2294:
122:
93:
that is model complete. Robinson proved that a theory has at most one model companion. Not every theory is model-companionable, e.g. theory of groups. However if
673:
521:
1348:
1431:
572:
299:(in a language with symbols for the identity, product, and inverses) has the amalgamation property but does not have a model companion.
1745:
252:
is the theory of equivalence relations with infinitely many equivalence classes, each containing an infinite number of elements.
1903:
470:
441:
409:
691:
1758:
1081:
1763:
1753:
1490:
1343:
696:
687:
1899:
503:
1241:
1996:
1740:
565:
1301:
994:
17:
735:
2257:
1959:
1722:
1717:
1542:
963:
647:
46:. Equivalently, every first-order formula is equivalent to a universal formula. This notion was introduced by
2252:
2035:
1952:
1665:
1596:
1473:
715:
495:
2177:
2003:
1689:
1323:
922:
156:
2318:
2055:
2050:
1660:
1399:
1328:
657:
558:
1984:
1574:
968:
936:
627:
242:
235:
222:
2274:
2223:
2120:
1618:
1579:
1056:
701:
538:
730:
2115:
2045:
1584:
1436:
1419:
1142:
622:
292:
The theory of dense linear orders with a first and last element is complete but not model complete.
1947:
1924:
1885:
1771:
1712:
1358:
1278:
1122:
1066:
679:
100:
2237:
1964:
1942:
1909:
1802:
1648:
1633:
1606:
1557:
1441:
1376:
1201:
1167:
1162:
1036:
867:
844:
2323:
2167:
1812:
1530:
1266:
1172:
1031:
1016:
897:
872:
425:
397:
203:
2140:
2102:
1979:
1783:
1623:
1547:
1525:
1353:
1311:
1210:
1177:
1041:
829:
740:
480:
249:
43:
8:
2269:
2160:
2145:
2125:
2082:
1969:
1919:
1845:
1790:
1727:
1520:
1515:
1463:
1231:
1220:
892:
792:
720:
711:
707:
642:
637:
453:
Hirschfeld, Joram; Wheeler, William H. (1975). "Model-completions and model-companions".
296:
279:
2298:
2067:
2030:
2015:
2008:
1991:
1795:
1777:
1643:
1569:
1552:
1505:
1318:
1227:
1061:
1046:
1006:
958:
943:
931:
887:
862:
632:
581:
433:
275:
245:
is the model completion of the theory of fields. It is model complete but not complete.
125:
1251:
2293:
2233:
2040:
1850:
1840:
1732:
1613:
1448:
1424:
1205:
1189:
1094:
1071:
948:
917:
882:
777:
612:
499:
466:
437:
405:
256:
35:
2247:
2242:
2135:
2092:
1914:
1875:
1870:
1855:
1681:
1638:
1535:
1333:
1283:
857:
819:
458:
421:
393:
303:
47:
2228:
2218:
2172:
2155:
2110:
2072:
1974:
1894:
1701:
1628:
1601:
1589:
1495:
1409:
1383:
1338:
1306:
1107:
909:
852:
802:
767:
725:
476:
268:
164:
404:. Studies in Logic and the Foundations of Mathematics (3rd ed.). Elsevier.
2213:
2192:
2150:
2130:
2025:
1880:
1478:
1468:
1458:
1453:
1387:
1261:
1137:
1026:
1021:
999:
600:
2312:
2187:
1865:
1372:
1157:
1147:
1117:
1102:
772:
264:
214:
also has universal axiomatization, both of the above are also equivalent to:
2087:
1934:
1835:
1827:
1707:
1655:
1564:
1500:
1483:
1414:
1273:
1132:
834:
617:
260:
31:
2197:
2077:
1256:
1246:
1193:
877:
797:
782:
662:
607:
457:. Lecture Notes in Mathematics. Vol. 454. Springer. pp. 44â54.
1127:
982:
953:
759:
536:
Simmons, H. (1976). "Large and Small
Existentially Closed Structures".
