2210:
31:
242:
The following statements are inconsistent with the axiom of choice, and therefore with ZFC. However they are probably independent of ZF, in a corresponding sense to the above: They cannot be proved in ZF, and few working set theorists expect to find a refutation in ZF. However ZF cannot prove that
218:
The following statements (none of which have been proved false) cannot be proved in ZFC (the
ZermeloâFraenkel set theory plus the axiom of choice) to be independent of ZFC, under the added hypothesis that ZFC is consistent.
589:
1264:
1347:
488:
308:
Paterek, T.; Kofler, J.; Prevedel, R.; Klimek, P.; Aspelmeyer, M.; Zeilinger, A.; Brukner, Ä. (2010), "Logical independence and quantum randomness",
1661:
354:
Székely, Gergely (2013), "The
Existence of Superluminal Particles is Consistent with the Kinematics of Einstein's Special Theory of Relativity",
1819:
446:
420:
279:
607:
1674:
997:
1259:
1679:
1669:
1406:
612:
1157:
603:
1912:
1656:
481:
1217:
910:
651:
188:
356:
2173:
1875:
1638:
1633:
1458:
879:
563:
224:
191:(ZF). The following statements in set theory are known to be independent of ZF, under the assumption that ZF is consistent:
2168:
1951:
1868:
1581:
1512:
1389:
631:
86:
79:
1239:
2093:
1919:
1605:
838:
1244:
2234:
1576:
1315:
573:
474:
94:
1971:
1966:
270:
Since 2000, logical independence has become understood as having crucial significance in the foundations of physics.
1900:
1490:
884:
852:
543:
617:
2190:
2139:
2036:
1534:
1495:
972:
254:
2031:
646:
1961:
1500:
1352:
1335:
1058:
538:
1863:
1840:
1801:
1687:
1628:
1274:
1194:
1038:
982:
595:
2153:
1880:
1858:
1825:
1718:
1564:
1549:
1522:
1473:
1357:
1292:
1117:
1083:
1078:
952:
783:
760:
310:
30:
2239:
2083:
1936:
1728:
1446:
1182:
1088:
947:
932:
813:
788:
2209:
2056:
2018:
1895:
1699:
1539:
1463:
1441:
1269:
1227:
1126:
1093:
957:
745:
656:
375:
329:
248:
206:
202:
2185:
2076:
2061:
2041:
1998:
1885:
1835:
1761:
1706:
1643:
1436:
1431:
1379:
1147:
1136:
808:
708:
636:
627:
623:
558:
553:
212:
121:
43:
35:
82:
from some specific set of other sentences. The sentences in this set are referred to as "axioms".
2214:
1983:
1946:
1931:
1924:
1907:
1693:
1559:
1485:
1468:
1421:
1234:
1143:
977:
962:
922:
874:
859:
847:
803:
778:
548:
497:
365:
319:
284:
71:
1711:
1167:
2149:
1956:
1766:
1756:
1648:
1529:
1364:
1340:
1121:
1105:
1010:
987:
864:
833:
798:
693:
528:
442:
416:
412:
404:
2163:
2158:
2051:
2008:
1830:
1791:
1786:
1771:
1597:
1554:
1451:
1249:
1199:
773:
735:
383:
337:
125:
171:
cannot refute Ï. These authors will sometimes say "Ï is independent of and consistent with
2144:
2134:
2088:
2071:
2026:
1988:
1890:
1810:
1617:
1544:
1517:
1505:
1411:
1325:
1299:
1254:
1222:
1023:
825:
768:
718:
683:
641:
438:
196:
379:
341:
333:
17:
2129:
2108:
2066:
2046:
1941:
1796:
1394:
1384:
1374:
1369:
1303:
1177:
1053:
942:
937:
915:
516:
460:
230:
387:
2228:
2103:
1781:
1288:
1073:
1063:
1033:
1018:
688:
431:
2003:
1850:
1751:
1743:
1623:
1571:
1480:
1416:
1399:
1330:
1189:
1048:
750:
533:
243:
they are independent of ZF, even with the added hypothesis that ZF is consistent.
