1983:
2004:
177:. 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
166:
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
362:
1037:
1120:
261:
2045:
1434:
1592:
380:
1447:
770:
81:
A first-order theory has the joint embedding property if the class of its models of has the joint embedding property. A
1452:
1442:
1179:
1032:
385:
376:
2064:
1588:
230:
930:
1685:
1429:
254:
990:
683:
424:
1946:
1648:
1411:
1406:
1231:
652:
336:
25:
1941:
1724:
1641:
1354:
1285:
1162:
404:
200:
Chang, C. C.; Keisler, H. Jerome (2012). Model Theory (Third edition ed.). Dover
Publications. pp. 672 pages.
2038:
1866:
1692:
1378:
1012:
611:
1744:
1739:
1349:
1088:
1017:
346:
247:
1673:
1263:
657:
625:
316:
222:
174:
1963:
1912:
1809:
1307:
1268:
745:
390:
419:
1804:
1734:
1273:
1125:
1108:
831:
311:
178:
2019:
2031:
1636:
1613:
1574:
1460:
1401:
1047:
967:
811:
755:
368:
1926:
1653:
1631:
1598:
1491:
1337:
1322:
1295:
1246:
1130:
1065:
890:
856:
851:
725:
556:
533:
2069:
1856:
1709:
1501:
1219:
955:
861:
720:
705:
586:
561:
93:
86:
1829:
1791:
1668:
1472:
1312:
1236:
1214:
1042:
1000:
899:
866:
730:
518:
429:
8:
1958:
1849:
1834:
1814:
1771:
1658:
1608:
1534:
1479:
1416:
1209:
1204:
1152:
920:
909:
581:
481:
409:
400:
396:
331:
326:
2011:
1987:
1756:
1719:
1704:
1697:
1680:
1484:
1466:
1332:
1258:
1241:
1194:
1007:
916:
750:
735:
695:
647:
632:
620:
576:
551:
321:
270:
75:
940:
1982:
1922:
1729:
1539:
1529:
1421:
1302:
1137:
1113:
894:
878:
783:
760:
637:
606:
571:
466:
301:
226:
17:
1936:
1931:
1824:
1781:
1603:
1564:
1559:
1544:
1370:
1327:
1224:
1022:
972:
546:
508:
1917:
1907:
1861:
1844:
1799:
1761:
1663:
1583:
1390:
1317:
1290:
1278:
1184:
1098:
1072:
1027:
995:
796:
598:
541:
491:
456:
414:
140:
does not have the amalgamation property. The counterexample for this starts with
82:
2015:
1902:
1881:
1839:
1819:
1714:
1569:
1167:
1157:
1147:
1142:
1076:
950:
826:
715:
710:
688:
289:
214:
129:
has the joint embedding property because all three models can be embedded into
2058:
1876:
1554:
1061:
846:
836:
806:
791:
461:
1776:
1623:
1524:
1516:
1396:
1344:
1253:
1189:
1172:
1103:
962:
821:
523:
306:
101:
21:
1886:
1766:
945:
935:
882:
566:
486:
471:
351:
296:
816:
671:
642:
448:
92:
A similar but different notion to the joint embedding property is the
1968:
1871:
924:
841:
801:
765:
701:
513:
503:
476:
239:
64:
1953:
1751:
1199:
904:
498:
1549:
341:
2003:
1093:
439:
284:
74:
It is one of the three properties used to define the
96:. To see the difference, first consider the class
100:(or simply the set) containing three models with
2056:
89:with the joint embedding property is complete.
85:has the joint embedding property. Conversely a
2039:
255:
196:
194:
2046:
2032:
447:
262:
248:
191:
162:is the smallest and the other in which
2057:
269:
213:
243:
151:and extends in two different ways to
1998:
13:
14:
2081:
2002:
1981:
1:
1942:History of mathematical logic
207:
2018:. You can help Knowledge by
1867:Primitive recursive function
147:containing a single element
7:
175:algebraically closed fields
10:
2086:
1997:
931:SchröderâBernstein theorem
658:Monadic predicate calculus
317:Foundations of mathematics
223:Cambridge University Press
173:Now consider the class of
125:of size three. This class
1977:
1964:Philosophy of mathematics
1913:Automated theorem proving
1895:
1790:
1622:
1515:
1367:
1084:
1060:
1038:Von NeumannâBernaysâGödel
983:
877:
781:
679:
670:
597:
532:
438:
360:
277:
2065:Mathematical logic stubs
184:
33:joint embedding property
1614:Self-verifying theories
1435:Tarski's axiomatization
386:Tarski's undefinability
381:incompleteness theorems
47:, there is a structure
35:if for all structures
2014:-related article is a
1988:Mathematics portal
1599:Proof of impossibility
1247:propositional variable
557:Propositional calculus
219:A shorter model theory
181:of the fields differ.
