2194:
149:
suffices), then the theory cannot prove its own consistency. There are some technical caveats as to what requirements the formal statement representing the metamathematical statement "The theory is consistent" needs to satisfy, but the outcome is that if a (sufficiently strong) theory can prove its
123:
at the beginning of the 20th century whose ultimate goal was to show, using mathematical methods, the consistency of mathematics. Since most mathematical disciplines can be reduced to
573:
213:. When discussing these issues of consistency strength, the metatheory in which the discussion takes places needs to be carefully addressed. For theories at the level of
357:
319:
1248:
154:
of the theory or not, or else the theory itself is inconsistent (in which case it can prove anything, including false statements such as its own consistency).
1331:
472:
1645:
257:
can be adopted as the metatheory in question, but even if the metatheory is ZFC or an extension of it, the notion is meaningful. The method of
1803:
410:
268:
When discussing fragments of ZFC or their extensions (for example, ZF, set theory without the axiom of choice, or ZF+AD, set theory with the
2298:
591:
1658:
981:
134:
2230:
1243:
1663:
1653:
1390:
596:
2293:
1141:
587:
2338:
1799:
445:
127:, the program quickly became the establishment of the consistency of arithmetic by methods formalizable within arithmetic itself.
1896:
1640:
465:
2323:
1201:
894:
635:
2157:
1859:
1622:
1617:
1442:
863:
547:
254:
109:. Since some theories are powerful enough to model different mathematical objects, it is natural to wonder about their own
2152:
1935:
1852:
1565:
1496:
1373:
615:
1223:
2077:
1903:
1589:
822:
2348:
1228:
2374:
1560:
1299:
557:
458:
35:
2343:
1955:
1950:
2281:
1884:
1474:
868:
836:
527:
601:
2174:
2123:
2020:
1518:
1479:
956:
138:
2266:
2015:
630:
272:), the notions described above are adapted accordingly. Thus, ZF is equiconsistent with ZFC, as shown by Gödel.
229:
theory that can certainly model most of mathematics. The most widely used set of axioms of set theory is called
2369:
1945:
1484:
1336:
1319:
1042:
522:
2286:
2223:
1847:
1824:
1785:
1671:
1612:
1258:
1178:
1022:
966:
579:
377:
364:
214:
2318:
2137:
1864:
1842:
1809:
1702:
1548:
1533:
1506:
1457:
1341:
1276:
1101:
1067:
1062:
936:
767:
744:
261:
allows one to show that the theories ZFC, ZFC+CH and ZFC+ÂŹCH are all equiconsistent (where CH denotes the
2067:
1920:
1712:
1430:
1166:
1072:
931:
916:
797:
772:
288:
258:
2193:
2313:
2308:
2260:
2040:
2002:
1879:
1683:
1523:
1447:
1425:
1253:
1211:
1110:
1077:
941:
729:
640:
269:
262:
150:
own consistency then either there is no computable way of identifying whether a statement is even an
120:
2254:
2169:
2060:
2045:
2025:
1982:
1869:
1819:
1745:
1690:
1627:
1420:
1415:
1363:
1131:
1120:
792:
692:
620:
611:
607:
542:
537:
436:
218:
146:
106:
145:(whether something is a proof or not), i.e. strong enough to model a weak fragment of arithmetic (
2216:
2198:
1967:
1930:
1915:
1908:
1891:
1677:
1543:
1469:
1452:
1405:
1218:
1127:
961:
946:
906:
858:
843:
831:
787:
762:
532:
481:
335:
297:
31:
1695:
1151:
2133:
1940:
1750:
1740:
1632:
1513:
1348:
1324:
1105:
1089:
994:
971:
848:
817:
782:
677:
512:
441:
406:
157:
Given this, instead of outright consistency, one usually considers relative consistency: Let
2276:
2147:
2142:
2035:
1992:
1814:
1775:
1770:
1755:
1581:
1538:
1435:
1233:
1183:
757:
719:
431:
416:
242:
2243:
2128:
2118:
2072:
2055:
2010:
1972:
1874:
1794:
1601:
1528:
1501:
1489:
1395:
1309:
1283:
1238:
1206:
1007:
809:
752:
667:
625:
420:
142:
2333:
2113:
2092:
2050:
2030:
1925:
1780:
1378:
1368:
1358:
1353:
1287:
1161:
1037:
926:
921:
899:
500:
360:
326:
322:
276:
226:
2363:
2087:
1765:
1272:
1057:
1047:
1017:
1002:
672:
398:
130:
116:
17:
1987:
1834:
1735:
1727:
1607:
1555:
1464:
1400:
1383:
1314:
1173:
1032:
734:
517:
275:
The consistency strength of numerous combinatorial statements can be calibrated by
177:. Two theories are equiconsistent if each one is consistent relative to the other.