2113:
1993:
1172:
1162:
1109:
793:
713:
698:
578:
523:
236:
1043:
898:
869:
675:
2195:
2098:
1151:
1068:
1028:
992:
928:
740:
730:
703:
2180:
1978:
1426:
1131:
725:
288:
104:
neither proves nor refutes Ï; that is, it is impossible to prove Ï from
1776:
568:
466:
1320:
666:
511:
370:
324:
167:
simply cannot prove Ï, and do not necessarily assert by this that
259:
470:
147:. A theory for which there is an independent set of axioms is
187:
Many interesting statements in set theory are independent of
112:
that Ï is false. Sometimes, Ï is said (synonymously) to be
2122:
2017:
1849:
1742:
1594:
1287:
1210:
1104:
1008:
897:
824:
759:
674:
665:
587:
504:
437:, Graduate Texts in Mathematics, Berlin, New York:
430:
482:
120:. (This concept is unrelated to the idea of "
8:
1308:
903:
671:
489:
475:
467:
159:Some authors say that Ï is independent of
108:, and it is also impossible to prove from
369:
323:
143:is provable from the remaining axioms in
29:
457:An introduction to mathematical thought
300:
78:is the unprovability of some specific
409:An Introduction to Mathematical Logic
280:List of statements independent of ZFC
58:, but also models (2,3) that satisfy
50:): there are models (1) that satisfy
7:
183:Independence results in set theory
42:) is independent of the remaining
25:
2208:
455:Stabler, Edward Russell (1948),
207:generalized continuum hypothesis
179:can neither prove nor refute Ï.
357:Reports on Mathematical Physics
266:Applications to physical theory
225:strongly inaccessible cardinals
1:
2169:History of mathematical logic
388:10.1016/S0034-4877(13)00021-9
342:10.1088/1367-2630/12/1/013019
2094:Primitive recursive function
189:ZermeloâFraenkel set theory
149:independently axiomatizable
2256:
1158:SchröderâBernstein theorem
885:Monadic predicate calculus
544:Foundations of mathematics
459:, Reading, Massachusetts:
27:Term in mathematical logic
2204:
2191:Philosophy of mathematics
2140:Automated theorem proving
1311:
1265:Von NeumannâBernaysâGödel
906:
255:axiom of real determinacy
429:Monk, J. Donald (1976),
411:(4th ed.), London:
18:Independent (Set theory)
1841:Self-verifying theories
1662:Tarski's axiomatization
613:Tarski's undefinability
608:incompleteness theorems
2215:Mathematics portal
1826:Proof of impossibility
1474:propositional variable
784:Propositional calculus
311:New Journal of Physics
67:
2084:Kolmogorov complexity
2037:Computably enumerable
1937:Model complete theory
1729:Principia Mathematica
789:Propositional formula
618:BanachâTarski paradox
235:The non-existence of
33:
2032:ChurchâTuring thesis
2019:Computability theory
1228:continuum hypothesis
746:Square of opposition
604:Gödel's completeness
249:axiom of determinacy
203:continuum hypothesis
2186:Mathematical object
2077:P versus NP problem
2042:Computable function
1836:Reverse mathematics
1762:Logical consequence
1639:primitive recursive
1634:elementary function
1407:Free/bound variable
1260:TarskiâGrothendieck
779:Logical connectives
709:Logical equivalence
559:Logical consequence
380:2013RpMP...72..133S
334:2010NJPh...12a3019P
175:" to indicate that
2235:Mathematical logic
1984:Transfer principle
1947:Semantics of logic
1932:Categorical theory
1908:Non-standard model
1422:Logical connective
549:Information theory
498:Mathematical logic
433:Mathematical Logic
413:Chapman & Hall
405:Mendelson, Elliott
287:for an example in
285:Parallel postulate
95:first-order theory
72:mathematical logic
68:
2222:
2221:
2154:Abstract category
1957:Theories of truth
1767:Rule of inference
