1971:
58:
Thus a structure is minimal only if the parametrically definable subsets of its domain cannot be avoided, because they are already parametrically definable in the pure language of equality. Strong minimality was one of the early notions in the new field of classification theory and
139:. A model of a strongly minimal theory is determined up to isomorphism by its dimension as a matroid. Totally categorical theories are controlled by a strongly minimal set; this fact explains (and is used in the proof of) Morley's theorem.
143:
conjectured that the only pregeometries that can arise from strongly minimal sets are those that arise in vector spaces, projective spaces, or algebraically closed fields. This conjecture was refuted by
68:
350:
1025:
1108:
249:
1422:
101:
shows, the parametrically definable subsets of the square of the domain of a minimal structure can be relatively complicated ("
1580:
148:, who developed a method known as "Hrushovski construction" to build new strongly minimal structures from finite structures.
368:
1435:
758:
1440:
1430:
1167:
1020:
373:
364:
1576:
918:
1673:
1417:
242:
64:
74:
The nontrivial standard examples of strongly minimal theories are the one-sorted theories of infinite-dimensional
978:
671:
412:
1934:
1636:
1399:
1394:
1219:
640:
324:
32:
1929:
1712:
1629:
1342:
1273:
1150:
392:
1854:
1680:
1366:
1000:
599:
1732:
1727:
1337:
1076:
1005:
334:
235:
1661:
1251:
645:
613:
304:
136:
85:
1951:
1900:
1797:
1295:
1256:
733:
378:
407:
1792:
1722:
1261:
1113:
1096:
819:
299:
1624:
1601:
1562:
1448:
1389:
1035:
955:
799:
743:
356:
108:
More generally, a subset of a structure that is defined as the set of realizations of a formula
1914:
1641:
1619:
1586:
1479:
1325:
1310:
1283:
1234:
1118:
1053:
878:
844:
839:
713:
544:
521:
89:
1995:
1844:
1697:
1489:
1207:
943:
849:
708:
693:
574:
549:
120:
if every parametrically definable subset of it is either finite or cofinite. It is called a
1817:
1779:
1656:
1460:
1300:
1224:
1202:
1030:
988:
887:
854:
718:
506:
417:
125:
8:
1946:
1837:
1822:
1802:
1759:
1646:
1596:
1522:
1467:
1404:
1197:
1192:
1140:
908:
897:
569:
469:
397:
388:
384:
319:
314:
1975:
1744:
1707:
1692:
1685:
1668:
1472:
1454:
1320:
1246:
1229:
1182:
995:
904:
738:
723:
683:
635:
620:
608:
564:
539:
309:
258:
192:
24:
928:
1970:
1910:
1717:
1527:
1517:
1409:
1290:
1125:
1101:
882:
866:
771:
748:
625:
594:
559:
454:
289:
220:
135:
given by algebraic closure in the model-theoretic sense, is an infinite matroid, or
1924:
1919:
1812:
1769:
1591:
1552:
1547:
1532:
1358:
1315:
1212:
1010:
960:
534:
496:
215:
184:
162:
157:
132:
1905:
1895:
1849:
1832:
1787:
1749:
1651:
1571:
1378:
1305:
1278:
1266:
1172:
1086:
1060:
1015:
983:
784:
586:
529:
479:
444:
402:
203:
145:
102:
48:
1890:
1869:
1827:
1807:
1702:
1557:
1155:
1145:
1135:
1130:
1064:
938:
814:
703:
698:
676:
277:
1989:
1864:
1542:
1049:
834:
824:
794:
779:
449:
60:
36:
1764:
1611:
1512:
1504:
1384:
1332:
1241:
1177:
1160:
1091:
950:
809:
511:
294:
175:
Baldwin, John T.; Lachlan, Alistair H. (1971), "On
Strongly Minimal Sets",
140:
75:
20:
1874:
1754:
933:
923:
870:
554:
474:
459:
339:
284:
804:
659:
630:
436:
196:
1956:
1859:
912:
829:
789:
753:
689:
501:
491:
464:
227:
188:
1941:
1739:
1187:
892:
486:
40:
1537:
329:
1081:
427:
272:
183:(1), The Journal of Symbolic Logic, Vol. 36, No. 1: 79â96,
55:is a structure whose theory is strongly minimal.
