2357:
430:
The standard definition above is restricted to the case when the cardinal can be well-ordered, i.e. is finite or an aleph. Without the axiom of choice, there are cardinals that cannot be well-ordered. Some mathematicians have defined the successor of such a cardinal as the cardinality of the least
146:
192:. Therefore, the successor operation on cardinals gains a lot of power in the infinite case (relative the ordinal successorship operation), and consequently the cardinal numbers are a very "sparse" subclass of the ordinals. We define the sequence of
523:
352:
156:
of ordinals. That is, the successor cardinal is the cardinality of the least ordinal into which a set of the given cardinality can be mapped one-to-one, but which cannot be mapped one-to-one back into that set.
286:
168:
cardinal, a larger such cardinal is constructible. The minimum actually exists because the ordinals are well-ordered. It is therefore immediate that there is no cardinal number in between
32:. The cardinal successor coincides with the ordinal successor for finite cardinals, but in the infinite case they diverge because every infinite ordinal and its successor have the same
62:
234:
425:
390:
437:
40:
can be set up between the two by simply sending the last element of the successor to 0, 0 to 1, etc., and fixing Ï and all the elements above; in the style of
Hilbert's
736:
301:
1411:
1494:
635:
1808:
41:
240:
1966:
754:
1821:
1144:
1826:
1816:
1553:
1406:
759:
188:. In the infinite case, the successor operation skips over many ordinal numbers; in fact, every infinite cardinal is a
750:
1962:
606:
588:
570:
1304:
2059:
1803:
628:
1364:
1057:
45:
798:
2320:
2022:
1785:
1780:
1605:
1026:
710:
141:{\displaystyle \kappa ^{+}=\left|\inf\{\lambda \in \mathrm {ON} \ :\ \kappa <\left|\lambda \right|\}\right|}
2315:
2098:
2015:
1728:
1659:
1536:
778:
2240:
2066:
1752:
1386:
985:
2118:
2113:
1723:
1462:
1391:
720:
621:
206:
403:
2047:
1637:
1031:
999:
690:
518:{\displaystyle \kappa ^{+}=\left|\inf\{\lambda \in \mathrm {ON} \ :\ |\lambda |\nleq \kappa \}\right|}
368:
2381:
2337:
2286:
2183:
1681:
1642:
1119:
764:
793:
2178:
2108:
1647:
1499:
1482:
1205:
685:
2010:
1987:
1948:
1834:
1775:
1421:
1341:
1185:
1129:
742:
2300:
2027:
2005:
1972:
1865:
1711:
1696:
1669:
1620:
1504:
1439:
1264:
1230:
1225:
1099:
930:
907:
565:. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974.
2230:
2083:
1875:
1593:
1329:
1235:
1094:
1079:
960:
935:
2203:
2165:
2042:
1846:
1686:
1610:
1588:
1416:
1374:
1273:
1240:
1104:
892:
803:
197:
8:
2386:
2332:
2223:
2208:
2188:
2145:
2032:
1982:
1908:
1853:
1790:
1583:
1578:
1526:
1294:
1283:
955:
855:
783:
774:
770:
705:
700:
545:
431:
ordinal that cannot be mapped one-to-one into a set of the given cardinality. That is:
2361:
2130:
2093:
2078:
2071:
2054:
1858:
1840:
1706:
1632:
1615:
1568:
1381:
1290:
1124:
1109:
1069:
1021:
