2517:
693:
385:
258:
164:
471:
896:
1571:
1654:
795:
1968:
2126:
914:
1981:
1304:
597:
1986:
1976:
1713:
1566:
919:
910:
35:
with certain properties. Algebraically closed fields with a (Krull) valuation are perhaps the most important example.
2122:
1464:
2219:
1963:
788:
1524:
1217:
958:
2480:
2182:
1945:
1940:
1765:
1186:
870:
481:
2475:
2258:
2175:
1888:
1819:
1696:
938:
2400:
2226:
1912:
1546:
1145:
2278:
2273:
1883:
1622:
1551:
880:
781:
2207:
1797:
1191:
1159:
850:
2497:
2446:
2343:
1841:
1802:
1279:
924:
553:
if its theory is C-minimal. One can construct C-minimal structures which are not strongly C-minimal.
264:
953:
170:
79:
2338:
2268:
1807:
1659:
1642:
1365:
845:
391:
2170:
2147:
2108:
1994:
1935:
1581:
1501:
1345:
1289:
902:
2460:
2187:
2165:
2132:
2025:
1871:
1856:
1829:
1780:
1664:
1599:
1424:
1390:
1385:
1259:
1090:
1067:
2541:
2390:
2243:
2035:
1753:
1489:
1395:
1254:
1239:
1120:
1095:
2363:
2325:
2202:
2006:
1846:
1770:
1748:
1576:
1534:
1433:
1400:
1264:
1052:
963:
8:
2492:
2383:
2368:
2348:
2305:
2192:
2142:
2068:
2013:
1950:
1743:
1738:
1686:
1454:
1443:
1115:
1015:
943:
934:
930:
865:
860:
2521:
2290:
2253:
2238:
2231:
2214:
2018:
2000:
1866:
1792:
1775:
1728:
1541:
1450:
1284:
1269:
1229:
1181:
1166:
1154:
1110:
1085:
855:
804:
21:
1474:
2516:
2456:
2263:
2073:
2063:
1955:
1836:
1671:
1647:
1428:
1412:
1317:
1294:
1171:
1140:
1105:
1000:
835:
767:
751:
743:
727:
488:
2470:
2465:
2358:
2315:
2137:
2098:
2093:
2078:
1904:
1861:
1758:
1556:
1506:
1080:
1042:
763:
739:
39:
29:
2451:
2441:
2395:
2378:
2333:
2295:
2197:
2117:
1924:
1851:
1824:
1812:
1718:
1632:
1606:
1561:
1529:
1330:
1132:
1075:
1025:
990:
948:
2436:
2415:
2373:
2353:
2248:
2103:
1701:
1691:
1681:
1676:
1610:
1484:
1360:
1249:
1244:
1222:
823:
2535:
2410:
2088:
1595:
1380:
1370:
1340:
1325:
995:
569:
2310:
2157:
2058:
2050:
1930:
1878:
1787:
1723:
1706:
1637:
1496:
1355:
1057:
840:
562:
17:
42:, which are "minimal" (in the same sense) with respect to a linear order.
2420:
2300:
1479:
1469:
1416:
1100:
1020:
1005:
885:
830:
1350:
1205:
1176:
982:
2502:
2405:
1458:
1375:
1335:
1299:
1235:
1047:
1037:
1010:
773:
2487:
2285:
1733:
1438:
1032:
2083:
875:
1627:
973:
818:
730:; Steinhorn, Charles (1996), "On variants of o-minimality",
708:
with addition and this relation is C-minimal. The theory of
549:
if all of its models are C-minimal. A structure is called
499:
satisfies the above axioms and every set of elements of
754:(1994), "Cell decompositions of C-minimal structures",
600:
394:
267:
173:
82:
726:
688:{\displaystyle C(a;b,c)\iff |b-c|_{p}<|a-c|_{p}}
594:-adic absolute value. Then the relation defined by
687:
465:
379:
252:
158:
749:
28:is a theory that is "minimal" with respect to a
2533:
789:
981:
796:
782:
629:
625:
38:This notion was defined in analogy to the
507:is a Boolean combination of instances of
432:
404:
283:
186:
95:
717:as a field, however, is not C-minimal.
2534:
803:
777:
503:that is definable with parameters in
73:that satisfies the following axioms.
