:
2123:
2120:
2118:
2115:
2113:
2110:
2108:
2105:
2103:
2100:
2099:
2097:
2093:
2092:
2090:
2085:
2079:
2074:Example
2071:
2063:
2058:
2057:
2056:
2053:
2051:
2048:
2044:
2041:
2039:
2036:
2034:
2031:
2029:
2026:
2025:
2024:
2021:
2019:
2016:
2014:
2011:
2009:
2006:
2002:
1999:
1997:
1994:
1993:
1992:
1989:
1985:
1982:
1980:
1977:
1975:
1972:
1970:
1967:
1966:
1965:
1962:
1960:
1957:
1953:
1950:
1948:
1945:
1943:
1940:
1939:
1938:
1935:
1931:
1928:
1926:
1923:
1921:
1918:
1916:
1913:
1911:
1908:
1906:
1903:
1902:
1901:
1898:
1896:
1893:
1891:
1888:
1886:
1883:
1879:
1876:
1874:
1871:
1869:
1866:
1864:
1861:
1860:
1859:
1856:
1854:
1851:
1849:
1846:
1844:
1841:
1837:
1834:
1832:
1831:by definition
1829:
1828:
1827:
1824:
1820:
1817:
1816:
1815:
1812:
1810:
1807:
1805:
1802:
1800:
1797:
1795:
1792:
1791:
1788:
1785:
1783:
1779:
1774:
1768:
1764:
1754:
1751:
1749:
1746:
1744:
1741:
1739:
1736:
1734:
1731:
1729:
1726:
1724:
1721:
1719:
1718:KripkeâPlatek
1716:
1714:
1711:
1707:
1704:
1702:
1699:
1698:
1697:
1694:
1693:
1691:
1687:
1679:
1676:
1675:
1674:
1671:
1669:
1666:
1662:
1659:
1658:
1657:
1654:
1652:
1649:
1647:
1644:
1642:
1639:
1637:
1634:
1631:
1627:
1623:
1620:
1616:
1613:
1611:
1608:
1606:
1603:
1602:
1601:
1597:
1594:
1593:
1591:
1589:
1585:
1581:
1573:
1570:
1568:
1565:
1563:
1562:constructible
1560:
1559:
1558:
1555:
1553:
1550:
1548:
1545:
1543:
1540:
1538:
1535:
1533:
1530:
1528:
1525:
1523:
1520:
1518:
1515:
1513:
1510:
1508:
1505:
1503:
1500:
1498:
1495:
1494:
1492:
1490:
1485:
1477:
1474:
1472:
1469:
1467:
1464:
1462:
1459:
1457:
1454:
1452:
1449:
1448:
1446:
1442:
1439:
1437:
1434:
1433:
1432:
1429:
1427:
1424:
1422:
1419:
1417:
1414:
1412:
1408:
1404:
1402:
1399:
1395:
1392:
1391:
1390:
1387:
1386:
1383:
1380:
1378:
1374:
1364:
1361:
1359:
1356:
1354:
1351:
1349:
1346:
1344:
1341:
1339:
1336:
1332:
1329:
1328:
1327:
1324:
1320:
1315:
1314:
1313:
1310:
1309:
1307:
1305:
1301:
1293:
1290:
1288:
1285:
1283:
1280:
1279:
1278:
1275:
1273:
1270:
1268:
1265:
1263:
1260:
1258:
1255:
1253:
1250:
1248:
1245:
1244:
1242:
1240:
1239:Propositional
1236:
1230:
1227:
1225:
1222:
1220:
1217:
1215:
1212:
1210:
1207:
1205:
1202:
1198:
1195:
1194:
1193:
1190:
1188:
1185:
1183:
1180:
1178:
1175:
1173:
1170:
1168:
1167:Logical truth
1165:
1163:
1160:
1159:
1157:
1155:
1151:
1148:
1146:
1142:
1136:
1133:
1131:
1128:
1126:
1123:
1121:
1118:
1116:
1113:
1111:
1107:
1103:
1099:
1097:
1094:
1092:
1089:
1087:
1083:
1080:
1079:
1077:
1075:
1069:
1064:
1058:
1055:
1053:
1050:
1048:
1045:
1043:
1040:
1038:
1035:
1033:
1030:
1028:
1025:
1023:
1020:
1018:
1015:
1013:
1010:
1008:
1005:
1003:
1000:
996:
993:
992:
991:
988:
987:
985:
981:
977:
970:
965:
963:
958:
956:
951:
950:
947:
943:
935:
930:
926:
922:
917:
914:
910:
906:
903:
899:
895:
891:
887:
885:0-521-58713-1
881:
877:
873:
869:
865:
864:
852:
847:
838:
829:
825:
815:
812:
810:
807:
805:
802:
800:
797:
796:
775:
769:
766:
763:
760:
754:
748:
742:
736:
730:
723:
722:
717:
713:
710:and function
709:
705:
704:
703:
702:
683:
674:
671:
667:
664:
657:
651:
642:
639:
635:
632:
625:
619:
612:
609:
605:
599:
592:
589:
581:
580:
574:
573:
572:
571:
555:
552:
548:
545:
541:
538:
535:
531:
528:
520:
519:
518:
516:
513: â
512:
509: â
508:
505:
501:
497:
493:
489:
485:
481:
477:
467:
465:
461:
456:
454:
450:
446:
439:
435:
428:
424:
417:
413:
406:
399:
396:of size one,
392:
388:
384:
376:
372:
368:
365:
349:
345:
341:
338:
335:
328:
324:
321:The class of
320:
317:
316:
310:
308:
303:
301:
285:
265:
241:
238:
235:
231:
228:
224:
221:
218:
214:
211:
203:
202:
201:
199:
196: â
195:
191:
188: â
187:
183:
179:
176: â
175:
171:
167:
164: â
163:
159:
154:
152:
148:
144:
140:
136:
132:
129: â
128:
124:
120:
117: â
116:
112:
109: â
108:
104:
100:
96:
92:
82:
80:
76:
72:
68:
64:
60:
55:
53:
49:
44:
42:
38:
34:
26:
21:
2713:Model theory
2684:
2482:Ultraproduct
2329:Model theory
2294:Independence
2230:Formal proof
2222:Proof theory
2205:
2178:
2135:real numbers
2107:second-order
2018:Substitution
1895:Metalanguage
1836:conservative
1809:Axiom schema
1753:Constructive
1723:MorseâKelley
1689:Set theories
1668:Aleph number
1661:inaccessible
1567:Grothendieck
1451:intersection
1338:Higher-order
1326:Second-order
1272:Truth tables
1229:Venn diagram
1012:Formal proof
941:
924:
920:
908:
871:
846:
837:
828:
715:
711:
707:
514:
510:
506:
503:
499:
495:
491:
487:
483:
479:
475:
473:
457:
452:
448:
444:
437:
433:
426:
422:
415:
411:
404:
397:
390:
386:
380:
364:free product
304:
257:
197:
193:
189:
185:
181:
177:
173:
169:
165:
161:
157:
155:
150:
146:
142:
138:
134:
130:
126:
122:
118:
114:
110:
106:
98:
97:) such that
94:
90:
88:
56:
45:
36:
33:model theory
30:
2592:Type theory
2540:undecidable
2472:Truth value
2359:equivalence
2038:non-logical
1651:Enumeration
1641:Isomorphism
1588:cardinality
1572:Von Neumann
1537:Ultrafilter
1502:Uncountable
1436:equivalence
1353:Quantifiers
1343:Fixed-point
1312:First-order
1192:Consistency
1177:Proposition
1154:Traditional
1125:Lindström's
1115:Compactness
1057:Type theory
1002:Cardinality
892:Entries on
421:. However,
323:free groups
77:having the
67:modal logic
2403:elementary
2096:arithmetic
1964:Quantifier
1942:functional
1814:Expression
1532:Transitive
1476:identities
1461:complement
1394:hereditary
1377:Set theory
861:References
820:References
517:such that
200:such that
123:embeddings
85:Definition
41:structures
2674:Supertask
2577:Recursion
2535:decidable
2369:saturated
2347:of models
2270:deductive
2265:axiomatic
2185:Hilbert's
2172:Euclidean
2153:canonical
2076:axiomatic
2008:Signature
1937:Predicate
1826:Extension
1748:Ackermann
1673:Operation
1552:Universal
1542:Recursive
1517:Singleton
1512:Inhabited
1497:Countable
1487:Types of
1471:power set
1441:partition
1358:Predicate
1304:Predicate
1219:Syllogism
1209:Soundness
1182:Inference
1172:Tautology
1074:paradoxes
767:∈
761:∣
672:∘
640:∘
606:∩
553:∘
536:∘
339:∗
236:∘
219:∘
135:embedding
127:f: A
107:f: A
103:signature
95:A,f,B,g,C
75:reduction
2707:Category
2659:Logicism
2652:timeline
2628:Concrete
2487:Validity
2457:T-schema
2450:Kripke's
2445:Tarski's
2440:semantic
2430:Strength
2379:submodel
2374:spectrum
2342:function
2190:Tarski's
2179:Elements
2166:geometry
2122:Robinson
2043:variable
2028:function
2001:spectrum
1991:Sentence
1947:variable
1890:Language
1843:Relation
1804:Automata
1794:Alphabet
1778:language
1632:-jection
1610:codomain
1596:Function
1557:Universe
1527:Infinite
1431:Relation
1214:Validity
1204:Argument
1102:theorem,
870:(1997).