1006:
994:
950:
925:
695:
644:
153:
1314:
2356:
2296:
2103:
1913:
1903:
1795:
1676:
1511:
1487:
1268:
1252:
1157:
1134:
1011:
980:
945:
840:
675:
602:
584:
566:
362:
2310:
2305:
2198:
2155:
1977:
1938:
1933:
1918:
1744:
1701:
1598:
1396:
1346:
920:
882:
2291:
2281:
2235:
2218:
2173:
2135:
2037:
1957:
1764:
1691:
1664:
1652:
1558:
1472:
1446:
1401:
1369:
1170:
972:
915:
865:
830:
788:
49:
25:
392:
is a successor cardinal. Cardinals that are not successor cardinals are called
347:{\displaystyle \aleph _{\lambda }=\bigcup _{\beta <\lambda }\aleph _{\beta }}
2276:
2255:
2213:
2193:
2088:
1943:
1541:
1531:
1521:
1516:
1450:
1324:
1200:
1089:
1084:
1062:
663:
529:
393:
161:
29:
2375:
2250:
1928:
1435:
1220:
1210:
1180:
1165:
835:
594:
189:
2150:
1997:
1898:
1890:
1770:
1718:
1627:
1563:
1546:
1477:
1336:
1195:
897:
680:
193:
2260:
2140:
1319:
1309:
1256:
940:
860:
845:
725:
670:
576:
558:
33:
52:(AC), this successor operation is easy to define: for a cardinal number
1190:
1045:
1016:
822:
165:
17:
2342:
2245:
1298:
1215:
1175:
1139:
1075:
887:
877:
850:
613:
37:
2327:
2125:
1573:
1278:
872:
200:) via this operation, through all the ordinal numbers as follows:
1923:
715:
581:
Set Theory: The Third
Millennium Edition, Revised and Expanded
1467:
813:
658:
281:{\displaystyle \aleph _{\alpha +1}=\aleph _{\alpha }^{+}}
440:
406:
371:
304:
243:
209:
65:
28:in a similar way to the successor operation on the
599:Set Theory: An Introduction to Independence Proofs
517:
419:
384:
346:
280:
228:
140:
2373:
459:
84:
629:
160:That the set above is nonempty follows from
507:
462:
130:
87:
821:
636:
622:
2374:
643:
617:
229:{\displaystyle \aleph _{0}=\omega }
13:
475:
472:
420:{\displaystyle \aleph _{\lambda }}
408:
396:; and by the above definition, if
373:
335:
306:
264:
245:
211:
100:
97:
14:
2398:
2355:
385:{\displaystyle \aleph _{\beta }}
46:von Neumann cardinal assignment
497:
489:
1:
2316:History of mathematical logic
551:
2241:Primitive recursive function
7:
539:
295:an infinite limit ordinal,
10:
2403:
1305:SchröderâBernstein theorem
1032:Monadic predicate calculus
691:Foundations of mathematics
573:(Springer-Verlag edition).
164:, which says that for any
2351:
2338:Philosophy of mathematics
2287:Automated theorem proving
2269:
2164:
1996:
1889:
1741:
1458:
1434:
1412:Von NeumannâBernaysâGödel
1357:
1251:
1155:
1053:
1044:
971:
906:
812:
734:
651:
400:is a limit ordinal, then
1988:Self-verifying theories
1809:Tarski's axiomatization
760:Tarski's undefinability
755:incompleteness theorems
2362:Mathematics portal
1973:Proof of impossibility
1621:propositional variable
931:Propositional calculus
519:
421:
386:
348:
282:
230:
180:is a cardinal that is
142:
2231:Kolmogorov complexity
2184:Computably enumerable
2084:Model complete theory
1876:Principia Mathematica
936:Propositional formula
765:BanachâTarski paradox
520:
427:is a limit cardinal.