1987:
174:
243:
35:such that every subset of its domain that is
435:
250:
236:
202:
131:A strongly minimal set, equipped with the
219:
1988:
257:
206:(1993), "A new strongly minimal set",
231:
51:all models of which are minimal. A
13:
14:
2007:
1969:
208:Annals of Pure and Applied Logic
16:Concept from mathematical logic
1:
1930:History of mathematical logic
177:The Journal of Symbolic Logic
168:
1855:Primitive recursive function
221:10.1016/0168-0072(93)90171-9
124:if this is true even in all
7:
151:
86:algebraically closed fields
10:
2012:
919:SchröderâBernstein theorem
646:Monadic predicate calculus
305:Foundations of mathematics
53:strongly minimal structure
1965:
1952:Philosophy of mathematics
1901:Automated theorem proving
1883:
1778:
1610:
1503:
1355:
1072:
1048:
1026:Von NeumannâBernaysâGödel
971:
865:
769:
667:
658:
585:
520:
426:
348:
265:
37:definable with parameters
1602:Self-verifying theories
1423:Tarski's axiomatization
374:Tarski's undefinability
369:incompleteness theorems
45:strongly minimal theory
1976:Mathematics portal
1587:Proof of impossibility
1235:propositional variable
545:Propositional calculus
78:, and the theories ACF
63:that was opened up by
1845:Kolmogorov complexity
1798:Computably enumerable
1698:Model complete theory
1490:Principia Mathematica
550:Propositional formula
379:BanachâTarski paradox
126:elementary extensions
1793:ChurchâTuring thesis
1780:Computability theory
989:continuum hypothesis
507:Square of opposition
365:Gödel's completeness
122:strongly minimal set
95:. As the example ACF
39:is either finite or
33:one-sorted structure
1947:Mathematical object
1838:P versus NP problem
1803:Computable function
1597:Reverse mathematics
1523:Logical consequence
1400:primitive recursive
1395:elementary function
1168:Free/bound variable
1021:TarskiâGrothendieck
540:Logical connectives
470:Logical equivalence
320:Logical consequence
69:totally categorical
23:—a branch of
1745:Transfer principle
1708:Semantics of logic
1693:Categorical theory
1669:Non-standard model
1183:Logical connective
310:Information theory
259:Mathematical logic
25:mathematical logic
1983:
1982:
1915:Abstract category
1718:Theories of truth
1528:Rule of inference
1518:Natural deduction
1499:
1498:
1044:
1043:
749:Cartesian product
654:
653:
560:Many-valued logic
535:Boolean functions
418:Russell's paradox
393:diagonal argument
290:First-order logic
29:minimal structure
2003:
1974:
1973:
1925:History of logic
1920:Category of sets
1813:Decision problem
1592:Ordinal analysis
1533:Sequent calculus
1431:Boolean algebras
1371:
1370:
1345:
1316:logical/constant
1070:
1069:
1056:
979:ZermeloâFraenkel
730:Set operations:
665:
664:
602:
433:
432:
413:LöwenheimâSkolem
300:Formal semantics
252:
245:
238:
229:
228:
224:
223:
204:Hrushovski, Ehud
199:
163:o-minimal theory
158:C-minimal theory
133:closure operator
65:Morley's theorem
61:stability theory
2011:
2010:
2006:
2005:
2004:
2002:
2001:
2000:
1986:
1985:
1984:
1979:
1968:
1961:
1906:Category theory
1896:Algebraic logic
1879:
1850:Lambda calculus
1788:Church encoding
1774:
1750:Truth predicate
1606:
1572:Complete theory
1495:
1364:
1360:
1356:
1351:
1343:
1063: and