13:
420:
395:
268:
217:
174:
83:
14:
2553:
2515:
756:Annals of Pure and Applied Logic
732:Annals of Pure and Applied Logic
54:-relation is a ternary relation
511:, i.e. of formulas of the form
380:{\displaystyle \forall xyzw\,,}
675:
660:
646:
631:
626:
622:
604:
457:
454:
436:
417:
405:
371:
368:
365:
347:
338:
320:
314:
311:
308:
290:
284:
253:{\displaystyle \forall xyz\,,}
244:
241:
223:
214:
211:
193:
187:
159:{\displaystyle \forall xyz\,,}
150:
147:
129:
123:
120:
102:
96:
1:
2476:History of mathematical logic
720:
699:-relation, and the theory of
466:{\displaystyle \forall xy\,.}
45:
2401:Primitive recursive function
768:10.1016/0168-0072(94)90064-7
744:10.1016/0168-0072(95)00037-2
7:
10:
2558:
1465:SchröderâBernstein theorem
1192:Monadic predicate calculus
851:Foundations of mathematics
556:
2511:
2498:Philosophy of mathematics
2447:Automated theorem proving
2429:
2324:
2156:
2049:
1901:
1618:
1594:
1572:Von NeumannâBernaysâGödel
1517:
1411:
1315:
1213:
1204:
1131:
1066:
972:
894:
811:
2148:Self-verifying theories
1969:Tarski's axiomatization
920:Tarski's undefinability
915:incompleteness theorems
2522:Mathematics portal
2133:Proof of impossibility
1781:propositional variable
1091:Propositional calculus
689:
491:containing the symbol
467:
381:
254:
160:
2391:Kolmogorov complexity
2344:Computably enumerable
2244:Model complete theory
2036:Principia Mathematica
1096:Propositional formula
925:BanachâTarski paradox
690:
468:
382:
255:
161:
2339:ChurchâTuring thesis
2326:Computability theory
1535:continuum hypothesis
1053:Square of opposition
911:Gödel's completeness
598:
392:
265:
171:
80:
2493:Mathematical object
2384:P versus NP problem
2349:Computable function
2143:Reverse mathematics
2069:Logical consequence
1946:primitive recursive
1941:elementary function
1714:Free/bound variable
1567:TarskiâGrothendieck
1086:Logical connectives
1016:Logical equivalence
866:Logical consequence
545:A theory is called
478:C-minimal structure
2291:Transfer principle
2254:Semantics of logic
2239:Categorical theory
2215:Non-standard model
1729:Logical connective
856:Information theory
805:Mathematical logic
752:Macpherson, Dugald
750:Haskell, Deirdre;
728:Macpherson, Dugald
685:
551:strongly C-minimal
463:
377:
250:
156:
40:o-minimal theories
22:mathematical logic
2529:
2528:
2461:Abstract category
2264:Theories of truth
2074:Rule of inference
2064:Natural deduction
2045:
2044:
1590:
1589:
1295:Cartesian product
1200:
1199:
1106:Many-valued logic
1081:Boolean functions
964:Russell's paradox
939:diagonal argument
836:First-order logic
2549:
2520:
2519:
2471:History of logic
2466:Category of sets
2359:Decision problem
2138:Ordinal analysis
2079:Sequent calculus
1977:Boolean algebras
1917:
1916:
1891:
1862:logical/constant
1616:
1615:
1602:
1525:ZermeloâFraenkel
1276:Set operations:
1211:
1210:
1148:
979:
978:
959:LöwenheimâSkolem
846:Formal semantics
798:
791:
784:
775:
774:
770:
746:
694:
692:
691:
686:
684:
683:
678:
663:
655:
654:
649:
634:
583:
538:are elements of
529:
472:
470:
469:
464:
386:
384:
383:
378:
259:
257:
256:
251:
165:
163:
162:
157:
72:
30:ternary relation
26:C-minimal theory
2557:
2556:
2552:
2551:
2550:
2548:
2547:
2546:
2532:
2531:
2530:
2525:
2514:
2507:
2452:Category theory
2442:Algebraic logic
2425:
2396:Lambda calculus
2334:Church encoding
2320:
2296:Truth predicate
2152:
2118:Complete theory
2041:
1910:
1906:
1902:
1897:
1889:
1609: and
1605:
1600:
1586:
1562:New Foundations
1530:axiom of choice
1513:
1475:Gödel numbering
1415: and
1407:
1311:
1196:
1146:
1127:
1076:Boolean algebra
1062:
1026:Equiconsistency
991:Classical logic
968:
949:Halting problem
937: and
913: and
901: and
900:
895:Theorems (
890:
807:
802:
723:
716:
707:
679:
674:
673:
659:
650:
645:
644:
630:
599:
596:
595:
589:
579:
559:
512:
393:
390:
389:
266:
263:
262:
172:
169:
168:
81:
78:
77:
55:
48:
12:
11:
5:
2555:
2545:
2544:
2527:
2526:
2512:
2509:
2508:
2506:
2505:
2500:
2495:
2490:
2485:
2484:
2483:
2473:
2468:
2463:
2454:
2449:
2444:
2439:
2437:Abstract logic
2433:
2431:
2427:
2426:
2424:
2423:
2418:
2416:Turing machine
2413:
2408:
2403:
2398:
2393:
2388:
2387:
2386:
2381:
2376:
2371:
2366:
2356:
2354:Computable set
2351:
2346:
2341:
2336:
2330:
2328:
2322:
2321:
2319:
2318:
2313:
2308:
2303:
2298:
2293:
2288:
2283:
2282:
2281:
2276:
2271:
2261:
2256:
2251:
2249:Satisfiability
2246:
2241:
2236:
2235:
2234:
2224:
2223:
2222:
2212:
2211:
2210:
2205:
2200:
2195:
2190:
2180:
2179:
2178:
2173:
2166:Interpretation
2162:
2160:
2154:
2153:
2151:
2150:
2145:
2140:
2135:
2130:
2120:
2115:
2114:
2113:
2112:
2111:
2101:
2096:
2086:
2081:
2076:
2071:
2066:
2061:
2055:
2053:
2047:
2046:
2043:
2042:
2040:
2039:
2031:
2030:
2029:
2028:
2023:
2022:
2021:
2016:
2011:
1991:
1990:
1989:
1987:minimal axioms
1984:
1973:
1972:
1971:
1960:
1959:
1958:
1953:
1948:
1943:
1938:
1933:
1920:
1918:
1899:
1898:
1896:
1895:
1894:
1893:
1881:
1876:
1875:
1874:
1869:
1864:
1859:
1849:
1844:
1839:
1834:
1833:
1832:
1827:
1817:
1816:
1815:
1810:
1805:
1800:
1790:
1785:
1784:
1783:
1778:
1773:
1763:
1762:
1761:
1756:
1751:
1746:
1741:
1736:
1726:
1721:
1716:
1711:
1710:
1709:
1704:
1699:
1694:
1684:
1679:
1677:Formation rule
1674:
1669:
1668:
1667:
1662:
1652:
1651:
1650:
1640:
1635:
1630:
1625:
1619:
1613:
1596:Formal systems
1592:
1591:
1588:
1587:
1585:
1584:
1579:
1574:
1569:
1564:
1559:
1554:
1549:
1544:
1539:
1538:
1537:
1532:
1521:
1519:
1515:
1514:
1512:
1511:
1510:
1509:
1499:
1494:
1493:
1492:
1485:Large cardinal
1482:
1477:
1472:
1467:
1462:
1448:
1447:
1446:
1441:
1436:
1421:
1419:
1409:
1408:
1406:
1405:
1404:
1403:
1398:
1393:
1383:
1378:
1373:
1368:
1363:
1358:
1353:
1348:
1343:
1338:
1333:
1328:
1322:
1320:
1313:
1312:
1310:
1309:
1308:
1307:
1302:
1297:
1292:
1287:
1282:
1274:
1273:
1272:
1267:
1257:
1252:
1250:Extensionality
1247:
1245:Ordinal number
1242:
1232:
1227:
1226:
1225:
1214:
1208:
1202:
1201:
1198:
1197:
1195:
1194:
1189:
1184:
1179:
1174:
1169:
1164:
1163:
1162:
1152:
1151:
1150:
1137:
1135:
1129:
1128:
1126:
1125:
1124:
1123:
1118:
1113:
1103:
1098:
1093:
1088:
1083:
1078:
1072:
1070:
1064:
1063:
1061:
1060:
1055:
1050:
1045:
1040:
1035:
1030:
1029:
1028:
1018:
1013:
1008:
1003:
998:
993:
987:
985:
976:
970:
969:
967:
966:
961:
956:
951:
946:
941:
929:Cantor's
927:
922:
917:
907:
905:
892:
891:
889:
888:
883:
878:
873:
868:
863:
858:
853:
848:
843:
838:
833:
828:
827:
826:
815:
813:
809:
808:
801:
800:
793:
786:
778:
772:
771:
762:(2): 113â162,
747:
738:(2): 165â209,
722:
719:
712:
703:
682:
677:
672:
669:
666:
662:
658:
653:
648:
643:
640:
637:
633:
628:
624:
621:
618:
615:
612:
609:
606:
603:
585:
558:
555:
474:
473:
462:
459:
456:
453:
450:
447:
444:
441:
438:
435:
431:
428:
425:
422:
419:
416:
413:
410:
407:
403:
400:
397:
387:
376:
373:
370:
367:
364:
361:
358:
355:
352:
349:
346:
343:
340:
337:
334:
331:
328:
325:
322:
319:
316:
313:
310:
307:
304:
301:
298:
295:
292:
289:
286:
282:
279:
276:
273:
270:
260:
249:
246:
243:
240:
237:
234:
231:
228:
225:
222:
219:
216:
213:
210:
207:
204:
201:
198:
195:
192:
189:
185:
182:
179:
176:
166:
155:
152:
149:
146:
143:
140:
137:
134:
131:
128:
125:
122:
119:
116:
113:
110:
107:
104:
101:
98:
94:
91:
88:
85:
47:
44:
20:, a branch of
9:
6:
4:
3:
2:
2554:
2543:
2540:
2539:
2537:
2524:
2523:
2518:
2510:
2504:
2501:
2499:
2496:
2494:
2491:
2489:
2486:
2482:
2479:
2478:
2477:
2474:
2472:
2469:
2467:
2464:
2462:
2458:
2455:
2453:
2450:
2448:
2445:
2443:
2440:
2438:
2435:
2434:
2432:
2428:
2422:
2419:
2417:
2414:
2412:
2411:Recursive set
2409:
2407:
2404:
2402:
2399:
2397:
2394:
2392:
2389:
2385:
2382:
2380:
2377:
2375:
2372:
2370:
2367:
2365:
2362:
2361:
2360:
2357:
2355:
2352:
2350:
2347:
2345:
2342:
2340:
2337:
2335:
2332:
2331:
2329:
2327:
2323:
2317:
2314:
2312:
2309:
2307:
2304:
2302:
2299:
2297:
2294:
2292:
2289:
2287:
2284:
2280:
2277:
2275:
2272:
2270:
2267:
2266:
2265:
2262:
2260:
2257:
2255:
2252:
2250:
2247:
2245:
2242:
2240:
2237:
2233:
2230:
2229:
2228:
2225:
2221:
2220:of arithmetic
2218:
2217:
2216:
2213:
2209:
2206:
2204:
2201:
2199:
2196:
2194:
2191:
2189:
2186:
2185:
2184:
2181:
2177:
2174:
2172:
2169:
2168:
2167:
2164:
2163:
2161:
2159:
2155:
2149:
2146:
2144:
2141:
2139:
2136:
2134:
2131:
2128:
2127:from ZFC
2124:
2121:
2119:
2116:
2110:
2107:
2106:
2105:
2102:
2100:
2097:
2095:
2092:
2091:
2090:
2087:
2085:
2082:
2080:
2077:
2075:
2072:
2070:
2067:
2065:
2062:
2060:
2057:
2056:
2054:
2052:
2048:
2038:
2037:
2033:
2032:
2027:
2026:non-Euclidean
2024:
2020:
2017:
2015:
2012:
2010:
2009:
2005:
2004:
2002:
1999:
1998:
1996:
1992:
1988:
1985:
1983:
1980:
1979:
1978:
1974:
1970:
1967:
1966:
1965:
1961:
1957:
1954:
1952:
1949:
1947:
1944:
1942:
1939:
1937:
1934:
1932:
1929:
1928:
1926:
1922:
1921:
1919:
1914:
1908:
1903:Example
1900:
1892:
1887:
1886:
1885:
1882:
1880:
1877:
1873:
1870:
1868:
1865:
1863:
1860:
1858:
1855:
1854:
1853:
1850:
1848:
1845:
1843:
1840:
1838:
1835:
1831:
1828:
1826:
1823:
1822:
1821:
1818:
1814:
1811:
1809:
1806:
1804:
1801:
1799:
1796:
1795:
1794:
1791:
1789:
1786:
1782:
1779:
1777:
1774:
1772:
1769:
1768:
1767:
1764:
1760:
1757:
1755:
1752:
1750:
1747:
1745:
1742:
1740:
1737:
1735:
1732:
1731:
1730:
1727:
1725:
1722:
1720:
1717:
1715:
1712:
1708:
1705:
1703:
1700:
1698:
1695:
1693:
1690:
1689:
1688:
1685:
1683:
1680:
1678:
1675:
1673:
1670:
1666:
1663:
1661:
1660:by definition
1658:
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1597:
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1575:
1573:
1570:
1568:
1565:
1563:
1560:
1558:
1555:
1553:
1550:
1548:
1547:KripkeâPlatek
1545:
1543:
1540:
1536:
1533:
1531:
1528:
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1523:
1522:
1520:
1516:
1508:
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1503:
1500:
1498:
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1486:
1483:
1481:
1478:
1476:
1473:
1471:
1468:
1466:
1463:
1460:
1456:
1452:
1449:
1445:
1442:
1440:
1437:
1435:
1432:
1431:
1430:
1426:
1423:
1422:
1420:
1418:
1414:
1410:
1402:
1399:
1397:
1394:
1392:
1391:constructible
1389:
1388:
1387:
1384:
1382:
1379:
1377:
1374:
1372:
1369:
1367:
1364:
1362:
1359:
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1332:
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1327:
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1306:
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1301:
1298:
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1278:
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1203:
1193:
1190:
1188:
1185:
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1178:
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1170:
1168:
1165:
1161:
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1149:
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1134:
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1122:
1119:
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1109:
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1104:
1102:
1099:
1097:
1094:
1092:
1089:
1087:
1084:
1082:
1079:
1077:
1074:
1073:
1071:
1069:
1068:Propositional
1065:
1059:
1056:
1054:
1051:
1049:
1046:
1044:
1041:
1039:
1036:
1034:
1031:
1027:
1024:
1023:
1022:
1019:
1017:
1014:
1012:
1009:
1007:
1004:
1002:
999:
997:
996:Logical truth
994:
992:
989:
988:
986:
984:
980:
977:
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965:
962:
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957:
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947:
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940:
936:
932:
928:
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923:
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918:
916:
912:
909:
908:
906:
904:
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877:
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872:
869:
867:
864:
862:
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844:
842:
839:
837:
834:
832:
829:
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822:
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820:
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806:
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792:
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753:
748:
745:
741:
737:
733:
729:
725:
724:
718:
715:
711:
706:
702:
698:
680:
670:
667:
664:
656:
651:
641:
638:
635:
619:
616:
613:
610:
607:
601:
593:
588:
582:
577:
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572:
567:
564:
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548:
543:
541:
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519:
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510:
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502:
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401:
398:
388:
374:
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344:
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329:
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317:
305:
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296:
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287:
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261:
247:
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183:
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167:
153:
144:
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138:
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132:
126:
117:
114:
111:
108:
105:
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92:
89:
86:
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74:
70:
66:
62:
58:
53:
43:
41:
36:
34:
31:
27:
23:
19:
2542:Model theory
2513:
2311:Ultraproduct
2158:Model theory
2123:Independence
2059:Formal proof
2051:Proof theory
2034:
2007:
1964:real numbers
1936:second-order
1847:Substitution
1724:Metalanguage
1665:conservative
1638:Axiom schema
1582:Constructive
1552:MorseâKelley
1518:Set theories
1497:Aleph number
1490:inaccessible
1396:Grothendieck
1280:intersection
1167:Higher-order
1155:Second-order
1101:Truth tables
1058:Venn diagram
841:Formal proof
759:
755:
735:
731:
713:
709:
704:
700:
696:
591:
586:
580:
578:, let |
575:
573:-adic number
570:
565:
563:prime number
560:
550:
546:
544:
539:
535:
531:
525:
521:
517:
513:
508:
504:
500:
496:
495:, such that
492:
484:
477:
475:
68:
64:
60:
56:
51:
49:
37:
32:
25:
18:model theory
15:
2421:Type theory
2369:undecidable
2301:Truth value
2188:equivalence
1867:non-logical
1480:Enumeration
1470:Isomorphism
1417:cardinality
1401:Von Neumann
1366:Ultrafilter
1331:Uncountable
1265:equivalence
1182:Quantifiers
1172:Fixed-point
1141:First-order
