3756:
27:
1443:
The compactness theorem says that if a formula φ is a logical consequence of a (possibly infinite) set of formulas Γ then it is a logical consequence of a finite subset of Γ. This is an immediate consequence of the completeness theorem, because only a finite number of axioms from Γ can be mentioned
1474:
without axiom of choice, and thus the completeness and compactness theorems for countable languages are provable in ZF. However the situation is different when the language is of arbitrary large cardinality since then, though the completeness and compactness theorems remain provably equivalent to
237:
if every logically valid formula is the conclusion of some formal deduction, and the completeness theorem for a particular deductive system is the theorem that it is complete in this sense. Thus, in a sense, there is a different completeness theorem for each deductive system. A converse to
247:
If some specific deductive system of first-order logic is sound and complete, then it is "perfect" (a formula is provable if and only if it is logically valid), thus equivalent to any other deductive system with the same quality (any proof in one system can be converted into the other).
1503:), and the set of logically-valid formulas in second-order logic is not recursively enumerable. The same is true of all higher-order logics. It is possible to produce sound deductive systems for higher-order logics, but no such system can be complete.
232:
for the language of the formula (i.e. for any assignment of values to the variables of the formula). To formally state, and then prove, the completeness theorem, it is necessary to also define a deductive system. A deductive system is called
1279:, i.e. it states that some finitistic property is true of all natural numbers; so if it is false, then some natural number is a counterexample. If this counterexample existed within the standard natural numbers, its existence would disprove
1483:. In particular, no theory extending ZF can prove either the completeness or compactness theorems over arbitrary (possibly uncountable) languages without also proving the ultrafilter lemma on a set of the same cardinality.
1392:
non-standard with a non-equivalent provability predicate and a non-equivalent way to interpret its own construction, so that this construction is non-recursive (as recursive definitions would be unambiguous).
1444:
in a formal deduction of φ, and the soundness of the deductive system then implies φ is a logical consequence of this finite set. This proof of the compactness theorem is originally due to Gödel.
877:
This comes in contrast with the direct meaning of the notion of semantic consequence, that quantifies over all structures in a particular language, which is clearly not a recursive definition.
273:, the fact that only logically valid formulae are provable in the deductive system. Together with soundness (whose verification is easy), this theorem implies that a formula is logically valid
880:
Also, it makes the concept of "provability", and thus of "theorem", a clear concept that only depends on the chosen system of axioms of the theory, and not on the choice of a proof system.
799:
256:
We first fix a deductive system of first-order predicate calculus, choosing any of the well-known equivalent systems. Gödel's original proof assumed the
Hilbert-Ackermann proof system.
891:
show that there are inherent limitations to what can be proven within any given first-order theory in mathematics. The "incompleteness" in their name refers to another meaning of
267:
Thus, the deductive system is "complete" in the sense that no additional inference rules are required to prove all the logically valid formulae. A converse to completeness is
1012:
758:
709:
680:
651:
483:
428:
371:
983:
825:
511:
454:
325:
1454:. When considered over a countable language, the completeness and compactness theorems are equivalent to each other and equivalent to a weak form of choice known as
1371:
1304:
1269:
1242:
1175:
1125:
1078:
729:
847:
by an arithmetical formula whose free variables are the arguments of the symbol. (In many cases, we will need to assume, as a hypothesis of the construction, that
874:
first-order theory, by enumerating all the possible formal deductions from the axioms of the theory, and use this to produce an enumeration of their conclusions.
1414:
1344:
1324:
1215:
1195:
1145:
1098:
1035:
957:
937:
917:
608:
588:
1569:
1533:
3780:
2135:
851:
is consistent, since Peano arithmetic may not prove that fact.) However, the definition expressed by this formula is not recursive (but is, in general,
852:
1845:
1447:
Conversely, for many deductive systems, it is possible to prove the completeness theorem as an effective consequence of the compactness theorem.
2810:
2893:
2034:
1656:
Gisbert F. R. Hasenjaeger (Mar 1953). "Eine
Bemerkung zu Henkin's Beweis für die Vollständigkeit des Prädikatenkalküls der Ersten Stufe".
1326:; but the incompleteness theorem showed this to be impossible, so the counterexample must not be a standard number, and thus any model of
1707:
175:
that the hard part of the proof can be presented as the Model
Existence Theorem (published in 1949). Henkin's proof was simplified by
1782:
The same material as the dissertation, except with briefer proofs, more succinct explanations, and omitting the lengthy introduction.
264:
The completeness theorem says that if a formula is logically valid then there is a finite deduction (a formal proof) of the formula.
1509:
states that first-order logic is the strongest (subject to certain constraints) logic satisfying both compactness and completeness.