2328:
2271:
2208:
2097:
1977:
1156:
1146:
1093:
777:
697:
682:
562:
507:
284:
110:
43:
1027:
882:
853:
659:
222:
124:
47:
405:, Studies in Logic, vol. 34, London: College Publications, p. 225,
53:
In general, it is not possible to prove the absolute consistency of a theory
2303:
2239:
2179:
2082:
1135:
1052:
1012:
976:
912:
724:
714:
687:
50:. In this case, they are, roughly speaking, "as consistent as each other".
61:, believed to be consistent, and try to prove the weaker statement that if
2164:
1962:
1410:
1115:
709:
221:
program has much to say. Consistency strength issues are a usual part of
1760:
552:
450:
1304:
650:
495:
151:
137:
show that
Hilbert's program cannot be realized: if a consistent
2212:
454:
46:
of one theory implies the consistency of the other theory, and
230:
241:, what is really being claimed is that in the metatheory (
105:
In mathematical logic, formal theories are studied as
69:
must also be consistentâif we can do this we say that
338:
300:
245:
in this case) it can be proven that the theories ZFC+
2106:
2001:
1833:
1726:
1578:
1271:
1194:
1088:
992:
881:
808:
743:
658:
649:
571:
488:
351:
313:
2224:
466:
141:theory is strong enough to formalize its own
8:
169:is a consistent theory. Does it follow that
287:is equiconsistent with the existence of an
2231:
2217:
2209:
1292:
887:
655:
473:
459:
451:
363:is equiconsistent with the existence of a
325:is equiconsistent with the existence of a
197:is not known to be consistent relative to
343:
337:
305:
299:
237:is said to be equiconsistent to another
389:
7:
57:. Instead we usually take a theory
25:
233:. When a set-theoretic statement
2192:
165:be formal theories. Assume that
2299:Gödel's incompleteness theorems
81:is also consistent relative to
255:primitive recursive arithmetic
1:
2153:History of mathematical logic
294:the non-existence of special
253:are equiconsistent. Usually,
175:T is consistent relative to S
2294:Gödel's completeness theorem
2078:Primitive recursive function
352:{\displaystyle \omega _{2}}
314:{\displaystyle \omega _{2}}
173:is consistent? If so, then
2391:
2282:Foundations of mathematics
1142:SchröderâBernstein theorem
869:Monadic predicate calculus
528:Foundations of mathematics
189:is consistent relative to
2250:
2188:
2175:Philosophy of mathematics
2124:Automated theorem proving
1295:
1249:Von NeumannâBernaysâGödel
890:
2324:LöwenheimâSkolem theorem
75:consistent relative to S
27:Being equally consistent
2349:Useâmention distinction
1825:Self-verifying theories
1646:Tarski's axiomatization
597:Tarski's undefinability
592:incompleteness theorems
378:Large cardinal property
365:weakly compact cardinal
215:second-order arithmetic
135:incompleteness theorems
2344:Typeâtoken distinction
2199:Mathematics portal
1810:Proof of impossibility
1458:propositional variable
768:Propositional calculus
353:
315:
2068:Kolmogorov complexity
2021:Computably enumerable
1921:Model complete theory
1713:Principia Mathematica
773:Propositional formula
602:BanachâTarski paradox
354:
332:the non-existence of
316:
289:inaccessible cardinal
139:computably enumerable
2267:ChurchâTuring thesis
2261:Entscheidungsproblem
2016:ChurchâTuring thesis
2003:Computability theory
1212:continuum hypothesis
730:Square of opposition
588:Gödel's completeness
336:
298:
270:axiom of determinacy
263:continuum hypothesis