1857:Kolmogorov complexity
1810:Computably enumerable
1710:Model complete theory
1502:Principia Mathematica
562:Propositional formula
391:BanachâTarski paradox
94:amalgamation property
87:model-complete theory
1805:ChurchâTuring thesis
1792:Computability theory
1001:continuum hypothesis
519:Square of opposition
377:Gödel's completeness
31:is said to have the
1959:Mathematical object
1850:P versus NP problem
1815:Computable function
1609:Reverse mathematics
1535:Logical consequence
1412:primitive recursive
1407:elementary function
1180:Free/bound variable
1033:TarskiâGrothendieck
552:Logical connectives
482:Logical equivalence
332:Logical consequence
2012:mathematical logic
1757:Transfer principle
1720:Semantics of logic
1705:Categorical theory
1681:Non-standard model
1195:Logical connective
322:Information theory
271:Mathematical logic
2027:
2026:
1995:
1994:
1927:Abstract category
1730:Theories of truth
1540:Rule of inference
1530:Natural deduction
1511:
1510:
1056:
1055:
761:Cartesian product
666:
665:
572:Many-valued logic
547:Boolean functions
430:Russell's paradox
405:diagonal argument
302:First-order logic
118:of size two, and
18:universal algebra
2077:
2048:
2041:
2034:
2006:
1999:
1986:
1985:
1937:History of logic
1932:Category of sets
1825:Decision problem
1604:Ordinal analysis
1545:Sequent calculus
1443:Boolean algebras
1383:
1382:
1357:
1328:logical/constant
1082:
1081:
1068:
991:ZermeloâFraenkel
742:Set operations:
677:
676:
614:
445:
444:
425:LöwenheimâSkolem
312:Formal semantics
264:
257:
250:
241:
240:
236:
201:
198:
78:of a structure.
2085:
2084:
2080:
2079:
2078:
2076:
2075:
2074:
2055:
2054:
2053:
2052:
1996:
1991:
1980:
1973:
1918:Category theory
1908:Algebraic logic
1891:
1862:Lambda calculus
1800:Church encoding
1786:
1762:Truth predicate
1618:
1584:Complete theory
1507:
1376:
1372:
1368:
1363:
1355:
1075: and
1071:
1066:
1052:
1028:New Foundations
996:axiom of choice
979:
941:Gödel numbering
881: and
873:
777:
662:
612:
593:
542:Boolean algebra
528:
492:Equiconsistency
457:Classical logic
434:
415:Halting problem
403: and
379: and
367: and
366:
361:Theorems (
356:
273:
268:
233:
215:Hodges, Wilfrid
210:
205:
204:
199:
192:
187:
158:, one in which
157:
146:
135:
124:
117:
110:
83:complete theory
55:such that both
12:
11:
5:
2083:
2073:
2072:
2067:
2051:
2050:
2043:
2036:
2028:
2025:
2024:
2007:
1993:
1992:
1978:
1975:
1974:
1972:
1971:
1966:
1961:
1956:
1951:
1950:
1949:
1939:
1934:
1929:
1920:
1915:
1910:
1905:
1903:Abstract logic
1899:
1897:
1893:
1892:
1890:
1889:
1884:
1882:Turing machine
1879:
1874:
1869:
1864:
1859:
1854:
1853:
1852:
1847:
1842:
1837:
1832:
1822:
1820:Computable set
1817:
1812:
1807:
1802:
1796:
1794:
1788:
1787:
1785:
1784:
1779:
1774:
1769:
1764:
1759:
1754:
1749:
1748:
1747:
1742:
1737:
1727:
1722:
1717:
1715:Satisfiability
1712:
1707:
1702:
1701:
1700:
1690:
1689:
1688:
1678:
1677:
1676:
1671:
1666:
1661:
1656:
1646:
1645:
1644:
1639:
1632:Interpretation
1628:
1626:
1620:
1619:
1617:
1616:
1611:
1606:
1601:
1596:
1586:
1581:
1580:
1579:
1578:
1577:
1567:
1562:
1552:
1547:
1542:
1537:
1532:
1527:
1521:
1519:
1513:
1512:
1509:
1508:
1506:
1505:
1497:
1496:
1495:
1494:
1489:
1488:
1487:
1482:
1477:
1457:
1456:
1455:
1453:minimal axioms
1450:
1439:
1438:
1437:
1426:
1425:
1424:
1419:
1414:
1409:
1404:
1399:
1386:
1384:
1365:
1364:
1362:
1361:
1360:
1359:
1347:
1342:
1341:
1340:
1335:
1330:
1325:
1315:
1310:
1305:
1300:
1299:
1298:
1293:
1283:
1282:
1281:
1276:
1271:
1266:
1256:
1251:
1250:
1249:
1244:
1239:
1229:
1228:
1227:
1222:
1217:
1212:
1207:
1202:
1192:
1187:
1182:
1177:
1176:
1175:
1170:
1165:
1160:
1150:
1145:
1143:Formation rule
1140:
1135:
1134:
1133:
1128:
1118:
1117:
1116:
1106:
1101:
1096:
1091:
1085:
1079:
1062:Formal systems
1058:
1057:
1054:
1053:
1051:
1050:
1045:
1040:
1035:
1030:
1025:
1020:
1015:
1010:
1005:
1004:
1003:
998:
987:
985:
981:
980:
978:
977:
976:
975:
965:
960:
959:
958:
951:Large cardinal
948:
943:
938:
933:
928:
914:
913:
912:
907:
902:
887:
885:
875:
874:
872:
871:
870:
869:
864:
859:
849:
844:
839:
834:
829:
824:
819:
814:
809:
804:
799:
794:
788:
786:
779:
778:
776:
775:
774:
773:
768:
763:
758:
753:
748:
740:
739:
738:
733:
723:
718:
716:Extensionality
713:
711:Ordinal number
708:
698:
693:
692:
691:
680:
674:
668:
667:
664:
663:
661:
660:
655:
650:
645:
640:
635:
630:
629:
628:
618:
617:
616:
603:
601:
595:
594:
592:
591:
590:
589:
584:
579:
569:
564:
559:
554:
549:
544:
538:
536:
530:
529:
527:
526:
521:
516:
511:
506:
501:
496:
495:
494:
484:
479:
474:
469:
464:
459:
453:
451:
442:
436:
435:
433:
432:
427:
422:
417:
412:
407:
395:Cantor's
393:
388:
383:
373:
371:
358:
357:
355:
354:
349:
344:
339:
334:
329:
324:
319:
314:
309:
304:
299:
294:
293:
292:
281:
279:
275:
274:
267:
266:
259:
252:
244:
238:
237:
231:
209:
206:
203:
202:
189:
188:
186:
183:
179:characteristic
155:
144:
133:
122:
115:
108:
9:
6:
4:
3:
2:
2082:
2071:
2068:
2066:
2063:
2062:
2060:
2049:
2044:
2042:
2037:
2035:
2030:
2029:
2023:
2021:
2017:
2013:
2008:
2005:
2001:
2000:
1990:
1989:
1984:
1976:
1970:
1967:
1965:
1962:
1960:
1957:
1955:
1952:
1948:
1945:
1944:
1943:
1940:
1938:
1935:
1933:
1930:
1928:
1924:
1921:
1919:
1916:
1914:
1911:
1909:
1906:
1904:
1901:
1900:
1898:
1894:
1888:
1885:
1883:
1880:
1878:
1877:Recursive set
1875:
1873:
1870:
1868:
1865:
1863:
1860:
1858:
1855:
1851:
1848:
1846:
1843:
1841:
1838:
1836:
1833:
1831:
1828:
1827:
1826:
1823:
1821:
1818:
1816:
1813:
1811:
1808:
1806:
1803:
1801:
1798:
1797:
1795:
1793:
1789:
1783:
1780:
1778:
1775:
1773:
1770:
1768:
1765:
1763:
1760:
1758:
1755:
1753:
1750:
1746:
1743:
1741:
1738:
1736:
1733:
1732:
1731:
1728:
1726:
1723:
1721:
1718:
1716:
1713:
1711:
1708:
1706:
1703:
1699:
1696:
1695:
1694:
1691:
1687:
1686:of arithmetic
1684:
1683:
1682:
1679:
1675:
1672:
1670:
1667:
1665:
1662:
1660:
1657:
1655:
1652:
1651:
1650:
1647:
1643:
1640:
1638:
1635:
1634:
1633:
1630:
1629:
1627:
1625:
1621:
1615:
1612:
1610:
1607:
1605:
1602:
1600:
1597:
1594:
1593:from ZFC
1590:
1587:
1585:
1582:
1576:
1573:
1572:
1571:
1568:
1566:
1563:
1561:
1558:
1557:
1556:
1553:
1551:
1548:
1546:
1543:
1541:
1538:
1536:
1533:
1531:
1528:
1526:
1523:
1522:
1520:
1518:
1514:
1504:
1503:
1499:
1498:
1493:
1492:non-Euclidean
1490:
1486:
1483:
1481:
1478:
1476:
1475:
1471:
1470:
1468:
1465:
1464:
1462:
1458:
1454:
1451:
1449:
1446:
1445:
1444:
1440:
1436:
1433:
1432:
1431:
1427:
1423:
1420:
1418:
1415:
1413:
1410:
1408:
1405:
1403:
1400:
1398:
1395:
1394:
1392:
1388:
1387:
1385:
1380:
1374:
1369:Example
1366:
1358:
1353:
1352:
1351:
1348:
1346:
1343:
1339:
1336:
1334:
1331:
1329:
1326:
1324:
1321:
1320:
1319:
1316:
1314:
1311:
1309:
1306:
1304:
1301:
1297:
1294:
1292:
1289:
1288:
1287:
1284:
1280:
1277:
1275:
1272:
1270:
1267:
1265:
1262:
1261:
1260:
1257:
1255:
1252:
1248:
1245:
1243:
1240:
1238:
1235:
1234:
1233:
1230:
1226:
1223:
1221:
1218:
1216:
1213:
1211:
1208:
1206:
1203:
1201:
1198:
1197:
1196:
1193:
1191:
1188:
1186:
1183:
1181:
1178:
1174:
1171:
1169:
1166:
1164:
1161:
1159:
1156:
1155:
1154:
1151:
1149:
1146:
1144:
1141:
1139:
1136:
1132:
1129:
1127:
1126:by definition
1124:
1123:
1122:
1119:
1115:
1112:
1111:
1110:
1107:
1105:
1102:
1100:
1097:
1095:
1092:
1090:
1087:
1086:
1083:
1080:
1078:
1074:
1069:
1063:
1059:
1049:
1046:
1044:
1041:
1039:
1036:
1034:
1031:
1029:
1026:
1024:
1021:
1019:
1016:
1014:
1013:KripkeâPlatek
1011:
1009:
1006:
1002:
999:
997:
994:
993:
992:
989:
988:
986:
982:
974:
971:
970:
969:
966:
964:
961:
957:
954:
953:
952:
949:
947:
944:
942:
939:
937:
934:
932:
929:
926:
922:
918:
915:
911:
908:
906:
903:
901:
898:
897:
896:
892:
889:
888:
886:
884:
880:
876:
868:
865:
863:
860:
858:
857:constructible
855:
854:
853:
850:
848:
845:
843:
840:
838:
835:
833:
830:
828:
825:
823:
820:
818:
815:
813:
810:
808:
805:
803:
800:
798:
795:
793:
790:
789:
787:
785:
780:
772:
769:
767:
764:
762:
759:
757:
754:
752:
749:
747:
744:
743:
741:
737:
734:
732:
729:
728:
727:
724:
722:
719:
717:
714:
712:
709:
707:
703:
699:
697:
694:
690:
687:
686:
685:
682:
681:
678:
675:
673:
669:
659:
656:
654:
651:
649:
646:
644:
641:
639:
636:
634:
631:
627:
624:
623:
622:
619:
615:
610:
609:
608:
605:
604:
602:
600:
596:
588:
585:
583:
580:
578:
575:
574:
573:
570:
568:
565:
563:
560:
558:
555:
553:
550:
548:
545:
543:
540:
539:
537:
535:
534:Propositional
531:
525:
522:
520:
517:
515:
512:
510:
507:
505:
502:
500:
497:
493:
490:
489:
488:
485:
483:
480:
478:
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2020:expanding it
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1777:Ultraproduct
1624:Model theory
1589:Independence
1525:Formal proof
1517:Proof theory
1500:
1473:
1430:real numbers
1402:second-order
1313:Substitution