1536:
proceeded by reducing the problem to a special case for formulas in a certain syntactic form, and then handling this form with an
3207:
1499:, for example, does not have a completeness theorem for its standard semantics (though does have the completeness property for
1480:
3365:
2153:
1951:
1564:
888:
20:
3220:
2543:
896:
126:
1883:
1827:
3225:
3215:
2952:
2805:
2158:
516:
This more general theorem is used implicitly, for example, when a sentence is shown to be provable from the axioms of
3361:
1991:
2703:
3458:
3202:
2027:
1374:
2763:
2456:
1471:
763:
3800:
2197:
1976:
614:
3719:
3421:
3184:
3179:
3004:
2425:
2109:
1798:
229:
196:
76:
1551:
for any consistent first-order theory. James
Margetson (2004) developed a computerized formal proof using the
3714:
3497:
3414:
3127:
3058:
2935:
2177:
1613:
1748:
3639:
3465:
3151:
2785:
2384:
125:
as axioms. One sometimes says this as "anything true in all models is provable". (This does not contradict
3517:
3512:
3122:
2861:
2790:
2119:
2020:
2001:
1436:
are two cornerstones of first-order logic. While neither of these theorems can be proven in a completely
110:
1842:
3446:
3036:
2430:
2398:
2089:
1996:
1934:
378:
57:
3736:
3685:
3582:
3080:
3041:
2518:
2163:
1552:
523:
Gödel's original formulation is deduced by taking the particular case of a theory without any axiom.
2192:
1506:
988:
3577:
3507:
3046:
2898:
2881:
2604:
2084:
1919:
734:
685:
656:
31:
1455:
3409:
3386:
3347:
3233:
3174:
2820:
2740:
2584:
2528:
2141:
1939:
1876:
1421:
839:. Precisely, we can systematically define a model of any consistent effective first-order theory
630:
462:
407:
350:
3699:
3426:
3404:
3371:
3264:
3110:
3095:
3068:
3019:
2903:
2838:
2663:
2629:
2624:
2498:
2329:
2306:
1971:
1437:
1272:
1042:
962:
871:
804:
490:
433:
304:
536:
219:, for example, or by hand) that a given sequence (or tree) of formulae is indeed a deduction.
3795:
3790:
3785:
3629:
3482:
3274:
2992:
2728:
2634:
2493:
2478:
2359:
2334:
1700:
520:
by considering an arbitrary group and showing that the sentence is satisfied by that group.
211:. The definition of a deduction is such that it is finite and that it is possible to verify
3602:
3564:
3441:
3245:
3085:
3009:
2987:
2815:
2773:
2672:
2639:
2503:
2291:
2202:
1966:
1961:
1913:
1517:
1349:
1282:
1247:
1220:
1153:
1103:
1056:
714:
563:
in our deductive system. The model existence theorem says that for any first-order theory
79:(only the simplest 8 are shown left). By Gödel's completeness result, it must hence have a
50:
1769:
8:
3731:
3622:
3607:
3587:
3544:
3431:
3381:
3307:
3252:
3189:
2982:
2977:
2925:
2693:
2682:
2354:
2254:
2182:
2173:
2169:
2104:
2099:
1907:
1451:
1433:
1046:
513:, and thus that syntactic and semantic consequence are equivalent for first-order logic.
286:
176:
1547:'s proof, rather than with Gödel's original proof. Henkin's proof directly constructs a
1388:
obtained by the systematic construction of the arithmetical model existence theorem, is
3760:
3529:
3492:
3477:
3470:
3453:
3257:
3239:
3105:
3031:
3014:
2967:
2780:
2689:
2523:
2508:
2468:
2420:
2405:
2393:
2349:
2324:
2094:
2043:
1869:
1773:
1681:
1673:
1638:
1630:
1590:
1496:
1399:
1329:
1309:
1200:
1180:
1130:
1083:
1020:
942:
922:
902:
593:
573:
91:
2713:
3755:
3695:
3502:
3312:
3302:
3194:
3075:
2910:
2886:
2667:
2651:
2556:
2533:
2410:
2379:
2344:
2239:
2074:
1804:
1794:
1777:
1492:
269:
240:
192:
103:
99:
80:
1685:
1642:
244:, the fact that only logically valid formulas are provable in the deductive system.
3709:
3704:
3597:
3554:
3376:
3337:
3332:
3317:
3143:
3100:
2997:
2795:
2745:
2319:
2281:
1929:
1835:
1765:
1757:
1665:
1622:
1611:
Leon Henkin (Sep 1949). "The completeness of the first-order functional calculus".
1521:
1500:
1463:
1450:
The ineffectiveness of the completeness theorem can be measured along the lines of
836:
224:
188:
26:
121:
is a model of φ, then there is a (first-order) proof of φ using the statements of
3690:
3680:
3634:
3617:
3572:
3534:
3436:
3356:
3163:
3090:
3063:
3051:
2957:
2871:
2845:
2800:
2768:
2569:
2371:
2314:
2264:
2229:
2187:
1896:
1849:
1476:
1861:
835:
The model existence theorem and its proof can be formalized in the framework of
117:
is such a theory, and φ is a sentence (in the same language) and every model of
3675:
3654:
3612:
3592:
3487:
3342:
2940:
2930:
2920:
2915:
2849:
2723:
2599:
2488:
2483:
2461:
2062:
1986:
867:
866:
An important consequence of the completeness theorem is that it is possible to
485:
274:
204:
627:
Given Henkin's theorem, the completeness theorem can be proved as follows: If
3774:
3649:
3327:
2834:
2619:
2609:
2579:
2564:
2234:
1808:
157:
1831:
1743:
1728:
164:
3549:
3396:
3297:
3289:
3169:
3117:
3026:
2962:
2945:
2876:
2735:
2594:
2296:
2079:
1543:
In modern logic texts, Gödel's completeness theorem is usually proved with
1053:") must be incomplete in this sense, by explicitly constructing a sentence
517:
172:
153:
149:
1746:(1930). "Die Vollständigkeit der Axiome des logischen Funktionenkalküls".