207:consistency strength
181:Consistency strength
107:mathematical objects
18:Consistency strength
2170:Mathematical object
2061:P versus NP problem
2026:Computable function
1820:Reverse mathematics
1746:Logical consequence
1623:primitive recursive
1618:elementary function
1391:Free/bound variable
1244:TarskiâGrothendieck
763:Logical connectives
693:Logical equivalence
543:Logical consequence
437:The Higher Infinite
285:Kurepa's hypothesis
219:reverse mathematics
201:, then we say that
147:Robinson arithmetic
65:is consistent then
2375:Mathematical logic
1968:Transfer principle
1931:Semantics of logic
1916:Categorical theory
1892:Non-standard model
1406:Logical connective
533:Information theory
482:Mathematical logic
349:
311:
225:, since this is a
32:mathematical logic
2357:
2356:
2206:
2205:
2138:Abstract category
1941:Theories of truth
1751:Rule of inference
1741:Natural deduction
1722:
1721:
1267:
1266:
972:Cartesian product
877:
876:
783:Many-valued logic
758:Boolean functions
641:Russell's paradox
616:diagonal argument
513:First-order logic
412:978-1-84890-050-9
85:then we say that
16:(Redirected from
2382:
2277:Effective method
2255:Cantor's theorem
2233:
2226:
2219:
2210:
2197:
2196:
2148:History of logic
2143:Category of sets
2036:Decision problem
1815:Ordinal analysis
1756:Sequent calculus
1654:Boolean algebras
1594:
1593:
1568:
1539:logical/constant
1293:
1279:
1202:ZermeloâFraenkel
953:Set operations:
888:
825:
656:
636:LöwenheimâSkolem
523:Formal semantics
475:
468:
461:
452:
432:Akihiro Kanamori
424:
423:
394:
358:
356:
355:
350:
348:
347:
320:
318:
317:
312:
310:
309:
283:the negation of
279:. For example:
243:Peano arithmetic
21:
2390:
2389:
2385:
2384:
2383:
2381:
2380:
2379:
2370:Large cardinals
2360:
2359:
2358:
2353:
2246:
2244:metamathematics
2237:
2207:
2202:
2191:
2184:
2129:Category theory
2119:Algebraic logic
2102:
2073:Lambda calculus
2011:Church encoding
1997:
1973:Truth predicate
1829:
1795:Complete theory
1718:
1587:
1583:
1579:
1574:
1566:
1286: and
1282:
1277:
1263:
1239:New Foundations
1207:axiom of choice
1190:
1152:Gödel numbering
1092: and
1084:
988:
873:
823:
804:
753:Boolean algebra
739:
703:Equiconsistency
668:Classical logic
645:
626:Halting problem
614: and
590: and
578: and
577:
572:Theorems (
567:
484:
479:
428:
427:
413:
397:
395:
391:
386:
374:
361:Aronszajn trees
339:
334:
333:
323:Aronszajn trees
301:
296:
295:
277:large cardinals
252:
248:
240:
236:
183:
143:metamathematics
103:
28:
23:
22:
15:
12:
11:
5:
2388:
2386:
2378:
2377:
2372:
2362:
2361:
2355:
2354:
2352:
2351:
2346:
2341:
2336:
2334:Satisfiability
2331:
2326:
2321:
2319:Interpretation
2316:
2311:
2306:
2301:
2296:
2291:
2290:
2289:
2279:
2274:
2269:
2264:
2257:
2251:
2248:
2247:
2238:
2236:
2235:
2228:
2221:
2213:
2204:
2203:
2189:
2186:
2185:
2183:
2182:
2177:
2172:
2167:
2162:
2161:
2160:
2150:
2145:
2140:
2131:
2126:
2121:
2116:
2114:Abstract logic
2110:
2108:
2104:
2103:
2101:
2100:
2095:
2093:Turing machine
2090:
2085:
2080:
2075:
2070:
2065:
2064:
2063:
2058:
2053:
2048:
2043:
2033:
2031:Computable set
2028:
2023:
2018:
2013:
2007:
2005:
1999:
1998:
1996:
1995:
1990:
1985:
1980:
1975:
1970:
1965:
1960:
1959:
1958:
1953:
1948:
1938:
1933:
1928:
1926:Satisfiability
1923:
1918:
1913:
1912:
1911:
1901:
1900:
1899:
1889:
1888:
1887:
1882:
1877:
1872:
1867:
1857:
1856:
1855:
1850:
1843:Interpretation
1839:
1837:
1831:
1830:
1828:
1827:
1822:
1817:
1812:
1807:
1797:
1792:
1791:
1790:
1789:
1788:
1778:
1773:
1763:
1758:
1753:
1748:
1743:
1738:
1732:
1730:
1724:
1723:
1720:
1719:
1717:
1716:
1708:
1707:
1706:
1705:
1700:
1699:
1698:
1693:
1688:
1668:
1667:
1666:
1664:minimal axioms
1661:
1650:
1649:
1648:
1637:
1636:
1635:
1630:
1625:
1620:
1615:
1610:
1597:
1595:
1576:
1575:
1573:
1572:
1571:
1570:
1558:
1553:
1552:
1551:
1546:
1541:
1536:
1526:
1521:
1516:
1511:
1510:
1509:
1504:
1494:
1493:
1492:
1487:
1482:
1477:
1467:
1462:
1461:
1460:
1455:
1450:
1440:
1439:
1438:
1433:
1428:
1423:
1418:
1413:
1403:
1398:
1393:
1388:
1387:
1386:
1381:
1376:
1371:
1361:
1356:
1354:Formation rule
1351:
1346:
1345:
1344:
1339:
1329:
1328:
1327:
1317:
1312:
1307:
1302:
1296:
1290:
1273:Formal systems
1269:
1268:
1265:
1264:
1262:
1261:
1256:
1251:
1246:
1241:
1236:
1231:
1226:
1221:
1216:
1215:
1214:
1209:
1198:
1196:
1192:
1191:
1189:
1188:
1187:
1186:
1176:
1171:
1170:
1169:
1162:Large cardinal
1159:
1154:
1149:
1144:
1139:
1125:
1124:
1123:
1118:
1113:
1098:
1096:
1086:
1085:
1083:
1082:
1081:
1080:
1075:
1070:
1060:
1055:
1050:
1045:
1040:
1035:
1030:
1025:
1020:
1015:
1010:
1005:
999:
997:
990:
989:
987:
986:
985:
984:
979:
974:
969:
964:
959:
951:
950:
949:
944:
934:
929:
927:Extensionality
924:
922:Ordinal number
919:
909:
904:
903:
902:
891:
885:
879:
878:
875:
874:
872:
871:
866:
861:
856:
851:
846:
841:
840:
839:
829:
828:
827:
814:
812:
806:
805:
803:
802:
801:
800:
795:
790:
780:
775:
770:
765:
760:
755:
749:
747:
741:
740:
738:
737:
732:
727:
722:
717:
712:
707:
706:
705:
695:
690:
685:
680:
675:
670:
664:
662:
653:
647:
646:
644:
643:
638:
633:
628:
623:
618:
606:Cantor's
604:
599:
594:
584:
582:
569:
568:
566:
565:
560:
555:
550:
545:
540:
535:
530:
525:
520:
515:
510:
505:
504:
503:
492:
490:
486:
485:
480:
478:
477:
470:
463:
455:
449:
448:
426:
425:
411:
399:Kunen, Kenneth
388:
387:
385:
382:
381:
380:
373:
370:
369:
368:
346:
342:
330:
327:Mahlo cardinal
308:
304:
292:
250:
246:
238:
234:
182:
179:
102:
99:
95:equiconsistent
40:equiconsistent
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2387:
2376:
2373:
2371:
2368:
2367:
2365:
2350:
2347:
2345:
2342:
2340:
2337:
2335:
2332:
2330:
2327:
2325:
2322:
2320:
2317:
2315:
2312:
2310:
2307:
2305:
2302:
2300:
2297:
2295:
2292:
2288:
2285:
2284:
2283:
2280:
2278:
2275:
2273:
2270:
2268:
2265:
2263:
2262:
2258:
2256:
2253:
2252:
2249:
2245:
2241:
2234:
2229:
2227:
2222:
2220:
2215:
2214:
2211:
2201:
2200:
2195:
2187:
2181:
2178:
2176:
2173:
2171:
2168:
2166:
2163:
2159:
2156:
2155:
2154:
2151:
2149:
2146:
2144:
2141:
2139:
2135:
2132:
2130:
2127:
2125:
2122:
2120:
2117:
2115:
2112:
2111:
2109:
2105:
2099:
2096:
2094:
2091:
2089:
2088:Recursive set
2086:
2084:
2081:
2079:
2076:
2074:
2071:
2069:
2066:
2062:
2059:
2057:
2054:
2052:
2049:
2047:
2044:
2042:
2039:
2038:
2037:
2034:
2032:
2029:
2027:
2024:
2022:
2019:
2017:
2014:
2012:
2009:
2008:
2006:
2004:
2000:
1994:
1991:
1989:
1986:
1984:
1981:
1979:
1976:
1974:
1971:
1969:
1966:
1964:
1961:
1957:
1954:
1952:
1949:
1947:
1944:
1943:
1942:
1939:
1937:
1934:
1932:
1929:
1927:
1924:
1922:
1919:
1917:
1914:
1910:
1907:
1906:
1905:
1902:
1898:
1897:of arithmetic
1895:
1894:
1893:
1890:
1886:
1883:
1881:
1878:
1876:
1873:
1871:
1868:
1866:
1863:
1862:
1861:
1858:
1854:
1851:
1849:
1846:
1845:
1844:
1841:
1840:
1838:
1836:
1832:
1826:
1823:
1821:
1818:
1816:
1813:
1811:
1808:
1805:
1804:from ZFC
1801:
1798:
1796:
1793:
1787:
1784:
1783:
1782:
1779:
1777:
1774:
1772:
1769:
1768:
1767:
1764:
1762:
1759:
1757:
1754:
1752:
1749:
1747:
1744:
1742:
1739:
1737:
1734:
1733:
1731:
1729:
1725:
1715:
1714:
1710:
1709:
1704:
1703:non-Euclidean
1701:
1697:
1694:
1692:
1689:
1687:
1686:
1682:
1681:
1679:
1676:
1675:
1673:
1669:
1665:
1662:
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1580:Example
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1337:by definition
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1224:KripkeâPlatek
1222:
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1210:
1208:
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1203:
1200:
1199:
1197:
1193:
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1100:
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1069:
1068:constructible
1066:
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983:
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835:
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826:
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750:
748:
746:
745:Propositional
742:
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731:
728:
726:
723:
721:
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713:
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708:
704:
701:
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689:
686:
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681:
679:
676:
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673:Logical truth
671:
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629:
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457:
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453:
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446:3-540-00384-3
443:
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429:
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408:
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393:
390:
383:
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366:
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344:
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331:
328:
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293:
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286:
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273:
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128:
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118:
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108:
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88:
84:
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76:
72:
68:
64:
60:
56:
51:
49:
45:
41:
37:
33:
19:
2339:Independence
2314:Decidability
2309:Completeness
2259:
2190:
1988:Ultraproduct
1835:Model theory
1800:Independence
1736:Formal proof
1728:Proof theory
1711:
1684:
1641:real numbers
1613:second-order
1524:Substitution
1401:Metalanguage
1342:conservative
1315:Axiom schema
1259:Constructive
1229:MorseâKelley
1195:Set theories
1174:Aleph number
1167:inaccessible
1073:Grothendieck
957:intersection
844:Higher-order
832:Second-order
778:Truth tables
735:Venn diagram
702:
518:Formal proof
440:. Springer.