462:
2279:
2182:
1235:
1152:
1112:
1076:
1012:
824:
814:
787:
550:
2264:
2062:
1510:
1215:
809:
1860:
652:
304:
Sufficient condition for completeness of model-complete theories
1404:
750:
595:
358:
53:
274:
The theory of real closed fields, in the language of
103:
515:
Saracino, D. (August 1973). "Model
Companions for â”
334:
312:is a model complete theory and there is a model of
346:
116:
452:
370:
2310:
522:Proceedings of the American Mathematical Society
186:then the following conditions are equivalent:
167:. Roughly speaking, this means every model of
566:
432:. Dover Books on Mathematics (3rd ed.).
420:
392:
364:
278:, is the model companion for the theory of
758:
573:
559:
263:, is a model completion of the theory of
514:
340:
128:, then it always has a model companion.
535:
352:
42:if every embedding of its models is an
14:
2311:
580:
489:
376:
248:The model completion of the theory of
554:
54:Model companion and model completion
455:Forcing, Arithmetic, Division Rings
24:
105:
25:
2335:
2292:
498:217. New York: Springer-Verlag.
282:, but is not a model completion.
286:
74:can be embedded in a model of
316:that embeds into any model of
13:
1:
2253:History of mathematical logic
496:Graduate Texts in Mathematics
492:Model Theory: An Introduction
386:
2178:Primitive recursive function
171:is embeddable in a model of
7:
243:algebraically closed fields
228:
193:* is a model completion of
117:{\displaystyle \aleph _{0}}
70:* such that every model of
10:
2340:
1242:SchröderâBernstein theorem
969:Monadic predicate calculus
628:Foundations of mathematics
236:elimination of quantifiers
223:elimination of quantifiers
182:* is a model companion of
143:* such that for any model
2288:
2275:Philosophy of mathematics
2224:Automated theorem proving
2206:
2101:
1933:
1826:
1678:
1395:
1371:
1349:Von NeumannâBernaysâGödel
1294:
1188:
1092:
990:
981:
908:
843:
749:
671:
588:
539:Journal of Symbolic Logic
519:-Categorical Theories".
365:Chang & Keisler 2012
327:
1925:Self-verifying theories
1746:Tarski's axiomatization
697:Tarski's undefinability
692:incompleteness theorems
27:Concept in model theory
2299:Mathematics portal
1910:Proof of impossibility
1558:propositional variable
868:Propositional calculus
490:Marker, David (2002).
118:
2168:Kolmogorov complexity
2121:Computably enumerable
2021:Model complete theory
1813:Principia Mathematica
873:Propositional formula
702:BanachâTarski paradox
259:, in the language of
250:equivalence relations
204:amalgamation property
139:is a model companion
119:
2116:ChurchâTuring thesis
2103:Computability theory
1312:continuum hypothesis
830:Square of opposition
688:Gödel's completeness
280:formally real fields
155:* together with the
101:
44:elementary embedding
2270:Mathematical object
2161:P versus NP problem
2126:Computable function
1920:Reverse mathematics
1846:Logical consequence
1723:primitive recursive
1718:elementary function
1491:Free/bound variable
1344:TarskiâGrothendieck
863:Logical connectives
793:Logical equivalence
643:Logical consequence
175:* in a unique way.
2319:Mathematical logic
2068:Transfer principle
2031:Semantics of logic
2016:Categorical theory
1992:Non-standard model
1506:Logical connective
633:Information theory
582:Mathematical logic
463:10.1007/BFb0064085
434:Dover Publications
426:Keisler, H. Jerome
398:Keisler, H. Jerome
257:real closed fields
238:is model complete.
126:categorical theory
114:
89:is a companion of
78:* and vice versa.