1475:
each other in ZF, they are also provably equivalent to a weak form of the
3659:
3539:
2718:
2708:
2655:
2339:
2259:
2244:
2124:
2069:
1981:
1924:
1786:
1544:
1513:
532:
168:
1732:
1100:. The second incompleteness theorem extends this result by showing that
2589:
2444:
2415:
2221:
1761:
1677:
1634:
1548:
1420:(e.g. if it includes induction for bounded existential formulas), then
1038:
390:
682:
does not have models. By the contrapositive of Henkin's theorem, then
385:. The completeness theorem then says that for any first-order theory
3741:
3644:
2697:
2614:
2574:
2538:
2474:
2286:
2276:
2249:
2012:
1956:
1892:
621:
212:
95:
1669:
1626:
3726:
3524:
2972:
2677:
2271:
1595:
216:
3322:
2114:
19:
For the subsequent theories about the limits of provability, see
1855:
1470:
formulas). Weak Kőnig's lemma is provable in ZF, the system of
1197:, the completeness theorem implies the existence of a model of
1080:
which is demonstrably neither provable nor disprovable within
897:
model theory – Using the compactness and completeness theorems
883:
2866:
2212:
2057:
1589:
1440:
manner, each one can be effectively obtained from the other.
531:
The completeness theorem can also be understood in terms of
459:
Since the converse (soundness) also holds, it follows that
1427:
156:, which studies what can be formally proven in particular
137:
but true in the "standard" model of the natural numbers: φ
1655:
152:, which deals with what is true in different models, and
285:
The theorem can be expressed more generally in terms of
1737:(Thesis). Doctoral dissertation. University Of Vienna.
1127:
can be chosen so that it expresses the consistency of
801:, and then by the properties of the deductive system,
1793:. Hochschultext (Springer-Verlag). London: Springer.
1402:
1352:
1332:
1312:
1285:
1250:
1223:
1203:
1183:
1156:
1133:
1106:
1086:
1059:
1023:
991:
965:
945:
925:
905:
807:
766:
737:
717:
688:
659:
633:
596:
576:
493:
465:
436:
410:
353:
307:
199:. Common to all deductive systems is the notion of a
1701:
1424:
shows that it has no recursive non-standard models.
148:
The completeness theorem makes a close link between
109:
The completeness theorem applies to any first-order
843:in Peano arithmetic by interpreting each symbol of
711:
is syntactically inconsistent. So a contradiction (
624:
first-order theory has a finite or countable model.
1491:The completeness theorem is a central property of
1408:
1365:
1338:
1318:
1298:
1263:
1236:
1209:
1189:
1169:
1139:
1119:
1092:
1072:
1029:
1006:
977:
951:
931:
911:
819:
793:
752:
723:
703:
674:
645:
602:
582:
505:
477:
448:
422:
365:
319:
203:. This is a sequence (or, in some cases, a finite
1891:
1017:The first incompleteness theorem states that any
141:is false in some other, "non-standard" models of
3772:
1698:
1486:
259:
1570:Original proof of Gödel's completeness theorem
2028:
1877:
1555:theorem prover. Other proofs are also known.
919:is complete (or decidable) if every sentence
830:
1739:The first proof of the completeness theorem.
277:it is the conclusion of a formal deduction.
191:for first-order logic, including systems of
1785:
1610:
884:Relationship to the incompleteness theorems
794:{\displaystyle (T\cup \lnot s)\vdash \bot }
3781:Theorems in the foundations of mathematics
2220:
2035:
2021:
1884:
1870:
526:
207:) of formulae with a specially designated
94:that establishes a correspondence between
1734:Über die Vollständigkeit des Logikkalküls
1594:
1588:
1512:A completeness theorem can be proved for
613:Another version, with connections to the
16:Fundamental theorem in mathematical logic
1854:Detlovs, Vilnis, and Podnieks, Karlis, "
25:
1428:Relationship to the compactness theorem
133:that is unprovable in a certain theory
3773:
2042:
1458:, with the equivalence provable in RCA
335:in our deductive system. We say that
2016:
1865:
1742:
1727:
167:in 1929. It was then simplified when
280:
1856:Introduction to mathematical logic.
1828:Stanford Encyclopedia of Philosophy
1495:that does not hold for all logics.
1416:is at least slightly stronger than
13:
1791:Introduction to Mathematical Logic
1721:
998:
788:
776:
744:
718:
695:
666:
590:is syntactically consistent, then
14:
3812:
1821:
1432:The completeness theorem and the
870:the semantic consequences of any
3754:
1713:from the original on 2006-02-22.