422:
387:
349:
283:
231:
143:
2179:ChurchâTuring thesis
2166:Computability theory
1375:continuum hypothesis
893:Square of opposition
751:Gödel's completeness
438:
404:
369:
302:
241:
207:
198:axiom of replacement
63:
2333:Mathematical object
2224:P versus NP problem
2189:Computable function
1983:Reverse mathematics
1909:Logical consequence
1786:primitive recursive
1781:elementary function
1554:Free/bound variable
1407:TarskiâGrothendieck
926:Logical connectives
856:Logical equivalence
706:Logical consequence
546:Cardinal assignment
277:
20:, one can define a
2131:Transfer principle
2094:Semantics of logic
2079:Categorical theory
2055:Non-standard model
1569:Logical connective
696:Information theory
645:Mathematical logic
515:
417:
382:
344:
333:
278:
263:
226:
184:for some cardinal
178:successor cardinal
138:
2369:
2368:
2301:Abstract category
2104:Theories of truth
1914:Rule of inference
1904:Natural deduction
1885:
1884:
1430:
1429:
1135:Cartesian product
1040:
1039:
946:Many-valued logic
921:Boolean functions
804:Russell's paradox
779:diagonal argument
676:First-order logic
487:
481:
363:successor ordinal
318:
112:
106:
2394:
2382:Cardinal numbers
2360:
2359:
2311:History of logic
2306:Category of sets
2199:Decision problem
1978:Ordinal analysis
1919:Sequent calculus
1817:Boolean algebras
1757:
1756:
1731:
1702:logical/constant
1456:
1455:
1442:
1365:ZermeloâFraenkel
1116:Set operations:
1051:
1050:
988:
819:
818:
799:LöwenheimâSkolem
686:Formal semantics
638:
631:
624:
615:
614:
563:Naive set theory
524:
522:
521:
516:
514:
510:
500:
492:
485:
479:
478:
450:
449:
426:
424:
423:
418:
416:
415:
391:
389:
388:
383:
381:
380:
353:
351:
350:
345:
343:
342:
332:
314:
313:
287:
285:
284:
279:
276:
271:
259:
258:
235:
233:
232:
227:
219:
218:
162:Hartogs' theorem
152:where ON is the
147:
145:
144:
139:
137:
133:
129:
110:
104:
103:
75:
74:
26:cardinal numbers
2402:
2401:
2397:
2396:
2395:
2393:
2392:
2391:
2372:
2371:
2370:
2365:
2354:
2347:
2292:Category theory
2282:Algebraic logic
2265:
2236:Lambda calculus
2174:Church encoding
2160:
2136:Truth predicate
1992:
1958:Complete theory
1881:
1750:
1746:
1742:
1737:
1729:
1449: and
1445:
1440:
1426:
1402:New Foundations
1370:axiom of choice
1353:
1315:Gödel numbering
1255: and
1247:
1151:
1036:
986:
967:
916:Boolean algebra
902:
866:Equiconsistency
831:Classical logic
808:
789:Halting problem
777: and
753: and
741: and
740:
735:Theorems (
730:
647:
642:
612:
554:
542:
496:
488:
471:
458:
454:
445:
441:
439:
436:
435:
411:
407:
405:
402:
401:
394:limit cardinals
376:
372:
370:
367:
366:
338:
334:
322:
309:
305:
303:
300:
299:
272:
267:
248:
244:
242:
239:
238:
214:
210:
208:
205:
204:
119:
96:
83:
79:
70:
66:
64:
61:
60:
50:axiom of choice
30:ordinal numbers
12:
11:
5:
2400:
2390:
2389:
2384:
2367:
2366:
2352:
2349:
2348:
2346:
2345:
2340:
2335:
2330:
2325:
2324:
2323:
2313:
2308:
2303:
2294:
2289:
2284:
2279:
2277:Abstract logic
2273:
2271:
2267:
2266:
2264:
2263:
2258:
2256:Turing machine
2253:
2248:
2243:
2238:
2233:
2228:
2227:
2226:
2221:
2216:
2211:
2206:
2196:
2194:Computable set
2191:
2186:
2181:
2176:
2170:
2168:
2162:
2161:
2159:
2158:
2153:
2148:
2143:
2138:
2133:
2128:
2123:
2122:
2121:
2116:
2111:
2101:
2096:
2091:
2089:Satisfiability
2086:
2081:
2076:
2075:
2074:
2064:
2063:
2062:
2052:
2051:
2050:
2045:
2040:
2035:
2030:
2020:
2019:
2018:
2013:
2006:Interpretation
2002:
2000:
1994:
1993:
1991:
1990:
1985:
1980:
1975:
1970:
1960:
1955:
1954:
1953:
1952:
1951:
1941:
1936:
1926:
1921:
1916:
1911:
1906:
1901:
1895:
1893:
1887:
1886:
1883:
1882:
1880:
1879:
1871:
1870:
1869:
1868:
1863:
1862:
1861:
1856:
1851:
1831:
1830:
1829:
1827:minimal axioms
1824:
1813:
1812:
1811:
1800:
1799:
1798:
1793:
1788:
1783:
1778:
1773:
1760:
1758:
1739:
1738:
1736:
1735:
1734:
1733:
1721:
1716:
1715:
1714:
1709:
1704:
1699:
1689:
1684:
1679:
1674:
1673:
1672:
1667:
1657:
1656:
1655:
1650:
1645:
1640:
1630:
1625:
1624:
1623:
1618:
1613:
1603:
1602:
1601:
1596:
1591:
1586:
1581:
1576:
1566:
1561:
1556:
1551:
1550:
1549:
1544:
1539:
1534:
1524:
1519:
1517:Formation rule
1514:
1509:
1508:
1507:
1502:
1492:
1491:
1490:
1480:
1475:
1470:
1465:
1459:
1453:
1436:Formal systems
1432:
1431:
1428:
1427:
1425:
1424:
1419:
1414:
1409:
1404:
1399:
1394:
1389:
1384:
1379:
1378:
1377:
1372:
1361:
1359:
1355:
1354:
1352:
1351:
1350:
1349:
1339:
1334:
1333:
1332:
1325:Large cardinal
1322:
1317:
1312:
1307:
1302:
1288:
1287:
1286:
1281:
1276:
1261:
1259:
1249:
1248:
1246:
1245:
1244:
1243:
1238:
1233:
1223:
1218:
1213:
1208:
1203:
1198:
1193:
1188:
1183:
1178:
1173:
1168:
1162:
1160:
1153:
1152:
1150:
1149:
1148:
1147:
1142:
1137:
1132:
1127:
1122:
1114:
1113:
1112:
1107:
1097:
1092:
1090:Extensionality
1087:
1085:Ordinal number
1082:
1072:
1067:
1066:
1065:
1054:
1048:
1042:
1041:
1038:
1037:
1035:
1034:
1029:
1024:
1019:
1014:
1009:
1004:
1003:
1002:
992:
991:
990:
977:
975:
969:
968:
966:
965:
964:
963:
958:
953:
943:
938:
933:
928:
923:
918:
912:
910:
904:
903:
901:
900:
895:
890:
885:
880:
875:
870:
869:
868:
858:
853:
848:
843:
838:
833:
827:
825:
816:
810:
809:
807:
806:
801:
796:
791:
786:
781:
769:Cantor's
767:
762:
757:
747:
745:
732:
731:
729:
728:
723:
718:
713:
708:
703:
698:
693:
688:
683:
678:
673:
668:
667:
666:
655:
653:
649:
648:
641:
640:
633:
626:
618:
611:
610:
595:Kunen, Kenneth
592:
583:. Springer.