435:
402:
392:
274:
267:
210:
206:
205:has greater
202:
198:
194:
190:
186:
184:
174:
170:
166:
162:
158:
156:
129:
115:
104:
94:
90:
86:
82:
78:
74:
70:
66:
62:
58:
54:
52:
39:
29:
2329:Metatheorem
2287:of geometry
2272:Consistency
2098:Type theory
2046:undecidable
1978:Truth value
1865:equivalence
1544:non-logical
1157:Enumeration
1147:Isomorphism
1094:cardinality
1078:Von Neumann
1043:Ultrafilter
1008:Uncountable
942:equivalence
859:Quantifiers
849:Fixed-point
818:First-order
698:Consistency
683:Proposition
660:Traditional
631:Lindström's
621:Compactness
563:Type theory
508:Cardinality
119:proposed a
111:consistency
101:Consistency
44:consistency
2364:Categories
1909:elementary
1602:arithmetic
1470:Quantifier
1448:functional
1320:Expression
1038:Transitive
982:identities
967:complement
900:hereditary
883:Set theory
421:1262.03001
403:Set theory
384:References
227:computable
223:set theory
125:arithmetic
48:vice versa
2304:Soundness
2240:Metalogic
2180:Supertask
2083:Recursion
2041:decidable
1875:saturated
1853:of models
1776:deductive
1771:axiomatic
1691:Hilbert's
1678:Euclidean
1659:canonical
1582:axiomatic
1514:Signature
1443:Predicate
1332:Extension
1254:Ackermann
1179:Operation
1058:Universal
1048:Recursive
1023:Singleton
1018:Inhabited
1003:Countable
993:Types of
977:power set
947:partition
864:Predicate
810:Predicate
725:Syllogism
715:Soundness
688:Inference
678:Tautology
580:paradoxes
341:ω
303:ω
2165:Logicism
2158:timeline
2134:Concrete
1993:Validity
1963:T-schema
1956:Kripke's
1951:Tarski's
1946:semantic
1936:Strength
1885:submodel
1880:spectrum
1848:function
1696:Tarski's
1685:Elements
1672:geometry
1628:Robinson
1549:variable
1534:function
1507:spectrum
1497:Sentence
1453:variable
1396:Language
1349:Relation
1310:Automata
1300:Alphabet
1284:language
1138:-jection
1116:codomain
1102:Function
1063:Universe
1033:Infinite
937:Relation
720:Validity
710:Argument
608:theorem,
434:(2003).
401:(2011),
372:See also
249:and ZFC+
36:theories
2107:Related
1904:Diagram
1802: (
1781:Hilbert
1766:Systems
1761:Theorem
1639:of the
1584:systems
1364:Formula
1359:Grammar
1275: (
1219:General
932:Forcing
917:Element
837:Monadic
612:paradox
553:Theorem
489:General
259:forcing
121:program
117:Hilbert
42:if the
1870:finite
1633:Skolem
1586:
1561:Theory
1529:Symbol
1519:String
1502:atomic
1379:ground
1374:closed
1369:atomic
1325:ground
1288:syntax
1184:binary
1111:domain
1028:Finite
793:finite
651:Logics
610:
558:Theory
444:
419:
409:
217:, the
193:, but
77:. If
34:, two
1860:Model
1608:Peano
1465:Proof
1305:Arity
1234:Naive
1121:image
1053:Fuzzy
1013:Empty
962:union
907:Class
548:Model
538:Lemma
496:Axiom
209:than
152:axiom
131:Gödel
2242:and
1983:Type
1786:list
1590:list
1567:list
1556:Term
1490:rank
1384:open
1278:list
1090:Maps
995:sets
854:Free
824:list
574:list
501:list
442:ISBN
407:ISBN
161:and
93:are
89:and
38:are
1670:of
1652:of
1600:of
1132:Sur
1106:Map
913:Ur-
895:Set
417:Zbl
265:).
231:ZFC
185:If
133:'s
73:is
30:In
2366::
2056:NP
1680::
1674::
1604::
1281:),
1136:Bi
1128:In
415:,
113:.
97:.
2232:e
2225:t
2218:v
2136:/
2051:P
1806:)
1592:)
1588:(
1485:â
1480:!
1475:â
1436:=
1431:â
1426:â
1421:â§
1416:âš
1411:ÂŹ
1134:/
1130:/
1104:/
915:)
911:(
798:â
788:3
576:)
474:e
467:t
460:v
396:*
367:.
359:-
345:2
329:,
321:-
307:2
291:,
251:B
247:A
239:B
235:A
211:T
203:S
199:T
195:S
191:S
187:T
171:T
167:S
163:T
159:S
91:T
87:S
83:T
79:S
71:T
67:T
63:S
59:S
55:T
20:)
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