793:See also
668:′
636:′
613:′
593:′
549:′
532:′
511:D, g': C
474:A class
313:Examples
307:pullback
232:′
215:′
156:A class
2601:Related
2398:Diagram
2296: (
2275:Hilbert
2260:Systems
2255:Theorem
2133:of the
2078:systems
1858:Formula
1853:Grammar
1769: (
1713:General
1426:Forcing
1411:Element
1331:Monadic
1106:paradox
1047:Theorem
983:General
113::
91:amalgam
59:diagram
2364:finite
2127:Skolem
2080:
2055:Theory
2023:Symbol
2013:String
1996:atomic
1873:ground
1868:closed
1863:atomic
1819:ground
1782:syntax
1678:binary
1605:domain
1522:Finite
1287:finite
1145:Logics
1104:
1052:Theory
882:
192:
190:D, g':
184:
133:is an
105:, and
35:, the
2354:Model
2102:Peano
1959:Proof
1799:Arity
1728:Naive
1615:image
1547:Fuzzy
1507:Empty
1456:union
1401:Class
1042:Model
1032:Lemma
990:Axiom
488:A,B,C
162:A,B,C
99:A,B,C
2477:Type
2280:list
2084:list
2061:list
2050:Term
1984:rank
1878:open
1772:list
1584:Maps
1489:sets
1348:Free
1318:list
1068:list
995:list
896:and
880:ISBN
168:and
147:f(A)
121:are
111:B, g
57:The
52:ages
2164:of
2146:of
2094:of
1626:Sur
1600:Map
1407:Ur-
1389:Set
929:doi
925:311
900:in
714:on
575:and
504:f':
182:f':
149:of
137:if
89:An
2709::
2550:NP
2174::
2168::
2098::
1775:),
1630:Bi
1622:In
923:,
878:.
874:.
716:X,
498:â
490:â
455:.
302:.
153:.
81:.
23:A
2630:/
2545:P
2300:)
2086:)
2082:(
1979:â
1974:!
1969:â
1930:=
1925:â
1920:â
1915:â§
1910:âš
1905:ÂŹ
1628:/
1624:/
1598:/
1409:)
1405:(
1292:â
1282:3
1070:)
968:e
961:t
954:v
938:.
931::
915:.
888:.
853:.
776:.
773:}
770:X
764:x
758:)
755:x
752:(
749:h
746:{
743:=
740:]
737:X
734:[
731:h
712:h
708:X
687:]
684:A
681:[
678:)
675:g
665:g
661:(
658:=
655:]
652:A
649:[
646:)
643:f
633:f
629:(
626:=
623:]
620:C
617:[
610:g
603:]
600:B
597:[
590:f
556:g
546:g
542:=
539:f
529:f
515:D
507:B
500:K
496:D
492:K
476:K
453:e
449:e
445:e
441:3
438:L
434:e
430:1
427:L
423:K
419:3
416:L
412:K
408:3
405:L
401:2
398:L
394:1
391:L
387:K
366:.
350:A
346:/
342:C
336:B
286:T
266:T
242:.
239:g
229:g
225:=
222:f
212:f
198:D
194:C
186:B
178:K
174:D
170:A
166:K
158:K
151:B
143:A
139:f
131:B
119:C
115:A
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