574:
555:
553:
550:
549:
548:
541:
538:
530:Hartogs number
526:
525:
513:
509:
506:
503:
499:
495:
491:
484:
477:
474:
470:
467:
464:
461:
457:
453:
448:
444:
414:
410:
379:
375:
355:
354:
341:
337:
331:
328:
325:
321:
317:
312:
308:
289:
288:
275:
270:
266:
262:
257:
254:
251:
247:
236:
225:
222:
217:
213:
166:well-orderable
150:
149:
136:
132:
128:
125:
122:
118:
115:
109:
102:
99:
95:
92:
89:
86:
82:
78:
73:
69:
42:Hotel Infinity
9:
6:
4:
3:
2:
2399:
2388:
2385:
2383:
2380:
2379:
2377:
2364:
2363:
2358:
2350:
2344:
2341:
2339:
2336:
2334:
2331:
2329:
2326:
2322:
2319:
2318:
2317:
2314:
2312:
2309:
2307:
2304:
2302:
2298:
2295:
2293:
2290:
2288:
2285:
2283:
2280:
2278:
2275:
2274:
2272:
2268:
2262:
2259:
2257:
2254:
2252:
2251:Recursive set
2249:
2247:
2244:
2242:
2239:
2237:
2234:
2232:
2229:
2225:
2222:
2220:
2217:
2215:
2212:
2210:
2207:
2205:
2202:
2201:
2200:
2197:
2195:
2192:
2190:
2187:
2185:
2182:
2180:
2177:
2175:
2172:
2171:
2169:
2167:
2163:
2157:
2154:
2152:
2149:
2147:
2144:
2142:
2139:
2137:
2134:
2132:
2129:
2127:
2124:
2120:
2117:
2115:
2112:
2110:
2107:
2106:
2105:
2102:
2100:
2097:
2095:
2092:
2090:
2087:
2085:
2082:
2080:
2077:
2073:
2070:
2069:
2068:
2065:
2061:
2060:of arithmetic
2058:
2057:
2056:
2053:
2049:
2046:
2044:
2041:
2039:
2036:
2034:
2031:
2029:
2026:
2025:
2024:
2021:
2017:
2014:
2012:
2009:
2008:
2007:
2004:
2003:
2001:
1999:
1995:
1989:
1986:
1984:
1981:
1979:
1976:
1974:
1971:
1968:
1967:from ZFC
1964:
1961:
1959:
1956:
1950:
1947:
1946:
1945:
1942:
1940:
1937:
1935:
1932:
1931:
1930:
1927:
1925:
1922:
1920:
1917:
1915:
1912:
1910:
1907:
1905:
1902:
1900:
1897:
1896:
1894:
1892:
1888:
1878:
1877:
1873:
1872:
1867:
1866:non-Euclidean
1864:
1860:
1857:
1855:
1852:
1850:
1849:
1845:
1844:
1842:
1839:
1838:
1836:
1832:
1828:
1825:
1823:
1820:
1819:
1818:
1814:
1810:
1807:
1806:
1805:
1801:
1797:
1794:
1792:
1789:
1787:
1784:
1782:
1779:
1777:
1774:
1772:
1769:
1768:
1766:
1762:
1761:
1759:
1754:
1748:
1743:Example
1740:
1732:
1727:
1726:
1725:
1722:
1720:
1717:
1713:
1710:
1708:
1705:
1703:
1700:
1698:
1695:
1694:
1693:
1690:
1688:
1685:
1683:
1680:
1678:
1675:
1671:
1668:
1666:
1663:
1662:
1661:
1658:
1654:
1651:
1649:
1646:
1644:
1641:
1639:
1636:
1635:
1634:
1631:
1629:
1626:
1622:
1619:
1617:
1614:
1612:
1609:
1608:
1607:
1604:
1600:
1597:
1595:
1592:
1590:
1587:
1585:
1582:
1580:
1577:
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1570:
1567:
1565:
1562:
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1557:
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1528:
1525:
1523:
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1513:
1510:
1506:
1503:
1501:
1500:by definition
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1471:
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1413:
1410:
1408:
1405:
1403:
1400:
1398:
1395:
1393:
1390:
1388:
1387:KripkeâPlatek
1385:
1383:
1380:
1376:
1373:
1371:
1368:
1367:
1366:
1363:
1362:
1360:
1356:
1348:
1345:
1344:
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1338:
1335:
1331:
1328:
1327:
1326:
1323:
1321:
1318:
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1311:
1308:
1306:
1303:
1300:
1296:
1292:
1289:
1285:
1282:
1280:
1277:
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1270:
1266:
1263:
1262:
1260:
1258:
1254:
1250:
1242:
1239:
1237:
1234:
1232:
1231:constructible
1229:
1228:
1227:
1224:
1222:
1219:
1217:
1214:
1212:
1209:
1207:
1204:
1202:
1199:
1197:
1194:
1192:
1189:
1187:
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1182:
1179:
1177:
1174:
1172:
1169:
1167:
1164:
1163:
1161:
1159:
1154:
1146:
1143:
1141:
1138:
1136:
1133:
1131:
1128:
1126:
1123:
1121:
1118:
1117:
1115:
1111:
1108:
1106:
1103:
1102:
1101:
1098:
1096:
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1086:
1083:
1081:
1077:
1073:
1071:
1068:
1064:
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1059:
1056:
1055:
1052:
1049:
1047:
1043:
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1025:
1023:
1020:
1018:
1015:
1013:
1010:
1008:
1005:
1001:
998:
997:
996:
993:
989:
984:
983:
982:
979:
978:
976:
974:
970:
962:
959:
957:
954:
952:
949:
948:
947:
944:
942:
939:
937:
934:
932:
929:
927:
924:
922:
919:
917:
914:
913:
911:
909:
908:Propositional
905:
899:
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891:
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886:
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881:
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876:
874:
871:
867:
864:
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859:
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852:
849:
847:
844:
842:
839:
837:
836:Logical truth
834:
832:
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826:
824:
820:
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815:
811:
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802:
800:
797:
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776:
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749:
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738:
733:
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724:
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707:
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702:
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694:
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639:
634:
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627:
625:
620:
619:
616:
608:
607:0-444-86839-9
604:
601:. Elsevier.
600:
596:
593:
590:
589:3-540-44085-2
586:
582:
578:
575:
572:
571:0-387-90092-6
568:
564:
560:
557:
556:
547:
544:
543:
537:
535:
531:
528:which is the
511:
504:
501:
493:
482:
468:
465:
455:
451:
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442:
434:
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432:
428:
412:
399:
395:
377:
364:
360:
339:
329:
326:
323:
319:
315:
310:
298:
297:
296:
294:
273:
268:
260:
255:
252:
249:
237:
223:
220:
215:
203:
202:
201:
199:
195:
191:
190:limit ordinal
187:
183:
179:
175:
171:
167:
163:
158:
155:
134:
126:
123:
120:
116:
113:
107:
93:
90:
80:
76:
71:
67:
59:
58:
57:
55:
51:
47:
44:). Using the
43:
39:
35:
31:
27:
24:operation on
23:
19:
2353:
2151:Ultraproduct
1998:Model theory
1963:Independence
1899:Formal proof
1891:Proof theory
1874:
1847:
1804:real numbers
1776:second-order
1687:Substitution
1564:Metalanguage
1505:conservative
1478:Axiom schema
1422:Constructive
1392:MorseâKelley
1358:Set theories
1337:Aleph number
1330:inaccessible
1236:Grothendieck
1120:intersection
1007:Higher-order
995:Second-order
941:Truth tables
898:Venn diagram
681:Formal proof
598:
580:
577:Jech, Thomas
562:
533:
527:
429:
397:
358:
356:
292:
290:
185:
181:
177:
173:
169:
159:
151:
53:
21:
15:
2261:Type theory
2209:undecidable
2141:Truth value
2028:equivalence
1707:non-logical
1320:Enumeration
1310:Isomorphism
