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160:, which asserts that for every theory that can be formulated in the logic, all semantically valid statements are provable theorems (for an appropriate sense of "semantically valid").
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involves use of only finitely many premises. In the case of modal logics, the collection of maximal consistent sets extending a theory
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that are consistent and rich enough to allow general mathematical reasoning to be formulated cannot be complete, as demonstrated by
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Complete theories are closed under a number of conditions internally modelling the
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402:. Their existence in a given case is usually a straightforward consequence of
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414:(closed under the necessitation rule) can be given the structure of a
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contains the sentence or its negation but not both (that is, either
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Maximal consistent sets are a fundamental tool in the
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in the theory's language, either that formula or its
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551:(Fourth ed.). Chapman & Hall. p. 86.
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164:is about this latter kind of completeness.
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156:is distinct from the notion of a complete
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430:Some examples of complete theories are:
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134:{\displaystyle T\vdash \neg \varphi }
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147:Gödel's first incompleteness theorem
2521:Analytic and synthetic propositions
2392:Formal semantics (natural language)
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549:Introduction to Mathematical Logic
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105:{\displaystyle T\vdash \varphi }
40:is provable. That is, for every
422:, called the canonical model.
1:
2268:History of mathematical logic
540:
220:{\displaystyle A\land B\in S}
141:). Recursively axiomatizable
2794:. You can help Knowledge by
2193:Primitive recursive function
327:{\displaystyle A\lor B\in S}
162:Gödel's completeness theorem
7:
547:Mendelson, Elliott (1997).
507:
460:algebraically closed fields
425:
406:, based on the idea that a
10:
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1257:SchröderâBernstein theorem
984:Monadic predicate calculus
643:Foundations of mathematics
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2239:Automated theorem proving
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59:{\displaystyle \varphi ,}
2851:Mathematical logic stubs
1940:Self-verifying theories
1761:Tarski's axiomatization
712:Tarski's undefinability
707:incompleteness theorems
493:group of three elements
476:uncountably categorical
2790:-related article is a
2314:Mathematics portal
1925:Proof of impossibility
1573:propositional variable
883:Propositional calculus
380:
379:{\displaystyle B\in S}
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353:{\displaystyle A\in S}
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282:For a set of formulas
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272:{\displaystyle B\in S}
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246:{\displaystyle A\in S}
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175:For a set of formulas
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2740:Philosophy portal
2183:Kolmogorov complexity
2136:Computably enumerable
2036:Model complete theory
1828:Principia Mathematica
888:Propositional formula
717:BanachâTarski paradox
486:countably categorical
435:Presburger arithmetic
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1327:continuum hypothesis
845:Square of opposition
703:Gödel's completeness
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143:first-order theories