620:Every syntactically consistent,
567:with a well-orderable language,
222:A first-order formula is called
182:
1952:Gödel's incompleteness theorems
1565:Gödel's incompleteness theorems
1466:restricted to induction over Σ
889:Gödel's incompleteness theorems
861:
760:in the deductive system. Hence
535:, as a consequence of Henkin's
127:Gödel's incompleteness theorem
21:Gödel's incompleteness theorems
1815:"Gödel's completeness theorem"
1692:
1649:
1604:
1582:
1007:{\displaystyle T\vdash \neg S}
782:
767:
1:
3715:History of mathematical logic
1658:The Journal of Symbolic Logic
1614:The Journal of Symbolic Logic
1575:
1534:original proof of the theorem
753:{\displaystyle T\cup \lnot s}
704:{\displaystyle T\cup \lnot s}
675:{\displaystyle T\cup \lnot s}
3640:Primitive recursive function
1947:Gödel's completeness theorem
1699:James Margetson (Sep 2004).
1601:(p.17). Accessed 2022-12-01.
1487:Completeness in other logics
260:Gödel's original formulation
251:
129:, which is about a formula φ
90:is a fundamental theorem in
88:Gödel's completeness theorem
7:
1558:
1472:Zermelo–Fraenkel set theory
1462:(a second-order variant of
393:language, and any sentence
10:
3817:
2704:Schröder–Bernstein theorem
2431:Monadic predicate calculus
2090:Foundations of mathematics
1935:Foundations of mathematics
1749:Monatshefte für Mathematik
831:As a theorem of arithmetic
646:{\displaystyle T\models s}
478:{\displaystyle T\models s}
423:{\displaystyle T\models s}
366:{\displaystyle T\models s}
289:. We say that a sentence
18:
3750:
3737:Philosophy of mathematics
3686:Automated theorem proving
3668:
3563:
3395:
3288:
3140:
2857:
2833:
2811:Von Neumann–Bernays–Gödel
2756:
2650:
2554:
2452:
2443:
2370:
2305:
2211:
2133:
2050:
1903:
1527:
978:{\displaystyle T\vdash S}
820:{\displaystyle T\vdash s}
506:{\displaystyle T\vdash s}
449:{\displaystyle T\vdash s}
320:{\displaystyle T\vdash s}
1977:Löwenheim–Skolem theorem
615:Löwenheim–Skolem theorem
547:if there is no sentence
545:syntactically consistent
539:. We say that a theory
3387:Self-verifying theories
3208:Tarski's axiomatization
2159:Tarski's undefinability
2154:incompleteness theorems
2002:Use–mention distinction
537:model existence theorem
527:Model existence theorem
228:if it is true in every
163:It was first proved by
3761:Mathematics portal
3372:Proof of impossibility
3020:propositional variable
2330:Propositional calculus
1997:Type–token distinction
1410:
1380:In fact, the model of
1373:is false must include
1367:
1340:
1320:
1300:
1265:
1238:
1211:
1191:
1171:
1141:
1121:
1094:
1074:
1031:
1008:
979:
953:
933:
913:
821:
795:
754:
725:
705:
676:
647:
604:
584:
507:
479:
450:
424:
367:
321:
84:
3630:Kolmogorov complexity
3583:Computably enumerable
3483:Model complete theory
3275:Principia Mathematica
2335:Propositional formula
2164:Banach–Tarski paradox
1481:the ultrafilter lemma
1411:
1368:
1366:{\displaystyle S_{T}}
1341:
1321:
1301:
1299:{\displaystyle S_{T}}
1266:
1264:{\displaystyle S_{T}}
1239:
1237:{\displaystyle S_{T}}
1212:
1192:
1172:
1170:{\displaystyle S_{T}}
1142:
1122:
1120:{\displaystyle S_{T}}
1095:
1075:
1073:{\displaystyle S_{T}}
1032:
1009:
980:
954:
934:
914:
868:recursively enumerate
822:
796:
755:
726:
724:{\displaystyle \bot }
706:
677:
648:
605:
585:
508:
480:
451:
425:
368:
322:
295:syntactic consequence
197:Hilbert-style systems
29:
3578:Church–Turing thesis
3565:Computability theory
2774:continuum hypothesis
2292:Square of opposition
2150:Gödel's completeness
1920:Church–Turing thesis
1914:Entscheidungsproblem
1841:MacTutor biography:
1706:(Technical Report).
1518:intuitionistic logic
1422:Tennenbaum's theorem
1400:
1375:non-standard numbers
1350:
1330:
1310:
1283:
1248:
1221:
1201:
1181:
1177:cannot be proven in
1154:
1131:
1104:
1084:
1057:
1021:
989:
963:
959:is either provable (
943:
923:
903:
805:
764:
735:
715:
686:
657:
631:
594:
574:
491:
463:
434:
408:
351:
341:semantic consequence
305:
98:truth and syntactic
83:proof (shown right).