1757:Natural deduction
1738:
1737:
1283:
1282:
988:Cartesian product
893:
892:
799:Many-valued logic
774:Boolean functions
657:Russell's paradox
632:diagonal argument
529:First-order logic
448:978-0-387-90170-1
422:978-0-412-80830-2
229:The existence of
223:The existence of
213:Suslin conjecture
16:(Redirected from
2247:
2213:
2212:
2164:History of logic
2159:Category of sets
2052:Decision problem
1831:Ordinal analysis
1772:Sequent calculus
1670:Boolean algebras
1610:
1609:
1584:
1555:logical/constant
1309:
1295:
1218:ZermeloâFraenkel
969:Set operations:
904:
841:
672:
652:LöwenheimâSkolem
539:Formal semantics
491:
484:
477:
468:
463:
451:
436:
425:
391:
390:
373:
351:
345:
344:
327:
305:
126:decision problem
21:
2255:
2254:
2250:
2249:
2248:
2246:
2245:
2244:
2225:
2224:
2223:
2218:
2207:
2200:
2145:Category theory
2135:Algebraic logic
2118:
2089:Lambda calculus
2027:Church encoding
2013:
1989:Truth predicate
1845:
1811:Complete theory
1734:
1603:
1599:
1595:
1590:
1582:
1302: and
1298:
1293:
1279:
1255:New Foundations
1223:axiom of choice
1206:
1168:Gödel numbering
1108: and
1100:
1004:
889:
839:
820:
769:Boolean algebra
755:
719:Equiconsistency
684:Classical logic
661:
642:Halting problem
630: and
606: and
594: and
593:
588:Theorems (
583:
500:
495:
454:
449:
439:Springer-Verlag
428:
423:
403:
400:
395:
394:
353:
352:
348:
307:
306:
302:
297:
276:
268:
231:large cardinals
197:axiom of choice
185:
157:
139:if no axiom in
44:geometry axioms
36:parallels axiom
28:
23:
22:
15:
12:
11:
5:
2253:
2251:
2243:
2242:
2237:
2227:
2226:
2220:
2219:
2205:
2202:
2201:
2199:
2198:
2193:
2188:
2183:
2178:
2177:
2176:
2166:
2161:
2156:
2147:
2142:
2137:
2132:
2130:Abstract logic
2126:
2124:
2120:
2119:
2117:
2116:
2111:
2109:Turing machine
2106:
2101:
2096:
2091:
2086:
2081:
2080:
2079:
2074:
2069:
2064:
2059:
2049:
2047:Computable set
2044:
2039:
2034:
2029:
2023:
2021:
2015:
2014:
2012:
2011:
2006:
2001:
1996:
1991:
1986:
1981:
1976:
1975:
1974:
1969:
1964:
1954:
1949:
1944:
1942:Satisfiability
1939:
1934:
1929:
1928:
1927:
1917:
1916:
1915:
1905:
1904:
1903:
1898:
1893:
1888:
1883:
1873:
1872:
1871:
1866:
1859:Interpretation
1855:
1853:
1847:
1846:
1844:
1843:
1838:
1833:
1828:
1823:
1813:
1808:
1807:
1806:
1805:
1804:
1794:
1789:
1779:
1774:
1769:
1764:
1759:
1754:
1748:
1746:
1740:
1739:
1736:
1735:
1733:
1732:
1724:
1723:
1722:
1721:
1716:
1715:
1714:
1709:
1704:
1684:
1683:
1682:
1680:minimal axioms
1677:
1666:
1665:
1664:
1653:
1652:
1651:
1646:
1641:
1636:
1631:
1626:
1613:
1611:
1592:
1591:
1589:
1588:
1587:
1586:
1574:
1569:
1568:
1567:
1562:
1557:
1552:
1542:
1537:
1532:
1527:
1526:
1525:
1520:
1510:
1509:
1508:
1503:
1498:
1493:
1483:
1478:
1477:
1476:
1471:
1466:
1456:
1455:
1454:
1449:
1444:
1439:
1434:
1429:
1419:
1414:
1409:
1404:
1403:
1402:
1397:
1392:
1387:
1377:
1372:
1370:Formation rule
1367:
1362:
1361:
1360:
1355:
1345:
1344:
1343:
1333:
1328:
1323:
1318:
1312:
1306:
1289:Formal systems
1285:
1284:
1281:
1280:
1278:
1277:
1272:
1267:
1262:
1257:
1252:
1247:
1242:
1237:
1232:
1231:
1230:
1225:
1214:
1212:
1208:
1207:
1205:
1204:
1203:
1202:
1192:
1187:
1186:
1185:
1178:Large cardinal
1175:
1170:
1165:
1160:
1155:
1141:
1140:
1139:
1134:
1129:
1114:
1112:
1102:
1101:
1099:
1098:
1097:
1096:
1091:
1086:
1076:
1071:
1066:
1061:
1056:
1051:
1046:
1041:
1036:
1031:
1026:
1021:
1015:
1013:
1006:
1005:
1003:
1002:
1001:
1000:
995:
990:
985:
980:
975:
967:
966:
965:
960:
950:
945:
943:Extensionality
940:
938:Ordinal number
935:
925:
920:
919:
918:
907:
901:
895:
894:
891:
890:
888:
887:
882:
877:
872:
867:
862:
857:
856:
855:
845:
844:
843:
830:
828:
822:
821:
819:
818:
817:
816:
811:
806:
796:
791:
786:
781:
776:
771:
765:
763:
757:
756:
754:
753:
748:
743:
738:
733:
728:
723:
722:
721:
711:
706:
701:
696:
691:
686:
680:
678:
669:
663:
662:
660:
659:
654:
649:
644:
639:
634:
622:Cantor's
620:
615:
610:
600:
598:
585:
584:
582:
581:
576:
571:
566:
561:
556:
551:
546:
541:
536:
531:
526:
521:
520:
519:
508:
506:
502:
501:
496:
494:
493:
486:
479:
471:
465:
464:
461:Addison-Wesley
452:
447:
426:
421:
399:
396:
393:
392:
364:(2): 133â152,
346:
299:
298:
296:
293:
292:
291:
282:
275:
272:
267:
264:
263:
262:
257:
251:
240:
239:
233:
227:
216:
215:
209:
199:
184:
181:
156:
153:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2252:
2241:
2238:
2236:
2233:
2232:
2230:
2217:
2216:
2211:
2203:
2197:
2194:
2192:
2189:
2187:
2184:
2182:
2179:
2175:
2172:
2171:
2170:
2167:
2165:
2162:
2160:
2157:
2155:
2151:
2148:
2146:
2143:
2141:
2138:
2136:
2133:
2131:
2128:
2127:
2125:
2121:
2115:
2112:
2110:
2107:
2105:
2104:Recursive set
2102:
2100:
2097:
2095:
2092:
2090:
2087:
2085:
2082:
2078:
2075:
2073:
2070:
2068:
2065:
2063:
2060:
2058:
2055:
2054:
2053:
2050:
2048:
2045:
2043:
2040:
2038:
2035:
2033:
2030:
2028:
2025:
2024:
2022:
2020:
2016:
2010:
2007:
2005:
2002:
2000:
1997:
1995:
1992:
1990:
1987:
1985:
1982:
1980:
1977:
1973:
1970:
1968:
1965:
1963:
1960:
1959:
1958:
1955:
1953:
1950:
1948:
1945:
1943:
1940:
1938:
1935:
1933:
1930:
1926:
1923:
1922:
1921:
1918:
1914:
1913:of arithmetic
1911:
1910:
1909:
1906:
1902:
1899:
1897:
1894:
1892:
1889:
1887:
1884:
1882:
1879:
1878:
1877:
1874:
1870:
1867:
1865:
1862:
1861:
1860:
1857:
1856:
1854:
1852:
1848:
1842:
1839:
1837:
1834:
1832:
1829:
1827:
1824:
1821:
1820:from ZFC
1817:
1814:
1812:
1809:
1803:
1800:
1799:
1798:
1795:
1793:
1790:
1788:
1785:
1784:
1783:
1780:
1778:
1775:
1773:
1770:
1768:
1765:
1763:
1760:
1758:
1755:
1753:
1750:
1749:
1747:
1745:
1741:
1731:
1730:
1726:
1725:
1720:
1719:non-Euclidean
1717:
1713:
1710:
1708:
1705:
1703:
1702:
1698:
1697:
1695:
1692:
1691:
1689:
1685:
1681:
1678:
1676:
1673:
1672:
1671:
1667:
1663:
1660:
1659:
1658:
1654:
1650:
1647:
1645:
1642:
1640:
1637:
1635:
1632:
1630:
1627:
1625:
1622:
1621:
1619:
1615:
1614:
1612:
1607:
1601:
1596:Example
1593:
1585:
1580:
1579:
1578:
1575:
1573:
1570:
1566:
1563:
1561:
1558:
1556:
1553:
1551:
1548:
1547:
1546:
1543:
1541:
1538:
1536:
1533:
1531:
1528:
1524:
1521:
1519:
1516:
1515:
1514:
1511:
1507:
1504:
1502:
1499:
1497:
1494:
1492:
1489:
1488:
1487:
1484:
1482:
1479:
1475:
1472:
1470:
1467:
1465:
1462:
1461:
1460:
1457:
1453:
1450:
1448:
1445:
1443:
1440:
1438:
1435:
1433:
1430:
1428:
1425:
1424:
1423:
1420:
1418:
1415:
1413:
1410:
1408:
1405:
1401:
1398:
1396:
1393:
1391:
1388:
1386:
1383:
1382:
1381:
1378:
1376:
1373:
1371:
1368:
1366:
1363:
1359:
1356:
1354:
1353:by definition
1351:
1350:
1349:
1346:
1342:
1339:
1338:
1337:
1334:
1332:
1329:
1327:
1324:
1322:
1319:
1317:
1314:
1313:
1310:
1307:
1305:
1301:
1296:
1290:
1286:
1276:
1273:
1271:
1268:
1266:
1263:
1261:
1258:
1256:
1253:
1251:
1248:
1246:
1243:
1241:
1240:KripkeâPlatek
1238:
1236:
1233:
1229:
1226:
1224:
1221:
1220:
1219:
1216:
1215:
1213:
1209:
1201:
1198:
1197:
1196:
1193:
1191:
1188:
1184:
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19:
2240:Proof theory
2206:
2004:Ultraproduct
1851:Model theory
1816:Independence
1815:
1752:Formal proof
1744:Proof theory
1727:
1700:
1657:real numbers
1629:second-order
1540:Substitution