1059:
1054:
1040:
1016:New Foundations
984:axiom of choice
967:
929:Gödel numbering
869: and
861:
765:
650:
600:
581:
530:Boolean algebra
516:
480:Equiconsistency
445:Classical logic
422:
403:Halting problem
391: and
367: and
355: and
354:
349:Theorems (
344:
261:
256:
189:10.2307/2271517
171:
154:
146:Ehud Hrushovski
100:
83:
49:complete theory
31:is an infinite
17:
12:
11:
5:
2009:
1999:
1998:
1981:
1980:
1966:
1963:
1962:
1960:
1959:
1954:
1949:
1944:
1939:
1938:
1937:
1927:
1922:
1917:
1908:
1903:
1898:
1893:
1891:Abstract logic
1887:
1885:
1881:
1880:
1878:
1877:
1872:
1870:Turing machine
1867:
1862:
1857:
1852:
1847:
1842:
1841:
1840:
1835:
1830:
1825:
1820:
1810:
1808:Computable set
1805:
1800:
1795:
1790:
1784:
1782:
1776:
1775:
1773:
1772:
1767:
1762:
1757:
1752:
1747:
1742:
1737:
1736:
1735:
1730:
1725:
1715:
1710:
1705:
1703:Satisfiability
1700:
1695:
1690:
1689:
1688:
1678:
1677:
1676:
1666:
1665:
1664:
1659:
1654:
1649:
1644:
1634:
1633:
1632:
1627:
1620:Interpretation
1616:
1614:
1608:
1607:
1605:
1604:
1599:
1594:
1589:
1584:
1574:
1569:
1568:
1567:
1566:
1565:
1555:
1550:
1540:
1535:
1530:
1525:
1520:
1515:
1509:
1507:
1501:
1500:
1497:
1496:
1494:
1493:
1485:
1484:
1483:
1482:
1477:
1476:
1475:
1470:
1465:
1445:
1444:
1443:
1441:minimal axioms
1438:
1427:
1426:
1425:
1414:
1413:
1412:
1407:
1402:
1397:
1392:
1387:
1374:
1372:
1353:
1352:
1350:
1349:
1348:
1347:
1335:
1330:
1329:
1328:
1323:
1318:
1313:
1303:
1298:
1293:
1288:
1287:
1286:
1281:
1271:
1270:
1269:
1264:
1259:
1254:
1244:
1239:
1238:
1237:
1232:
1227:
1217:
1216:
1215:
1210:
1205:
1200:
1195:
1190:
1180:
1175:
1170:
1165:
1164:
1163:
1158:
1153:
1148:
1138:
1133:
1131:Formation rule
1128:
1123:
1122:
1121:
1116:
1106:
1105:
1104:
1094:
1089:
1084:
1079:
1073:
1067:
1050:Formal systems
1046:
1045:
1042:
1041:
1039:
1038:
1033:
1028:
1023:
1018:
1013:
1008:
1003:
998:
993:
992:
991:
986:
975:
973:
969:
968:
966:
965:
964:
963:
953:
948:
947:
946:
939:Large cardinal
936:
931:
926:
921:
916:
902:
901:
900:
895:
890:
875:
873:
863:
862:
860:
859:
858:
857:
852:
847:
837:
832:
827:
822:
817:
812:
807:
802:
797:
792:
787:
782:
776:
774:
767:
766:
764:
763:
762:
761:
756:
751:
746:
741:
736:
728:
727:
726:
721:
711:
706:
704:Extensionality
701:
699:Ordinal number
696:
686:
681:
680:
679:
668:
662:
656:
655:
652:
651:
649:
648:
643:
638:
633:
628:
623:
618:
617:
616:
606:
605:
604:
591:
589:
583:
582:
580:
579:
578:
577:
572:
567:
557:
552:
547:
542:
537:
532:
526:
524:
518:
517:
515:
514:
509:
504:
499:
494:
489:
484:
483:
482:
472:
467:
462:
457:
452:
447:
441:
439:
430:
424:
423:
421:
420:
415:
410:
405:
400:
395:
383:Cantor's
381:
376:
371:
361:
359:
346:
345:
343:
342:
337:
332:
327:
322:
317:
312:
307:
302:
297:
292:
287:
282:
281:
280:
269:
267:
263:
262:
255:
254:
247:
240:
232:
226:
225:
200:
170:
167:
166:
165:
160:
153:
150:
116:) is called a
96:
90:characteristic
79:
15:
9:
6:
4:
3:
2:
2008:
1997:
1994:
1993:
1991:
1978:
1977:
1972:
1964:
1958:
1955:
1953:
1950:
1948:
1945:
1943:
1940:
1936:
1933:
1932:
1931:
1928:
1926:
1923:
1921:
1918:
1916:
1912:
1909:
1907:
1904:
1902:
1899:
1897:
1894:
1892:
1889:
1888:
1886:
1882:
1876:
1873:
1871:
1868:
1866:
1865:Recursive set
1863:
1861:
1858:
1856:
1853:
1851:
1848:
1846:
1843:
1839:
1836:
1834:
1831:
1829:
1826:
1824:
1821:
1819:
1816:
1815:
1814:
1811:
1809:
1806:
1804:
1801:
1799:
1796:
1794:
1791:
1789:
1786:
1785:
1783:
1781:
1777:
1771:
1768:
1766:
1763:
1761:
1758:
1756:
1753:
1751:
1748:
1746:
1743:
1741:
1738:
1734:
1731:
1729:
1726:
1724:
1721:
1720:
1719:
1716:
1714:
1711:
1709:
1706:
1704:
1701:
1699:
1696:
1694:
1691:
1687:
1684:
1683:
1682:
1679:
1675:
1674:of arithmetic
1672:
1671:
1670:
1667:
1663:
1660:
1658:
1655:
1653:
1650:
1648:
1645:
1643:
1640:
1639:
1638:
1635:
1631:
1628:
1626:
1623:
1622:
1621:
1618:
1617:
1615:
1613:
1609:
1603:
1600:
1598:
1595:
1593:
1590:
1588:
1585:
1582:
1581:from ZFC
1578:
1575:
1573:
1570:
1564:
1561:
1560:
1559:
1556:
1554:
1551:
1549:
1546:
1545:
1544:
1541:
1539:
1536:
1534:
1531:
1529:
1526:
1524:
1521:
1519:
1516:
1514:
1511:
1510:
1508:
1506:
1502:
1492:
1491:
1487:
1486:
1481:
1480:non-Euclidean
1478:
1474:
1471:
1469:
1466:
1464:
1463:
1459:
1458:
1456:
1453:
1452:
1450:
1446:
1442:
1439:
1437:
1434:
1433:
1432:
1428:
1424:
1421:
1420:
1419:
1415:
1411:
1408:
1406:
1403:
1401:
1398:
1396:
1393:
1391:
1388:
1386:
1383:
1382:
1380:
1376:
1375:
1373:
1368:
1362:
1357:Example
1354:
1346:
1341:
1340:
1339:
1336:
1334:
1331:
1327:
1324:
1322:
1319:
1317:
1314:
1312:
1309:
1308:
1307:
1304:
1302:
1299:
1297:
1294:
1292:
1289:
1285:
1282:
1280:
1277:
1276:
1275:
1272:
1268:
1265:
1263:
1260:
1258:
1255:
1253:
1250:
1249:
1248:
1245:
1243:
1240:
1236:
1233:
1231:
1228:
1226:
1223:
1222:
1221:
1218:
1214:
1211:
1209:
1206:
1204:
1201:
1199:
1196:
1194:
1191:
1189:
1186:
1185:
1184:
1181:
1179:
1176:
1174:
1171:
1169:
1166:
1162:
1159:
1157:
1154:
1152:
1149:
1147:
1144:
1143:
1142:
1139:
1137:
1134:
1132:
1129:
1127:
1124:
1120:
1117:
1115:
1114:by definition
1112:
1111:
1110:
1107:
1103:
1100:
1099:
1098:
1095:
1093:
1090:
1088:
1085:
1083:
1080:
1078:
1075:
1074:
1071:
1068:
1066:
1062:
1057:
1051:
1047:
1037:
1034:
1032:
1029:
1027:
1024:
1022:
1019:
1017:
1014:
1012:
1009:
1007:
1004:
1002:
1001:KripkeâPlatek
999:
997:
994:
990:
987:
985:
982:
981:
980:
977:
976:
974:
970:
962:
959:
958:
957:
954:
952:
949:
945:
942:
941:
940:
937:
935:
932:
930:
927:
925:
922:
920:
917:
914:
910:
906:
903:
899:
896:
894:
891:
889:
886:
885:
884:
880:
877:
876:
874:
872:
868:
864:
856:
853:
851:
848:
846:
845:constructible
843:
842:
841:
838:
836:
833:
831:
828:
826:
823:
821:
818:
816:
813:
811:
808:
806:
803:
801:
798:
796:
793:
791:
788:
786:
783:
781:
778:
777:
775:
773:
768:
760:
757:
755:
752:
750:
747:
745:
742:
740:
737:
735:
732:
731:
729:
725:
722:
720:
717:
716:
715:
712:
710:
707:
705:
702:
700:
697:
695:
691:
687:
685:
682:
678:
675:
674:
673:
670:
669:
666:
663:
661:
657:
647:
644:
642:
639:
637:
634:
632:
629:
627:
624:
622:
619:
615:
612:
611:
610:
607:
603:
598:
597:
596:
593:
592:
590:
588:
584:
576:
573:
571:
568:
566:
563:
562:
561:
558:
556:
553:
551:
548:
546:
543:
541:
538:
536:
533:
531:
528:
527:
525:
523:
522:Propositional
519:
513:
510:
508:
505:
503:
500:
498:
495:
493:
490:
488:
485:
481:
478:
477:
476:
473:
471:
468:
466:
463:
461:
458:
456:
453:
451:
450:Logical truth
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1612:Model theory
1577:Independence
1513:Formal proof
1505:Proof theory
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1418:real numbers
1390:second-order
1301:Substitution
1178:Metalanguage
1119:conservative
1092:Axiom schema
1036:Constructive
1006:MorseâKelley
972:Set theories
951:Aleph number
944:inaccessible
850:Grothendieck
734:intersection
621:Higher-order
609:Second-order
555:Truth tables
512:Venn diagram
295:Formal proof
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1823:undecidable
1755:Truth value
1642:equivalence
1321:non-logical
934:Enumeration
924:Isomorphism
871:cardinality
855:Von Neumann
820:Ultrafilter
785:Uncountable
719:equivalence
636:Quantifiers
626:Fixed-point
595:First-order
475:Consistency
460:Proposition
437:Traditional
408:Lindström's
398:Compactness
340:Type theory
285:Cardinality
137:pregeometry
118:minimal set
1686:elementary
1379:arithmetic
1247:Quantifier
1225:functional
1097:Expression
815:Transitive
759:identities
744:complement
677:hereditary
660:Set theory
214:(2): 147,
169:References
1957:Supertask
1860:Recursion
1818:decidable
1652:saturated
1630:of models
1553:deductive
1548:axiomatic
1468:Hilbert's
1455:Euclidean
1436:canonical
1359:axiomatic
1291:Signature
1220:Predicate
1109:Extension
1031:Ackermann
956:Operation
835:Universal
825:Recursive
800:Singleton
795:Inhabited
780:Countable
770:Types of
754:power set
724:partition
641:Predicate
587:Predicate
502:Syllogism
492:Soundness
465:Inference
455:Tautology
357:paradoxes
27:—a
1990:Category
1942:Logicism
1935:timeline
1911:Concrete
1770:Validity
1740:T-schema
1733:Kripke's
1728:Tarski's
1723:semantic
1713:Strength
1662:submodel
1657:spectrum
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1462:Elements
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1326:variable
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1173:Language
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1087:Automata
1077:Alphabet
1061:language
915:-jection
893:codomain
879:Function
840:Universe
810:Infinite
714:Relation
497:Validity
487:Argument
385:theorem,
152:See also
41:cofinite
1884:Related
1681:Diagram
1579: (
1558:Hilbert
1543:Systems
1538:Theorem
1416:of the
1361:systems
1141:Formula
1136:Grammar
1052: (
996:General
709:Forcing
694:Element
614:Monadic
389:paradox
330:Theorem
266:General
197:2271517
1647:finite
1410:Skolem
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1338:Theory
1306:Symbol
1296:String
1279:atomic
1156:ground
1151:closed
1146:atomic
1102:ground
1065:syntax
961:binary
888:domain
805:Finite
570:finite
428:Logics
387:
335:Theory
195:
103:curves
1637:Model
1385:Peano
1242:Proof
1082:Arity
1011:Naive
898:image
830:Fuzzy
790:Empty
739:union
684:Class
325:Model
315:Lemma
273:Axiom
193:JSTOR
47:is a
1760:Type
1563:list
1367:list
1344:list
1333:Term
1267:rank
1161:open
1055:list
867:Maps
772:sets
631:Free
601:list
351:list
278:list
105:").
43:. A
1447:of
1429:of
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909:Sur
883:Map
690:Ur-
672:Set
216:doi
185:doi
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