3801:Works by Kurt Gödel
3732:Mathematical object
3623:P versus NP problem
3588:Computable function
3382:Reverse mathematics
3308:Logical consequence
3185:primitive recursive
3180:elementary function
2953:Free/bound variable
2806:Tarski–Grothendieck
2325:Logical connectives
2255:Logical equivalence
2105:Logical consequence
1507:Lindström's theorem
1452:reverse mathematics
1434:compactness theorem
1244:is false. In fact,
1047:Robinson arithmetic
939:in the language of
731:) is provable from
397:in the language of
287:logical consequence
187:There are numerous
177:Gisbert Hasenjaeger
3530:Transfer principle
3493:Semantics of logic
3478:Categorical theory
3454:Non-standard model
2968:Logical connective
2095:Information theory
2044:Mathematical logic
1848:2005-10-13 at the
1762:10.1007/BF01696781
1497:Second-order logic
1456:weak Kőnig's lemma
1406:
1384:theory containing
1363:
1336:
1316:
1296:
1261:
1234:
1207:
1187:
1167:
1137:
1117:
1090:
1070:
1027:
1004:
985:) or disprovable (
975:
949:
929:
909:
817:
791:
750:
721:
701:
672:
643:
600:
580:
559:are provable from
555:and its negation ¬
503:
475:
446:
420:
363:
317:
92:mathematical logic
85:
3768:
3767:
3700:Abstract category
3503:Theories of truth
3313:Rule of inference
3303:Natural deduction
3284:
3283:
2829:
2828:
2534:Cartesian product
2439:
2438:
2345:Many-valued logic
2320:Boolean functions
2203:Russell's paradox
2178:diagonal argument
2075:First-order logic
2010:
2009:
1493:first-order logic
1409:{\displaystyle T}
1339:{\displaystyle T}
1319:{\displaystyle T}
1210:{\displaystyle T}
1190:{\displaystyle T}
1140:{\displaystyle T}
1093:{\displaystyle T}
1030:{\displaystyle T}
952:{\displaystyle T}
932:{\displaystyle S}
912:{\displaystyle T}
603:{\displaystyle T}
583:{\displaystyle T}
331:is provable from
281:More general form
193:natural deduction
189:deductive systems
104:first-order logic
81:natural deduction
3808:
3759:
3758:
3710:History of logic
3705:Category of sets
3598:Decision problem
3377:Ordinal analysis
3318:Sequent calculus
3216:Boolean algebras
3156:
3155:
3130:
3101:logical/constant
2855:
2854:
2841:
2764:Zermelo–Fraenkel
2515:Set operations:
2450:
2449:
2387:
2218:
2217:
2198:Löwenheim–Skolem
2085:Formal semantics
2037:
2030:
2023:
2014:
2013:
1930:Effective method
1908:Cantor's theorem
1886:
1879:
1872:
1863:
1862:
1836:Juliette Kennedy
1812:
1781:
1738:
1715:
1714:
1712:
1705:
1696:
1690:
1689:
1653:
1647:
1646:
1608:
1602:
1600:
1598:
1586:
1522:Kripke semantics
1520:with respect to
1501:Henkin semantics
1464:Peano arithmetic
1415:
1413:
1412:
1407:
1372:
1370:
1369:
1364:
1362:
1361:
1345:
1343:
1342:
1337:
1325:
1323:
1322:
1317:
1305:
1303:
1302:
1297:
1295:
1294:
1270:
1268:
1267:
1262:
1260:
1259:
1243:
1241:
1240:
1235:
1233:
1232:
1216:
1214:
1213:
1208:
1196:
1194:
1193:
1188:
1176:
1174:
1173:
1168:
1166:
1165:
1146:
1144:
1143:
1138:
1126:
1124:
1123:
1118:
1116:
1115:
1099:
1097:
1096:
1091:
1079:
1077:
1076:
1071:
1069:
1068:
1036:
1034:
1033:
1028:
1013:
1011:
1010:
1005:
984:
982:
981:
976:
958:
956:
955:
950:
938:
936:
935:
930:
918:
916:
915:
910:
837:Peano arithmetic
826:
824:
823:
818:
800:
798:
797:
792:
759:
757:
756:
751:
730:
728:
727:
722:
710:
708:
707:
702:
681:
679:
678:
673:
652:
650:
649:
644:
609:
607:
606:
601:
589:
587:
586:
581:
512:
510:
509:
504:
484:
482:
481:
476:
455:
453:
452:
447:
429:
427:
426:
421:
372:
370:
369:
364:
326:
324:
323:
318:
238:completeness is
201:formal deduction
171:observed in his
75:)) holds in all
3816:
3815:
3811:
3810:
3809:
3807:
3806:
3805:
3771:
3770:
3769:
3764:
3753:
3746:
3691:Category theory
3681:Algebraic logic
3664:
3635:Lambda calculus
3573:Church encoding
3559:
3535:Truth predicate
3391:
3357:Complete theory
3280:
3149:
3145:
3141:
3136:
3128:
2848: and
2844:
2839:
2825:
2801:New Foundations
2769:axiom of choice
2752:
2714:Gödel numbering
2654: and
2646:
2550:
2435:
2385:
2366:
2315:Boolean algebra
2301:
2265:Equiconsistency
2230:Classical logic
2207:
2188:Halting problem
2176: and
2152: and
2140: and
2139:
2134:Theorems (
2129:
2046:
2041:
2011:
2006:
1899:
1897:metamathematics
1890:
1850:Wayback Machine
1824:
1801:
1724:
1722:Further reading
1719:
1718:
1710:
1703:
1697:
1693:
1670:10.2307/2266326
1654:
1650:
1627:10.2307/2267044
1609:
1605:
1587:
1583:
1578:
1561:
1530:
1489:
1477:axiom of choice
1469:
1461:
1430:
1401:
1398:
1397:
1357:
1353:
1351:
1348:
1347:
1331:
1328:
1327:
1311:
1308:
1307:
1290:
1286:
1284:
1281:
1280:
1276:
1255:
1251:
1249:
1246:
1245:
1228:
1224:
1222:
1219:
1218:
1202:
1199:
1198:
1182:
1179:
1178:
1161:
1157:
1155:
1152:
1151:
1132:
1129:
1128:
1111:
1107:
1105:
1102:
1101:
1085:
1082:
1081:
1064:
1060:
1058:
1055:
1054:
1022:
1019:
1018:
990:
987:
986:
964:
961:
960:
944:
941:
940:
924:
921:
920:
904:
901:
900:
886:
864:
856:
833:
806:
803:
802:
765:
762:
761:
736:
733:
732:
716:
713:
712:
687:
684:
683:
658:
655:
654:
632:
629:
628:
625:
611:
595:
592:
591:
575:
572:
571:
551:such that both
529:
492:
489:
488:
464:
461:
460:
457:
435:
432:
431:
409:
406:
405:
377:holds in every
352:
349:
348:
306:
303:
302:
283:
262:
254:
225:logically valid
213:algorithmically
185:
140:
132:
24:
17:
12:
11:
5:
3814:
3804:
3803:
3798:
3793:
3788:
3783:
3766:
3765:
3751:
3748:
3747:
3745:
3744:
3739:
3734:
3729:
3724:
3723:
3722:
3712:
3707:
3702:
3693:
3688:
3683:
3678:
3676:Abstract logic
3672:
3670:
3666:
3665:
3663:
3662:
3657:
3655:Turing machine
3652:
3647:
3642:
3637:
3632:
3627:
3626:
3625:
3620:
3615:
3610:
3605:
3595:
3593:Computable set
3590:
3585:
3580:
3575:
3569:
3567:
3561:
3560:
3558:
3557:
3552:
3547:
3542:
3537:
3532:
3527:
3522:
3521:
3520:
3515:
3510:
3500:
3495:
3490:
3488:Satisfiability
3485:
3480:
3475:
3474:
3473:
3463:
3462:
3461:
3451:
3450:
3449:
3444:
3439:
3434:
3429:
3419:
3418:
3417:
3412:
3405:Interpretation
3401:
3399:
3393:
3392:
3390:
3389:
3384:
3379:
3374:
3369:
3359:
3354:
3353:
3352:
3351:
3350:
3340:
3335:
3325:
3320:
3315:
3310:
3305:
3300:
3294:
3292:
3286:
3285:
3282:
3281:
3279:
3278:
3270:
3269:
3268:
3267:
3262:
3261:
3260:
3255:
3250:
3230:
3229:
3228:
3226:minimal axioms
3223:
3212:
3211:
3210:
3199:
3198:
3197:
3192:
3187:
3182:
3177:
3172:
3159:
3157:
3138:
3137:
3135:
3134:
3133:
3132:
3120:
3115:
3114:
3113:
3108:
3103:
3098:
3088:
3083:
3078:
3073:
3072:
3071:
3066:
3056:
3055:
3054:
3049:
3044:
3039:
3029:
3024:
3023:
3022:
3017:
3012:
3002:
3001:
3000:
2995:
2990:
2985:
2980:
2975:
2965:
2960:
2955:
2950:
2949:
2948:
2943:
2938:
2933:
2923:
2918:
2916:Formation rule
2913:
2908:
2907:
2906:
2901:
2891:
2890:
2889:
2879:
2874:
2869:
2864:
2858:
2852:
2835:Formal systems
2831:
2830:
2827:
2826:
2824:
2823:
2818:
2813:
2808:
2803:
2798:
2793:
2788:
2783:
2778:
2777:
2776:
2771:
2760:
2758:
2754:
2753:
2751:
2750:
2749:
2748:
2738:
2733:
2732:
2731:
2724:Large cardinal
2721:
2716:
2711:
2706:
2701:
2687:
2686:
2685:
2680:
2675:
2660:
2658:
2648:
2647:
2645:
2644:
2643:
2642:
2637:
2632:
2622:
2617:
2612:
2607:
2602:
2597:
2592:
2587:
2582:
2577:
2572:
2567:
2561:
2559:
2552:
2551:
2549:
2548:
2547:
2546:
2541:
2536:
2531:
2526:
2521:
2513:
2512:
2511:
2506:
2496:
2491:
2489:Extensionality
2486:
2484:Ordinal number
2481:
2471:
2466:
2465:
2464:
2453:
2447:
2441:
2440:
2437:
2436:
2434:
2433:
2428:
2423:
2418:
2413:
2408:
2403:
2402:
2401:
2391:
2390:
2389:
2376:
2374:
2368:
2367:
2365:
2364:
2363:
2362:
2357:
2352:
2342:
2337:
2332:
2327:
2322:
2317:
2311:
2309:
2303:
2302:
2300:
2299:
2294:
2289:
2284:
2279:
2274:
2269:
2268:
2267:
2257:
2252:
2247:
2242:
2237:
2232:
2226:
2224:
2215:
2209:
2208:
2206:
2205:
2200:
2195:
2190:
2185:
2180:
2168:Cantor's
2166:
2161:
2156:
2146:
2144:
2131:
2130:
2128:
2127:
2122:
2117:
2112:
2107:
2102:
2097:
2092:
2087:
2082:
2077:
2072:
2067:
2066:
2065:
2054:
2052:
2048:
2047:
2040:
2039:
2032:
2025:
2017:
2008:
2007:
2005:
2004:
1999:
1994:
1989:
1987:Satisfiability
1984:
1979:
1974:
1972:Interpretation
1969:
1964:
1959:
1954:
1949:
1944:
1943:
1942:
1932:
1927:
1922:
1917:
1910:
1904:
1901:
1900:
1889:
1888:
1881:
1874:
1866:
1860:
1859:
1852:
1839:
1823:
1822:External links
1820:
1819:
1818:
1799:
1783:
1756:(1): 349–360.