1021:Consistency
1006:Proposition
983:Traditional
954:Lindström's
944:Compactness
886:Type theory
831:Cardinality
590:denote its
2232:elementary
1925:arithmetic
1793:Quantifier
1771:functional
1643:Expression
1361:Transitive
1305:identities
1290:complement
1223:hereditary
1206:Set theory
721:References
46:Definition
2503:Supertask
2406:Recursion
2364:decidable
2198:saturated
2176:of models
2099:deductive
2094:axiomatic
2014:Hilbert's
2001:Euclidean
1982:canonical
1905:axiomatic
1837:Signature
1766:Predicate
1655:Extension
1577:Ackermann
1502:Operation
1381:Universal
1371:Recursive
1346:Singleton
1341:Inhabited
1326:Countable
1316:Types of
1300:power set
1270:partition
1187:Predicate
1133:Predicate
1048:Syllogism
1038:Soundness
1011:Inference
1001:Tautology
903:paradoxes
668:−
639:−
627:⟺
547:C-minimal
489:signature
482:structure
427:≠
421:∃
418:→
412:≠
396:∀
342:∨
312:→
269:∀
218:¬
215:→
175:∀
124:→
84:∀
2536:Category
2488:Logicism
2481:timeline
2457:Concrete
2316:Validity
2286:T-schema
2279:Kripke's
2274:Tarski's
2269:semantic
2259:Strength
2208:submodel
2203:spectrum
2171:function
2019:Tarski's
2008:Elements
1995:geometry
1951:Robinson
1872:variable
1857:function
1830:spectrum
1820:Sentence
1776:variable
1719:Language
1672:Relation
1633:Automata
1623:Alphabet
1607:language
1461:-jection
1439:codomain
1425:Function
1386:Universe
1356:Infinite
1260:Relation
1043:Validity
1033:Argument
931:theorem,
530:, where
2430:Related
2227:Diagram
2125: (
2104:Hilbert
2089:Systems
2084:Theorem
1962:of the
1907:systems
1687:Formula
1682:Grammar
1598: (
1542:General
1255:Forcing
1240:Element
1160:Monadic
935:paradox
876:Theorem
812:General
557:Example
487:, in a
2193:finite
1956:Skolem
1909:
1884:Theory
1852:Symbol
1842:String
1825:atomic
1702:ground
1697:closed
1692:atomic
1648:ground
1611:syntax
1507:binary
1434:domain
1351:Finite
1116:finite
974:Logics
933:
881:Theory
584:|
568:and a
561:For a
2183:Model
1931:Peano
1788:Proof
1628:Arity
1557:Naive
1444:image
1376:Fuzzy
1336:Empty
1285:union
1230:Class
871:Model
861:Lemma
819:Axiom
695:is a
480:is a
2306:Type
2109:list
1913:list
1890:list
1879:Term
1813:rank
1707:open
1601:list
1413:Maps
1318:sets
1177:Free
1147:list
897:list
824:list
657:<
534:and
24:, a
1993:of
1975:of
1923:of
1455:Sur
1429:Map
1236:Ur-
1218:Set
764:doi
740:doi
16:In
2538::
2379:NP
2003::
1997::
1927::
1604:),
1459:Bi
1451:In
760:66
758:,
736:79
734:,
542:.
524:,
520:;
476:A
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63:;
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2459:/
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1911:(
1808:â
1803:!
1798:â
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1739:âš
1734:ÂŹ
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1234:(
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790:t
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714:p
710:Q
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681:p
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671:c
665:a
661:|
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647:|
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632:|
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620:c
617:,
614:b
611:;
608:a
605:(
602:C
592:p
587:p
581:a
576:a
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566:p
540:M
536:c
532:b
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518:x
516:(
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509:C
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501:M
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461:.
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