2306:
2305:
2238:Abstract category
2041:Theories of truth
1851:Rule of inference
1841:Natural deduction
1822:
1821:
1367:
1366:
1072:Cartesian product
977:
976:
883:Many-valued logic
858:Boolean functions
741:Russell's paradox
716:diagonal argument
613:First-order logic
472:978-3-540-07157-0
443:978-0-486-48821-9
422:Chang, Chen Chung
411:978-0-444-88054-3
394:Chang, Chen Chung
267:(or even ordered
38:theory is called
16:(Redirected from
2331:
2297:
2296:
2248:History of logic
2243:Category of sets
2136:Decision problem
1915:Ordinal analysis
1856:Sequent calculus
1754:Boolean algebras
1694:
1693:
1668:
1639:logical/constant
1393:
1392:
1379:
1302:ZermeloâFraenkel
1053:Set operations:
988:
987:
925:
756:
755:
736:LöwenheimâSkolem
623:Formal semantics
575:
568:
561:
552:
551:
547:
530:
509:
484:
447:
415:
380:
374:
368:
362:
356:
350:
344:
338:
234:Any theory with
151:, the theory of
133:model completion
123:
121:
120:
115:
113:
112:
48:Abraham Robinson
21:
2339:
2338:
2334:
2333:
2332:
2330:
2329:
2328:
2309:
2308:
2307:
2302:
2291:
2284:
2229:Category theory
2219:Algebraic logic
2202:
2173:Lambda calculus
2111:Church encoding
2097:
2073:Truth predicate
1929:
1895:Complete theory
1818:
1687:
1683:
1679:
1674:
1666:
1386: and
1382:
1377:
1363:
1339:New Foundations
1307:axiom of choice
1290:
1252:Gödel numbering
1192: and
1184:
1088:
973:
923:
904:
853:Boolean algebra
839:
803:Equiconsistency
768:Classical logic
745:
726:Halting problem
714: and
690: and
678: and
677:
672:Theorems (
667:
584:
579:
518:
506:
473:
444:
436:. p. 672.
412:
389:
384:
383:
375:
371:
363:
359:
351:
347:
339:
335:
330:
306:
289:
231:
108:
104:
102:
99:
98:
83:model companion
56:
28:
23:
22:
18:Model companion
15:
12:
11:
5:
2337:
2327:
2326:
2321:
2304:
2303:
2289:
2286:
2285:
2283:
2282:
2277:
2272:
2267:
2262:
2261:
2260:
2250:
2245:
2240:
2231:
2226:
2221:
2216:
2214:Abstract logic
2210:
2208:
2204:
2203:
2201:
2200:
2195:
2193:Turing machine
2190:
2185:
2180:
2175:
2170:
2165:
2164:
2163:
2158:
2153:
2148:
2143:
2133:
2131:Computable set
2128:
2123:
2118:
2113:
2107:
2105:
2099:
2098:
2096:
2095:
2090:
2085:
2080:
2075:
2070:
2065:
2060:
2059:
2058:
2053:
2048:
2038:
2033:
2028:
2026:Satisfiability
2023:
2018:
2013:
2012:
2011:
2001:
2000:
1999:
1989:
1988:
1987:
1982:
1977:
1972:
1967:
1957:
1956:
1955:
1950:
1943:Interpretation
1939:
1937:
1931:
1930:
1928:
1927:
1922:
1917:
1912:
1907:
1897:
1892:
1891:
1890:
1889:
1888:
1878:
1873:
1863:
1858:
1853:
1848:
1843:
1838:
1832:
1830:
1824:
1823:
1820:
1819:
1817:
1816:
1808:
1807:
1806:
1805:
1800:
1799:
1798:
1793:
1788:
1768:
1767:
1766:
1764:minimal axioms
1761:
1750:
1749:
1748:
1737:
1736:
1735:
1730:
1725:
1720:
1715:
1710:
1697:
1695:
1676:
1675:
1673:
1672:
1671:
1670:
1658:
1653:
1652:
1651:
1646:
1641:
1636:
1626:
1621:
1616:
1611:
1610:
1609:
1604:
1594:
1593:
1592:
1587:
1582:
1577:
1567:
1562:
1561:
1560:
1555:
1550:
1540:
1539:
1538:
1533:
1528:
1523:
1518:
1513:
1503:
1498:
1493:
1488:
1487:
1486:
1481:
1476:
1471:
1461:
1456:
1454:Formation rule
1451:
1446:
1445:
1444:
1439:
1429:
1428:
1427:
1417:
1412:
1407:
1402:
1396:
1390:
1373:Formal systems
1369:
1368:
1365:
1364:
1362:
1361:
1356:
1351:
1346:
1341:
1336:
1331:
1326:
1321:
1316:
1315:
1314:
1309:
1298:
1296:
1292:
1291:
1289:
1288:
1287:
1286:
1276:
1271:
1270:
1269:
1262:Large cardinal
1259:
1254:
1249:
1244:
1239:
1225:
1224:
1223:
1218:
1213:
1198:
1196:
1186:
1185:
1183:
1182:
1181:
1180:
1175:
1170:
1160:
1155:
1150:
1145:
1140:
1135:
1130:
1125:
1120:
1115:
1110:
1105:
1099:
1097:
1090:
1089:
1087:
1086:
1085:
1084:
1079:
1074:
1069:
1064:
1059:
1051:
1050:
1049:
1044:
1034:
1029:
1027:Extensionality
1024:
1022:Ordinal number
1019:
1009:
1004:
1003:
1002:
991:
985:
979:
978:
975:
974:
972:
971:
966:
961:
956:
951:
946:
941:
940:
939:
929:
928:
927:
914:
912:
906:
905:
903:
902:
901:
900:
895:
890:
880:
875:
870:
865:
860:
855:
849:
847:
841:
840:
838:
837:
832:
827:
822:
817:
812:
807:
806:
805:
795:
790:
785:
780:
775:
770:
764:
762:
753:
747:
746:
744:
743:
738:
733:
728:
723:
718:
706:Cantor's
704:
699:
694:
684:
682:
669:
668:
666:
665:
660:
655:
650:
645:
640:
635:
630:
625:
620:
615:
610:
605:
604:
603:
592:
590:
586:
585:
578:
577:
570:
563:
555:
549:
548:
532:
531:
516:
511:
510:
504:
486:
485:
471:
449:
448:
442:
417:
416:
410:
388:
385:
382:
381:
369:
357:
345:
332:
331:
329:
326:
305:
302:
301:
300:
295:The theory of
293:
288:
285:
284:
283:
272:
265:ordered fields
255:The theory of
253:
246:
241:The theory of
239:
230:
227:
226:
225:
208:
207:
197:
111:
107:
55:
52:
40:model complete
26:
9:
6:
4:
3:
2:
2336:
2325:
2322:
2320:
2317:
2316:
2314:
2301:
2300:
2295:
2287:
2281:
2278:
2276:
2273:
2271:
2268:
2266:
2263:
2259:
2256:
2255:
2254:
2251:
2249:
2246:
2244:
2241:
2239:
2235:
2232:
2230:
2227:
2225:
2222:
2220:
2217:
2215:
2212:
2211:
2209:
2205:
2199:
2196:
2194:
2191:
2189:
2188:Recursive set
2186:
2184:
2181:
2179:
2176:
2174:
2171:
2169:
2166:
2162:
2159:
2157:
2154:
2152:
2149:
2147:
2144:
2142:
2139:
2138:
2137:
2134:
2132:
2129:
2127:
2124:
2122:
2119:
2117:
2114:
2112:
2109:
2108:
2106:
2104:
2100:
2094:
2091:
2089:
2086:
2084:
2081:
2079:
2076:
2074:
2071:
2069:
2066:
2064:
2061:
2057:
2054:
2052:
2049:
2047:
2044:
2043:
2042:
2039:
2037:
2034:
2032:
2029:
2027:
2024:
2022:
2019:
2017:
2014:
2010:
2007:
2006:
2005:
2002:
1998:
1997:of arithmetic
1995:
1994:
1993:
1990:
1986:
1983:
1981:
1978:
1976:
1973:
1971:
1968:
1966:
1963:
1962:
1961:
1958:
1954:
1951:
1949:
1946:
1945:
1944:
1941:
1940:
1938:
1936:
1932:
1926:
1923:
1921:
1918:
1916:
1913:
1911:
1908:
1905:
1904:from ZFC
1901:
1898:
1896:
1893:
1887:
1884:
1883:
1882:
1879:
1877:
1874:
1872:
1869:
1868:
1867:
1864:
1862:
1859:
1857:
1854:
1852:
1849:
1847:
1844:
1842:
1839:
1837:
1834:
1833:
1831:
1829:
1825:
1815:
1814:
1810:
1809:
1804:
1803:non-Euclidean
1801:
1797:
1794:
1792:
1789:
1787:
1786:
1782:
1781:
1779:
1776:
1775:
1773:
1769:
1765:
1762:
1760:
1757:
1756:
1755:
1751:
1747:
1744:
1743:
1742:
1738:
1734:
1731:
1729:
1726:
1724:
1721:
1719:
1716:
1714:
1711:
1709:
1706:
1705:
1703:
1699:
1698:
1696:
1691:
1685:
1680:Example
1677:
1669:
1664:
1663:
1662:
1659:
1657:
1654:
1650:
1647:
1645:
1642:
1640:
1637:
1635:
1632:
1631:
1630:
1627:
1625:
1622:
1620:
1617:
1615:
1612:
1608:
1605:
1603:
1600:
1599:
1598:
1595:
1591:
1588:
1586:
1583:
1581:
1578:
1576:
1573:
1572:
1571:
1568:
1566:
1563:
1559:
1556:
1554:
1551:
1549:
1546:
1545:
1544:
1541:
1537:
1534:
1532:
1529:
1527:
1524:
1522:
1519:
1517:
1514:
1512:
1509:
1508:
1507:
1504:
1502:
1499:
1497:
1494:
1492:
1489:
1485:
1482:
1480:
1477:
1475:
1472:
1470:
1467:
1466:
1465:
1462:
1460:
1457:
1455:
1452:
1450:
1447:
1443:
1440:
1438:
1437:by definition
1435:
1434:
1433:
1430:
1426:
1423:
1422:
1421:
1418:
1416:
1413:
1411:
1408:
1406:
1403:
1401:
1398:
1397:
1394:
1391:
1389:
1385:
1380:
1374:
1370:
1360:
1357:
1355:
1352:
1350:
1347:
1345:
1342:
1340:
1337:
1335:
1332:
1330:
1327:
1325:
1324:KripkeâPlatek
1322:
1320:
1317:
1313:
1310:
1308:
1305:
1304:
1303:
1300:
1299:
1297:
1293:
1285:
1282:
1281:
1280:
1277:
1275:
1272:
1268:
1265:
1264:
1263:
1260:
1258:
1255:
1253:
1250:
1248:
1245:
1243:
1240:
1237:
1233:
1229:
1226:
1222:
1219:
1217:
1214:
1212:
1209:
1208:
1207:
1203:
1200:
1199:
1197:
1195:
1191:
1187:
1179:
1176:
1174:
1171:
1169:
1168:constructible
1166:
1165:
1164:
1161:
1159:
1156:
1154:
1151:
1149:
1146:
1144:
1141:
1139:
1136:
1134:
1131:
1129:
1126:
1124:
1121:
1119:
1116:
1114:
1111:
1109:
1106:
1104:
1101:
1100:
1098:
1096:
1091:
1083:
1080:
1078:
1075:
1073:
1070:
1068:
1065:
1063:
1060:
1058:
1055:
1054:
1052:
1048:
1045:
1043:
1040:
1039:
1038:
1035:
1033:
1030:
1028:
1025:
1023:
1020:
1018:
1014:
1010:
1008:
1005:
1001:
998:
997:
996:
993:
992:
989:
986:
984:
980:
970:
967:
965:
962:
960:
957:
955:
952:
950:
947:
945:
942:
938:
935:
934:
933:
930:
926:
921:
920:
919:
916:
915:
913:
911:
907:
899:
896:
894:
891:
889:
886:
885:
884:
881:
879:
876:
874:
871:
869:
866:
864:
861:
859:
856:
854:
851:
850:
848:
846:
845:Propositional
842:
836:
833:
831:
828:
826:
823:
821:
818:
816:
813:
811:
808:
804:
801:
800:
799:
796:
794:
791:
789:
786:
784:
781:
779:
776:
774:
773:Logical truth
771:
769:
766:
765:
763:
761:
757:
754:
752:
748:
742:
739:
737:
734:
732:
729:
727:
724:
722:
719:
717:
713:
709:
705:
703:
700:
698:
695:
693:
689:
686:
685:
683:
681:
675:
670:
664:
661:
659:
656:
654:
651:
649:
646:
644:
641:
639:
636:
634:
631:
629:
626:
624:
621:
619:
616:
614:
611:
609:
606:
602:
599:
598:
597:
594:
593:
591:
587:
583:
576:
571:
569:
564:
562:
557:
556:
553:
546:(2): 379â390.
545:
541:
540:
534:
533:
529:(3): 591â598.
528:
524:
523:
513:
512:
507:
505:0-387-98760-6
501:
497:
493:
488:
487:
482:
478:
474:
468:
464:
460:
456:
451:
450:
445:
439:
435:
431:
427:
423:
419:
418:
413:
407:
403:
399:
395:
391:
390:
378:
373:
366:
361:
354:
349:
342:
341:Saracino 1973
337:
333:
325:
324:is complete.
323:
319:
315:
311:
298:
294:
291:
290:
281:
277:
273:
270:
266:
262:
261:ordered rings
258:
254:
251:
247:
244:
240:
237:
233:
232:
224:
220:
217:
216:
215:
213:
205:
201:
198:
196:
192:
189:
188:
187:
185:
181:
176:
174:
170:
166:
162:
158:
154:
150:
146:
142:
138:
135:for a theory
134:
129:
127:
109:
96:
92:
88:
84:
79:
77:
73:
69:
66:is a theory
65:
62:of a theory
61:
51:
49:
45:
41:
37:
33:
19:
2324:Model theory
2290:
2088:Ultraproduct
2020:
1935:Model theory
1900:Independence
1836:Formal proof
1828:Proof theory
1811:
1784:
1741:real numbers
1713:second-order
1624:Substitution
1501:Metalanguage
1442:conservative
1415:Axiom schema
1359:Constructive
1329:MorseâKelley
1295:Set theories
1274:Aleph number
1267:inaccessible
1173:Grothendieck
1057:intersection
944:Higher-order
932:Second-order
878:Truth tables
835:Venn diagram
618:Formal proof
543:
537:
526:
520:
491:
454:
430:Model Theory
429:
402:Model Theory
401:
372:
360:
353:Simmons 1976
348:
336:
321:
317:
313:
309:
307:
287:Non-examples
218:
211:
209:
199:
194:
190:
183:
179:
177:
172:
168:
160:
152:
148:
144:
140:
136:
132:
130:
94:
90:
86:
85:of a theory
82:
80:
75:
71:
67:
63:
59:
57:
39:
32:model theory
29:
2198:Type theory
2146:undecidable
2078:Truth value
1965:equivalence
1644:non-logical
1257:Enumeration
1247:Isomorphism
1194:cardinality
1178:Von Neumann
1143:Ultrafilter
1108:Uncountable
1042:equivalence
959:Quantifiers
949:Fixed-point
918:First-order
798:Consistency
783:Proposition
760:Traditional
731:Lindström's
721:Compactness
663:Type theory
608:Cardinality
377:Marker 2002
36:first-order
2313:Categories
2009:elementary
1702:arithmetic
1570:Quantifier
1548:functional
1420:Expression
1138:Transitive
1082:identities
1067:complement
1000:hereditary
983:Set theory
387:References
2280:Supertask
2183:Recursion
2141:decidable
1975:saturated
1953:of models
1876:deductive
1871:axiomatic
1791:Hilbert's
1778:Euclidean
1759:canonical
1682:axiomatic
1614:Signature
1543:Predicate
1432:Extension
1354:Ackermann
1279:Operation
1158:Universal
1148:Recursive
1123:Singleton
1118:Inhabited
1103:Countable
1093:Types of
1077:power set
1047:partition
964:Predicate
910:Predicate
825:Syllogism
815:Soundness
788:Inference
778:Tautology
680:paradoxes
428:(2012) .