1417:Metalanguage
1358:conservative
1331:Axiom schema
1275:Constructive
1245:MorseâKelley
1211:Set theories
1190:Aleph number
1183:inaccessible
1089:Grothendieck
973:intersection
860:Higher-order
848:Second-order
794:Truth tables
751:Venn diagram
534:Formal proof
456:
432:
408:
361:
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269:
241:
237:Kurepa trees
217:
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122:decidability
117:
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109:
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97:
90:
84:
76:independence
75:
69:
63:
59:
55:
51:
47:
39:
2114:Type theory
2062:undecidable
1994:Truth value
1881:equivalence
1560:non-logical
1173:Enumeration
1163:Isomorphism
1110:cardinality
1094:Von Neumann
1059:Ultrafilter
1024:Uncountable
958:equivalence
875:Quantifiers
865:Fixed-point
834:First-order
714:Consistency
699:Proposition
676:Traditional
647:Lindström's
637:Compactness
579:Type theory
524:Cardinality
137:independent
114:undecidable
93:of a given
91:independent
2229:Categories
1925:elementary
1618:arithmetic
1486:Quantifier
1464:functional
1336:Expression
1054:Transitive
998:identities
983:complement
916:hereditary
899:Set theory
398:References
318:: 013019,
155:Usage note
124:" as in a
62:, but not
2196:Supertask
2099:Recursion
2057:decidable
1891:saturated
1869:of models
1792:deductive
1787:axiomatic
1707:Hilbert's
1694:Euclidean
1675:canonical
1598:axiomatic
1530:Signature
1459:Predicate
1348:Extension
1270:Ackermann
1195:Operation
1074:Universal
1064:Recursive
1039:Singleton
1034:Inhabited
1019:Countable
1009:Types of
993:power set
963:partition
880:Predicate
826:Predicate
741:Syllogism
731:Soundness
704:Inference
694:Tautology
596:paradoxes
371:1202.5790
325:0811.4542
131:A theory
2181:Logicism
2174:timeline
2150:Concrete
2009:Validity
1979:T-schema
1972:Kripke's
1967:Tarski's
1962:semantic
1952:Strength
1901:submodel
1896:spectrum
1864:function
1712:Tarski's
1701:Elements
1688:geometry
1644:Robinson
1565:variable
1550:function
1523:spectrum
1513:Sentence
1469:variable
1412:Language
1365:Relation
1326:Automata
1316:Alphabet
1300:language
1154:-jection
1132:codomain
1118:Function
1079:Universe
1049:Infinite
953:Relation
736:Validity
726:Argument
624:theorem,
407:(1997),
289:geometry
274:See also
205:and the
87:sentence
80:sentence
2123:Related
1920:Diagram
1818: (
1797:Hilbert
1782:Systems
1777:Theorem
1655:of the
1600:systems
1380:Formula
1375:Grammar
1291: (
1235:General
948:Forcing
933:Element
853:Monadic
628:paradox
569:Theorem
505:General
376:Bibcode
330:Bibcode
1886:finite
1649:Skolem
1602:
1577:Theory
1545:Symbol
1535:String
1518:atomic
1395:ground
1390:closed
1385:atomic
1341:ground
1304:syntax
1200:binary
1127:domain
1044:Finite
809:finite
667:Logics
626:
574:Theory
445:
419:
1876:Model
1624:Peano
1481:Proof
1321:Arity
1250:Naive
1137:image
1069:Fuzzy
1029:Empty
978:union
923:Class
564:Model
554:Lemma
512:Axiom
366:arXiv
320:arXiv
295:Notes
163:when
116:from
89:Ï is
1999:Type
1802:list
1606:list
1583:list
1572:Term
1506:rank
1400:open
1294:list
1106:Maps
1011:sets
870:Free
840:list
590:list
517:list
443:ISBN
417:ISBN
253:The
247:The
211:The
201:The
195:The
54:and
34:The
1686:of
1668:of
1616:of
1148:Sur
1122:Map
929:Ur-
911:Set
384:doi
338:doi
260:AD+
135:is
128:.)
100:if
70:In
2231::
2072:NP
1696::
1690::
1620::
1297:),
1152:Bi
1144:In
441:,
415:,
382:,
374:,
362:72
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336:,
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