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2402:Philosophy of logic
2285:Mathematical object
2176:P versus NP problem
2141:Computable function
1935:Reverse mathematics
1861:Logical consequence
1738:primitive recursive
1733:elementary function
1506:Free/bound variable
1359:TarskiâGrothendieck
878:Logical connectives
808:Logical equivalence
658:Logical consequence
2841:Mathematical logic
2788:mathematical logic
2701:Rules of inference
2670:Mathematical logic
2412:Semantics of logic
2083:Transfer principle
2046:Semantics of logic
2031:Categorical theory
2007:Non-standard model
1521:Logical connective
648:Information theory
597:Mathematical logic
530:Lindenbaum's lemma
502:elementary diagram
470:real closed fields
444:Euclidean geometry
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2056:Theories of truth
1866:Rule of inference
1856:Natural deduction
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1087:Cartesian product
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898:Many-valued logic
873:Boolean functions
756:Russell's paradox
731:diagonal argument
628:First-order logic
558:978-0-412-80830-2
522:Philosophy portal
455:without endpoints
295:{\displaystyle S}
188:{\displaystyle S}
79:{\displaystyle T}
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2258:Category of sets
2151:Decision problem
1930:Ordinal analysis
1871:Sequent calculus
1769:Boolean algebras
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1654:logical/constant
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2088:Truth predicate
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1910:Complete theory
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1401: and
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1354:New Foundations
1322:axiom of choice
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1267:Gödel numbering
1207: and
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868:Boolean algebra
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818:Equiconsistency
783:Classical logic
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741:Halting problem
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687:Theorems (
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535:ĆoĆâVaught test
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498:True arithmetic
440:Tarski's axioms
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396:classical logic
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2041:Satisfiability
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34:closed formula
32:and for every
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2693:
2691:
2687:
2681:
2678:
2676:
2673:
2671:
2668:
2667:
2665:
2663:
2659:
2656:
2652:
2646:
2643:
2641:
2638:
2636:
2633:
2631:
2628:
2626:
2623:
2621:
2618:
2616:
2613:
2611:
2608:
2606:
2603:
2601:
2598:
2596:
2595:Logical truth
2593:
2591:
2588:
2586:
2583:
2579:
2576:
2575:
2574:
2571:
2569:
2566:
2564:
2561:
2559:
2556:
2554:
2551:
2547:
2544:
2542:
2539:
2538:
2537:
2536:Contradiction
2534:
2532:
2529:
2527:
2524:
2522:
2519:
2517:
2514:
2513:
2511:
2507:
2497:
2494:
2492:
2489:
2487:
2484:
2482:
2481:Argumentation
2479:
2478:
2476:
2472:
2466:
2465:Philosophical
2463:
2461:
2460:Non-classical
2458:
2456:
2453:
2449:
2446:
2444:
2441:
2440:
2439:
2436:
2434:
2431:
2430:
2428:
2424:
2418:
2415:
2413:
2410:
2408:
2405:
2403:
2400:
2398:
2395:
2393:
2390:
2388:
2385:
2384:
2382:
2378:
2372:
2369:
2367:
2364:
2363:
2360:
2356:
2349:
2344:
2342:
2337:
2335:
2330:
2329:
2326:
2316:
2315:
2310:
2302:
2296:
2293:
2291:
2288:
2286:
2283:
2281:
2278:
2274:
2271:
2270:
2269:
2266:
2264:
2261:
2259:
2256:
2254:
2250:
2247:
2245:
2242:
2240:
2237:
2235:
2232:
2230:
2227:
2226:
2224:
2220:
2214:
2211:
2209:
2206:
2204:
2203:Recursive set
2201:
2199:
2196:
2194:
2191:
2189:
2186:
2184:
2181:
2177:
2174:
2172:
2169:
2167:
2164:
2162:
2159:
2157:
2154:
2153:
2152:
2149:
2147:
2144:
2142:
2139:
2137:
2134:
2132:
2129:
2127:
2124:
2123:
2121:
2119:
2115:
2109:
2106:
2104:
2101:
2099:
2096:
2094:
2091:
2089:
2086:
2084:
2081:
2079:
2076:
2072:
2069:
2067:
2064:
2062:
2059:
2058:
2057:
2054:
2052:
2049:
2047:
2044:
2042:
2039:
2037:
2034:
2032:
2029:
2025:
2022:
2021:
2020:
2017:
2013:
2012:of arithmetic
2010:
2009:
2008:
2005:
2001:
1998:
1996:
1993:
1991:
1988:
1986:
1983:
1981:
1978:
1977:
1976:
1973:
1969:
1966:
1964:
1961:
1960:
1959:
1956:
1955:
1953:
1951:
1947:
1941:
1938:
1936:
1933:
1931:
1928:
1926:
1923:
1920:
1919:from ZFC
1916:
1913:
1911:
1908:
1902:
1899:
1898:
1897:
1894:
1892:
1889:
1887:
1884:
1883:
1882:
1879:
1877:
1874:
1872:
1869:
1867:
1864:
1862:
1859:
1857:
1854:
1852:
1849:
1848:
1846:
1844:
1840:
1830:
1829:
1825:
1824:
1819:
1818:non-Euclidean
1816:
1812:
1809:
1807:
1804:
1802:
1801:
1797:
1796:
1794:
1791:
1790:
1788:
1784:
1780:
1777:
1775:
1772:
1771:
1770:
1766:
1762:
1759:
1758:
1757:
1753:
1749:
1746:
1744:
1741:
1739:
1736:
1734:
1731:
1729:
1726:
1724:
1721:
1720:
1718:
1714:
1713:
1711:
1706:
1700:
1695:Example
1692:
1684:
1679:
1678:
1677:
1674:
1672:
1669:
1665:
1662:
1660:
1657:
1655:
1652:
1650:
1647:
1646:
1645:
1642:
1640:
1637:
1635:
1632:
1630:
1627:
1623:
1620:
1618:
1615:
1614:
1613:
1610:
1606:
1603:
1601:
1598:
1596:
1593:
1591:
1588:
1587:
1586:
1583:
1581:
1578:
1574:
1571:
1569:
1566:
1564:
1561:
1560:
1559:
1556:
1552:
1549:
1547:
1544:
1542:
1539:
1537:
1534:
1532:
1529:
1527:
1524:
1523:
1522:
1519:
1517:
1514:
1512:
1509:
1507:
1504:
1500:
1497:
1495:
1492:
1490:
1487:
1485:
1482:
1481:
1480:
1477:
1475:
1472:
1470:
1467:
1465:
1462:
1458:
1455:
1453:
1452:by definition
1450:
1449:
1448:
1445:
1441:
1438:
1437:
1436:
1433:
1431:
1428:
1426:
1423:
1421:
1418:
1416:
1413:
1412:
1409:
1406:
1404:
1400:
1395:
1389:
1385:
1375:
1372:
1370:
1367:
1365:
1362:
1360:
1357:
1355:
1352:
1350:
1347:
1345:
1342:
1340:
1339:KripkeâPlatek
1337:
1335:
1332:
1328:
1325:
1323:
1320:
1319:
1318:
1315:
1314:
1312:
1308:
1300:
1297:
1296:
1295:
1292:
1290:
1287:
1283:
1280:
1279:
1278:
1275:
1273:
1270:
1268:
1265:
1263:
1260:
1258:
1255:
1252:
1248:
1244:
1241:
1237:
1234:
1232:
1229:
1227:
1224:
1223:
1222:
1218:
1215:
1214:
1212:
1210:
1206:
1202:
1194:
1191:
1189:
1186:
1184:
1183:constructible
1181:
1180:
1179:
1176:
1174:
1171:
1169:
1166:
1164:
1161:
1159:
1156:
1154:
1151:
1149:
1146:
1144:
1141:
1139:
1136:
1134:
1131:
1129:
1126:
1124:
1121:
1119:
1116:
1115:
1113:
1111:
1106:
1098:
1095:
1093:
1090:
1088:
1085:
1083:
1080:
1078:
1075:
1073:
1070:
1069:
1067:
1063:
1060:
1058:
1055:
1054:
1053:
1050:
1048:
1045:
1043:
1040:
1038:
1035:
1033:
1029:
1025:
1023:
1020:
1016:
1013:
1012:
1011:
1008:
1007:
1004:
1001:
999:
995:
985:
982:
980:
977:
975:
972:
970:
967:
965:
962:
960:
957:
953:
950:
949:
948:
945:
941:
936:
935:
934:
931:
930:
928:
926:
922:
914:
911:
909:
906:
904:
901:
900:
899:
896:
894:
891:
889:
886:
884:
881:
879:
876:
874:
871:
869:
866:
865:
863:
861:
860:Propositional
857:
851:
848:
846:
843:
841:
838:
836:
833:
831:
828:
826:
823:
819:
816:
815:
814:
811:
809:
806:
804:
801:
799:
796:
794:
791:
789:
788:Logical truth
786:
784:
781:
780:
778:
776:
772:
769:
767:
763:
757:
754:
752:
749:
747:
744:
742:
739:
737:
734:
732:
728:
724:
720:
718:
715:
713:
710:
708:
704:
701:
700:
698:
696:
690:
685:
679:
676:
674:
671:
669:
666:
664:
661:
659:
656:
654:
651:
649:
646:
644:
641:
639:
636:
634:
631:
629:
626:
624:
621:
617:
614:
613:
612:
609:
608:
606:
602:
598:
591:
586:
584:
579:
577:
572:
571:
568:
560:
554:
550:
545:
544:
536:
533:
531:
528:
527:
523:
512:
503:
500:or any other
499:
496:
494:
490:
487:
483:
480:
477:
473:
471:
467:
465:
461:
457:
454:
453:linear orders
451:
447:
445:
441:
438:
436:
433:
432:
431:
423:
421:
417:
413:
409:
408:contradiction
405:
401:
397:
393:
373:
370:
367:
347:
344:
341:
321:
318:
315:
312:
309:
289:
281:
266:
263:
260:
240:
237:
234:
214:
211:
208:
205:
202:
182:
174:
173:
172:
170:
165:
163:
159:
155:
150:
148:
144:
128:
122:
119:
99:
96:
93:
73:
53:
50:
43:
39:
35:
31:
27:
23:
19:
2846:Model theory
2796:expanding it
2785:
2770:
2635:Substitution
2455:Mathematical
2380:Major fields
2305:
2103:Ultraproduct
1950:Model theory
1915:Independence
1909:
1851:Formal proof
1843:Proof theory
1826:
1799:
1756:real numbers
1728:second-order
1639:Substitution
1516:Metalanguage
1457:conservative