400:(1990) .
106:ℵ
60:companion
2265:Logicism
2258:timeline
2234:Concrete
2093:Validity
2063:T-schema
2056:Kripke's
2051:Tarski's
2046:semantic
2036:Strength
1985:submodel
1980:spectrum
1948:function
1796:Tarski's
1785:Elements
1772:geometry
1728:Robinson
1649:variable
1634:function
1607:spectrum
1597:Sentence
1553:variable
1496:Language
1449:Relation
1410:Automata
1400:Alphabet
1384:language
1238:-jection
1216:codomain
1202:Function
1163:Universe
1133:Infinite
1037:Relation
820:Validity
810:Argument
708:theorem,
229:Examples
202:has the
165:complete
2207:Related
2004:Diagram
1902: (
1881:Hilbert
1866:Systems
1861:Theorem
1739:of the
1684:systems
1464:Formula
1459:Grammar
1375: (
1319:General
1032:Forcing
1017:Element
937:Monadic
712:paradox
653:Theorem
589:General
481:0389581
320:, then
269:domains
157:diagram
1970:finite
1733:Skolem
1686:
1661:Theory
1629:Symbol
1619:String
1602:atomic
1479:ground
1474:closed
1469:atomic
1425:ground
1388:syntax
1284:binary
1211:domain
1128:Finite
893:finite
751:Logics
710:
658:Theory
502:
479:
469:
440:
408:
297:groups
221:* has
97:is an
1960:Model
1708:Peano
1565:Proof
1405:Arity
1334:Naive
1221:image
1153:Fuzzy
1113:Empty
1062:union
1007:Class
648:Model
638:Lemma
596:Axiom
328:Notes
276:rings
2083:Type
1886:list
1690:list
1667:list
1656:Term
1590:rank
1484:open
1378:list
1190:Maps
1095:sets
954:Free
924:list
674:list
601:list
500:ISBN
467:ISBN
438:ISBN
406:ISBN
178:If
34:, a
1770:of
1752:of
1700:of
1232:Sur
1206:Map
1013:Ur-
995:Set
459:doi
308:If
210:If
163:is
159:of
147:of
30:In
2315::
2156:NP
1780::
1774::
1704::
1381:),
1236:Bi
1228:In
544:41
542:.
527:39
525:.
494:.
477:MR
475:.
465:.
424:;
396:;
271:).
131:A
81:A
58:A
50:.
2236:/
2151:P
1906:)
1692:)
1688:(
1585:â
1580:!
1575:â
1536:=
1531:â
1526:â
1521:â§
1516:âš
1511:ÂŹ
1234:/
1230:/
1204:/
1015:)
1011:(
898:â
888:3
676:)
574:e
567:t
560:v
517:0
508:.
483:.
461::
446:.
414:.
379:.
367:.
355:.
343:.
322:T
318:T
314:T
310:T
219:T
212:T
206:.
200:T
195:T
191:T
184:T
180:T
173:T
169:T
161:M
153:T
149:T
145:M
141:T
137:T
124:-
110:0
95:T
91:T
87:T
76:T
72:T
68:T
64:T
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.