1257:cardinality
1241:Von Neumann
1206:Ultrafilter
1171:Uncountable
1105:equivalence
1022:Quantifiers
1012:Fixed-point
981:First-order
861:Consistency
846:Proposition
823:Traditional
794:Lindström's
784:Compactness
726:Type theory
671:Cardinality
559:Paul Halmos
34:cardinality
2387:Set theory
2376:Categories
2072:elementary
1765:arithmetic
1633:Quantifier
1611:functional
1483:Expression
1201:Transitive
1145:identities
1130:complement
1063:hereditary
1046:Set theory
552:References
18:set theory
2343:Supertask
2246:Recursion
2204:decidable
2038:saturated
2016:of models
1939:deductive
1934:axiomatic
1854:Hilbert's
1841:Euclidean
1822:canonical
1745:axiomatic
1677:Signature
1606:Predicate
1495:Extension
1417:Ackermann
1342:Operation
1221:Universal
1211:Recursive
1186:Singleton
1181:Inhabited
1166:Countable
1156:Types of
1140:power set
1110:partition
1027:Predicate
973:Predicate
888:Syllogism
878:Soundness
851:Inference
841:Tautology
743:paradoxes
505:κ
502:≰
494:λ
469:∈
466:λ
443:κ
413:λ
409:ℵ
378:β
374:ℵ
340:β
336:ℵ
330:λ
324:β
320:⋃
311:λ
307:ℵ
269:α
265:ℵ
250:α
246:ℵ
224:ω
212:ℵ
196:(via the
124:λ
114:κ
94:∈
91:λ
68:κ
38:bijection
22:successor
2328:Logicism
2321:timeline
2297:Concrete
2156:Validity
2126:T-schema
2119:Kripke's
2114:Tarski's
2109:semantic
2099:Strength
2048:submodel
2043:spectrum
2011:function
1859:Tarski's
1848:Elements
1835:geometry
1791:Robinson
1712:variable
1697:function
1670:spectrum
1660:Sentence
1616:variable
1559:Language
1512:Relation
1473:Automata
1463:Alphabet
1447:language
1301:-jection
1279:codomain
1265:Function
1226:Universe
1196:Infinite
1100:Relation
883:Validity
873:Argument
771:theorem,
597:, 1980.
579:, 2003.
540:See also
291:and for
56:we have
48:and the
2270:Related
2067:Diagram
1965: (
1944:Hilbert
1929:Systems
1924:Theorem
1802:of the
1747:systems
1527:Formula
1522:Grammar
1438: (
1382:General
1095:Forcing
1080:Element
1000:Monadic
775:paradox
716:Theorem
652:General
365:, then
2033:finite
1796:Skolem
1749:
1724:Theory
1692:Symbol
1682:String
1665:atomic
1542:ground
1537:closed
1532:atomic
1488:ground
1451:syntax
1347:binary
1274:domain
1191:Finite
956:finite
814:Logics
773:
721:Theory
605:
587:
569:
486:
480:
194:alephs
111:
105:
2023:Model
1771:Peano
1628:Proof
1468:Arity
1397:Naive
1284:image
1216:Fuzzy
1176:Empty
1125:union
1070:Class
711:Model
701:Lemma
659:Axiom
361:is a
154:class
2146:Type
1949:list
1753:list
1730:list
1719:Term
1653:rank
1547:open
1441:list
1253:Maps
1158:sets
1017:Free
987:list
737:list
664:list
603:ISBN
585:ISBN
567:ISBN
327:<
176:. A
172:and
117:<
1833:of
1815:of
1763:of
1295:Sur
1269:Map
1076:Ur-
1058:Set
532:of
460:inf
357:If
85:inf
36:(a
16:In
2378::
2219:NP
1843::
1837::
1767::
1444:),
1299:Bi
1291:In
561:,
536:.
2299:/
2214:P
1969:)
1755:)
1751:(
1648:â
1643:!
1638:â
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1594:â
1589:â
1584:â§
1579:âš
1574:ÂŹ
1297:/
1293:/
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1074:(
961:â
951:3
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637:e
630:t
623:v
609:.
591:.
534:Îș
512:|
508:}
498:|
490:|
483::
476:N
473:O
463:{
456:|
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447:+
398:λ
359:ÎČ
316:=
293:λ
274:+
261:=
256:1
253:+
221:=
216:0
186:Îș
182:Îș
174:Îș
170:Îș
148:,
135:|
131:}
127:|
121:|
108::
101:N
98:O
88:{
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54:Îș
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