1430:Axiom schema
1374:Constructive
1344:MorseâKelley
1310:Set theories
1289:Aleph number
1282:inaccessible
1188:Grothendieck
1072:intersection
959:Higher-order
947:Second-order
893:Truth tables
850:Venn diagram
633:Formal proof
548:
429:
419:
411:
404:Zorn's lemma
392:model theory
389:
166:
157:
153:
151:
25:
15:
2750:WikiProject
2620:Proposition
2615:Probability
2568:Description
2509:Foundations
2213:Type theory
2161:undecidable
2093:Truth value
1980:equivalence
1659:non-logical
1272:Enumeration
1262:Isomorphism
1209:cardinality
1193:Von Neumann
1158:Ultrafilter
1123:Uncountable
1057:equivalence
974:Quantifiers
964:Fixed-point
933:First-order
813:Consistency
798:Proposition
775:Traditional
746:Lindström's
736:Compactness
678:Type theory
623:Cardinality
462:of a given
400:modal logic
66:the theory
2835:Categories
2680:Set theory
2578:Linguistic
2573:Entailment
2563:Definition
2531:Consequent
2526:Antecedent
2024:elementary
1717:arithmetic
1585:Quantifier
1563:functional
1435:Expression
1153:Transitive
1097:identities
1082:complement
1015:hereditary
998:Set theory
541:References
30:consistent
2711:Fallacies
2706:Paradoxes
2696:Logicians
2630:Statement
2625:Reference
2590:Induction
2553:Deduction
2516:Abduction
2486:Metalogic
2433:Classical
2397:Inference
2295:Supertask
2198:Recursion
2156:decidable
1990:saturated
1968:of models
1891:deductive
1886:axiomatic
1806:Hilbert's
1793:Euclidean
1774:canonical
1697:axiomatic
1629:Signature
1558:Predicate
1447:Extension
1369:Ackermann
1294:Operation
1173:Universal
1163:Recursive
1138:Singleton
1133:Inhabited
1118:Countable
1108:Types of
1092:power set
1062:partition
979:Predicate
925:Predicate
840:Syllogism
830:Soundness
803:Inference
793:Tautology
695:paradoxes
479:countable
371:∈
345:∈
319:∈
313:∨
264:∈
238:∈
212:∈
206:∧
129:φ
126:¬
123:⊢
100:φ
97:⊢
51:φ
28:if it is
2745:Category
2645:Validity
2546:Antinomy
2474:Theories
2438:Informal
2280:Logicism
2273:timeline
2249:Concrete
2108:Validity
2078:T-schema
2071:Kripke's
2066:Tarski's
2061:semantic
2051:Strength
2000:submodel
1995:spectrum
1963:function
1811:Tarski's
1800:Elements
1787:geometry
1743:Robinson
1664:variable
1649:function
1622:spectrum
1612:Sentence
1568:variable
1511:Language
1464:Relation
1425:Automata
1415:Alphabet
1399:language
1253:-jection
1231:codomain
1217:Function
1178:Universe
1148:Infinite
1052:Relation
835:Validity
825:Argument
723:theorem,
508:See also
426:Examples
169:T-schema
154:complete
42:sentence
38:negation
26:complete
2760:changes
2752: (
2610:Premise
2541:Paradox
2371:History
2366:Outline
2222:Related
2019:Diagram
1917: (
1896:Hilbert
1881:Systems
1876:Theorem
1754:of the
1699:systems
1479:Formula
1474:Grammar
1390: (
1334:General
1047:Forcing
1032:Element
952:Monadic
727:paradox
668:Theorem
604:General
2662:topics
2448:Reason
2426:Logics
2417:Syntax
1985:finite
1748:Skolem
1701:
1676:Theory
1644:Symbol
1634:String
1617:atomic
1494:ground
1489:closed
1484:atomic
1440:ground
1403:syntax
1299:binary
1226:domain
1143:Finite
908:finite
766:Logics
725:
673:Theory
555:
484:Every
481:theory
474:Every
22:theory
2786:This
2689:other
2654:Lists
2640:Truth
2407:Proof
2355:Logic
1975:Model
1723:Peano
1580:Proof
1420:Arity
1349:Naive
1236:image
1168:Fuzzy
1128:Empty
1077:union
1022:Class
663:Model
653:Lemma
611:Axiom
450:dense
416:model
158:logic
2792:stub
2754:talk
2600:Name
2585:Form
2098:Type
1901:list
1705:list
1682:list
1671:Term
1605:rank
1499:open
1393:list
1205:Maps
1110:sets
969:Free
939:list
689:list
616:list
553:ISBN
442:for
398:and
253:and
20:, a
2496:Set
1785:of
1767:of
1715:of
1247:Sur
1221:Map
1028:Ur-
1010:Set
418:of
394:of
360:or
112:or
24:is
16:In
2837::
2171:NP
1795::
1789::
1719::
1396:),
1251:Bi
1243:In
491:A
302::
195::
171::
149:.
2823:e
2816:t
2809:v
2798:.
2756:)
2347:e
2340:t
2333:v
2251:/
2166:P
1921:)
1707:)
1703:(
1600:â
1595:!
1590:â
1551:=
1546:â
1541:â
1536:â§
1531:âš
1526:ÂŹ
1249:/
1245:/
1219:/
1030:)
1026:(
913:â
903:3
691:)
589:e
582:t
575:v
561:.
420:T
412:T
386:.
374:S
368:B
348:S
342:A
322:S
316:B
310:A
290:S
279:,
267:S
261:B
241:S
235:A
215:S
209:B
203:A
183:S
120:T
94:T
74:T
54:,
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