1190:Metalanguage
1131:conservative
1104:Axiom schema
1048:Constructive
1018:MorseâKelley
984:Set theories
963:Aleph number
956:inaccessible
862:Grothendieck
746:intersection
633:Higher-order
621:Second-order
567:Truth tables
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307:Formal proof
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1835:undecidable
1767:Truth value
1654:equivalence
1333:non-logical
946:Enumeration
936:Isomorphism
883:cardinality
867:Von Neumann
832:Ultrafilter
797:Uncountable
731:equivalence
648:Quantifiers
638:Fixed-point
607:First-order
487:Consistency
472:Proposition
449:Traditional
420:Lindström's
410:Compactness
352:Type theory
297:Cardinality
136:. However,
2059:Categories
1698:elementary
1391:arithmetic
1259:Quantifier
1237:functional
1109:Expression
827:Transitive
771:identities
756:complement
689:hereditary
672:Set theory
208:References
65:embeddings
26:structures
1969:Supertask
1872:Recursion
1830:decidable
1664:saturated
1642:of models
1565:deductive
1560:axiomatic
1480:Hilbert's
1467:Euclidean
1448:canonical
1371:axiomatic
1303:Signature
1232:Predicate
1121:Extension
1043:Ackermann
968:Operation
847:Universal
837:Recursive
812:Singleton
807:Inhabited
792:Countable
782:Types of
766:power set
736:partition
653:Predicate
599:Predicate
514:Syllogism
504:Soundness
477:Inference
467:Tautology
369:paradoxes
1954:Logicism
1947:timeline
1923:Concrete
1782:Validity
1752:T-schema
1745:Kripke's
1740:Tarski's
1735:semantic
1725:Strength
1674:submodel
1669:spectrum
1637:function
1485:Tarski's
1474:Elements
1461:geometry
1417:Robinson
1338:variable
1323:function
1296:spectrum
1286:Sentence
1242:variable
1185:Language
1138:Relation
1099:Automata
1089:Alphabet
1073:language
927:-jection
905:codomain
891:Function
852:Universe
822:Infinite
726:Relation
509:Validity
499:Argument
397:theorem,
217:(1997).
1896:Related
1693:Diagram
1591: (
1570:Hilbert
1555:Systems
1550:Theorem
1428:of the
1373:systems
1153:Formula
1148:Grammar
1064: (
1008:General
721:Forcing
706:Element
626:Monadic
401:paradox
342:Theorem
278:General
1659:finite
1422:Skolem
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1350:Theory
1318:Symbol
1308:String
1291:atomic
1168:ground
1163:closed
1158:atomic
1114:ground
1077:syntax
973:binary
900:domain
817:Finite
582:finite
440:Logics
399:
347:Theory
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2010:This
1649:Model
1397:Peano
1254:Proof
1094:Arity
1023:Naive
910:image
842:Fuzzy
802:Empty
751:union
696:Class
337:Model
327:Lemma
285:Axiom
185:Notes
67:into
63:have
2016:stub
1772:Type
1575:list
1379:list
1356:list
1345:Term
1279:rank
1173:open
1067:list
879:Maps
784:sets
643:Free
613:list
363:list
290:list
227:ISBN
59:and
39:and
20:and
1459:of
1441:of
1389:of
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895:Map
702:Ur-
684:Set
76:age
51:in
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