1740:
1723:
1720:
1717:
1716:
1691:
1648:
1621:(3): 159–166.
1603:
1580:
1579:
1577:
1574:
1573:
1572:
1567:
1560:
1557:
1529:
1526:
1488:
1485:
1467:
1459:
1429:
1426:
1405:
1360:
1356:
1335:
1315:
1293:
1289:
1274:
1258:
1254:
1231:
1227:
1206:
1186:
1164:
1160:
1136:
1114:
1110:
1089:
1067:
1063:
1026:
1003:
1000:
997:
994:
974:
971:
968:
948:
928:
908:
885:
882:
863:
860:
854:
832:
829:
816:
813:
810:
790:
787:
784:
781:
778:
775:
772:
769:
749:
746:
743:
740:
720:
700:
697:
694:
691:
671:
668:
665:
662:
642:
639:
636:
619:
599:
579:
569:
528:
525:
502:
499:
496:
486:if and only if
474:
471:
468:
445:
442:
439:
419:
416:
413:
403:
391:well-orderable
362:
359:
356:
316:
313:
310:
282:
279:
275:if and only if
261:
258:
253:
250:
184:
181:
158:formal systems
138:
130:
15:
9:
6:
4:
3:
2:
3813:
3802:
3799:
3797:
3794:
3792:
3789:
3787:
3784:
3782:
3779:
3778:
3776:
3763:
3762:
3757:
3749:
3743:
3740:
3738:
3735:
3733:
3730:
3728:
3725:
3721:
3718:
3717:
3716:
3713:
3711:
3708:
3706:
3703:
3701:
3697:
3694:
3692:
3689:
3687:
3684:
3682:
3679:
3677:
3674:
3673:
3671:
3667:
3661:
3658:
3656:
3653:
3651:
3650:Recursive set
3648:
3646:
3643:
3641:
3638:
3636:
3633:
3631:
3628:
3624:
3621:
3619:
3616:
3614:
3611:
3609:
3606:
3604:
3601:
3600:
3599:
3596:
3594:
3591:
3589:
3586:
3584:
3581:
3579:
3576:
3574:
3571:
3570:
3568:
3566:
3562:
3556:
3553:
3551:
3548:
3546:
3543:
3541:
3538:
3536:
3533:
3531:
3528:
3526:
3523:
3519:
3516:
3514:
3511:
3509:
3506:
3505:
3504:
3501:
3499:
3496:
3494:
3491:
3489:
3486:
3484:
3481:
3479:
3476:
3472:
3469:
3468:
3467:
3464:
3460:
3459:of arithmetic
3457:
3456:
3455:
3452:
3448:
3445:
3443:
3440:
3438:
3435:
3433:
3430:
3428:
3425:
3424:
3423:
3420:
3416:
3413:
3411:
3408:
3407:
3406:
3403:
3402:
3400:
3398:
3394:
3388:
3385:
3383:
3380:
3378:
3375:
3373:
3370:
3367:
3366:from ZFC
3363:
3360:
3358:
3355:
3349:
3346:
3345:
3344:
3341:
3339:
3336:
3334:
3331:
3330:
3329:
3326:
3324:
3321:
3319:
3316:
3314:
3311:
3309:
3306:
3304:
3301:
3299:
3296:
3295:
3293:
3291:
3287:
3277:
3276:
3272:
3271:
3266:
3265:non-Euclidean
3263:
3259:
3256:
3254:
3251:
3249:
3248:
3244:
3243:
3241:
3238:
3237:
3235:
3231:
3227:
3224:
3222:
3219:
3218:
3217:
3213:
3209:
3206:
3205:
3204:
3200:
3196:
3193:
3191:
3188:
3186:
3183:
3181:
3178:
3176:
3173:
3171:
3168:
3167:
3165:
3161:
3160:
3158:
3153:
3147:
3142:Example
3139:
3131:
3126:
3125:
3124:
3121:
3119:
3116:
3112:
3109:
3107:
3104:
3102:
3099:
3097:
3094:
3093:
3092:
3089:
3087:
3084:
3082:
3079:
3077:
3074:
3070:
3067:
3065:
3062:
3061:
3060:
3057:
3053:
3050:
3048:
3045:
3043:
3040:
3038:
3035:
3034:
3033:
3030:
3028:
3025:
3021:
3018:
3016:
3013:
3011:
3008:
3007:
3006:
3003:
2999:
2996:
2994:
2991:
2989:
2986:
2984:
2981:
2979:
2976:
2974:
2971:
2970:
2969:
2966:
2964:
2961:
2959:
2956:
2954:
2951:
2947:
2944:
2942:
2939:
2937:
2934:
2932:
2929:
2928:
2927:
2924:
2922:
2919:
2917:
2914:
2912:
2909:
2905:
2902:
2900:
2899:by definition
2897:
2896:
2895:
2892:
2888:
2885:
2884:
2883:
2880:
2878:
2875:
2873:
2870:
2868:
2865:
2863:
2860:
2859:
2856:
2853:
2851:
2847:
2842:
2836:
2832:
2822:
2819:
2817:
2814:
2812:
2809:
2807:
2804:
2802:
2799:
2797:
2794:
2792:
2789:
2787:
2786:Kripke–Platek
2784:
2782:
2779:
2775:
2772:
2770:
2767:
2766:
2765:
2762:
2761:
2759:
2755:
2747:
2744:
2743:
2742:
2739:
2737:
2734:
2730:
2727:
2726:
2725:
2722:
2720:
2717:
2715:
2712:
2710:
2707:
2705:
2702:
2699:
2695:
2691:
2688:
2684:
2681:
2679:
2676:
2674:
2671:
2670:
2669:
2665:
2662:
2661:
2659:
2657:
2653:
2649:
2641:
2638:
2636:
2633:
2631:
2630:constructible
2628:
2627:
2626:
2623:
2621:
2618:
2616:
2613:
2611:
2608:
2606:
2603:
2601:
2598:
2596:
2593:
2591:
2588:
2586:
2583:
2581:
2578:
2576:
2573:
2571:
2568:
2566:
2563:
2562:
2560:
2558:
2553:
2545:
2542:
2540:
2537:
2535:
2532:
2530:
2527:
2525:
2522:
2520:
2517:
2516:
2514:
2510:
2507:
2505:
2502:
2501:
2500:
2497:
2495:
2492:
2490:
2487:
2485:
2482:
2480:
2476:
2472:
2470:
2467:
2463:
2460:
2459:
2458:
2455:
2454:
2451:
2448:
2446:
2442:
2432:
2429:
2427:
2424:
2422:
2419:
2417:
2414:
2412:
2409:
2407:
2404:
2400:
2397:
2396:
2395:
2392:
2388:
2383:
2382:
2381:
2378:
2377:
2375:
2373:
2369:
2361:
2358:
2356:
2353:
2351:
2348:
2347:
2346:
2343:
2341:
2338:
2336:
2333:
2331:
2328:
2326:
2323:
2321:
2318:
2316:
2313:
2312:
2310:
2308:
2307:Propositional
2304:
2298:
2295:
2293:
2290:
2288:
2285:
2283:
2280:
2278:
2275:
2273:
2270:
2266:
2263:
2262:
2261:
2258:
2256:
2253:
2251:
2248:
2246:
2243:
2241:
2238:
2236:
2235:Logical truth
2233:
2231:
2228:
2227:
2225:
2223:
2219:
2216:
2214:
2210:
2204:
2201:
2199:
2196:
2194:
2191:
2189:
2186:
2184:
2181:
2179:
2175:
2171:
2167:
2165:
2162:
2160:
2157:
2155:
2151:
2148:
2147:
2145:
2143:
2137:
2132:
2126:
2123:
2121:
2118:
2116:
2113:
2111:
2108:
2106:
2103:
2101:
2098:
2096:
2093:
2091:
2088:
2086:
2083:
2081:
2078:
2076:
2073:
2071:
2068:
2064:
2061:
2060:
2059:
2056:
2055:
2053:
2049:
2045:
2038:
2033:
2031:
2026:
2024:
2019:
2018:
2015:
2003:
2000:
1998:
1995:
1993:
1990:
1988:
1985:
1983:
1980:
1978:
1975:
1973:
1970:
1968:
1965:
1963:
1960:
1958:
1955:
1953:
1950:
1948:
1945:
1941:
1938:
1937:
1936:
1933:
1931:
1928:
1926:
1923:
1921:
1918:
1916:
1915:
1911:
1909:
1906:
1905:
1902:
1898:
1894:
1887:
1882:
1880:
1875:
1873:
1868:
1867:
1864:
1857:
1853:
1851:
1847:
1844:
1840:
1837:
1833:
1829:
1826:
1825:
1816:
1810:
1806:
1802:
1796:
1792:
1788:
1784:
1779:
1775:
1771:
1767:
1763:
1759:
1755:
1752:(in German).
1751:
1750:
1745:
1741:
1736:
1735:
1730:
1726:
1725:
1709:
1702:
1695:
1687:
1683:
1679:
1675:
1671:
1667:
1663:
1659:
1652:
1644:
1640:
1636:
1632:
1628:
1624:
1620:
1616:
1615:
1607:
1597:
1592:
1585:
1581:
1571:
1568:
1566:
1563:
1562:
1556:
1554:
1550:
1546:
1541:
1539:
1535:
1525:
1523:
1519:
1515:
1510:
1508:
1504:
1502:
1498:
1494:
1484:
1482:
1478:
1473:
1465:
1457:
1453:
1448:
1445:
1441:
1439:
1435:
1425:
1423:
1419:
1403:
1394:
1391:
1387:
1383:
1378:
1376:
1358:
1354:
1333:
1313:
1291:
1287:
1278:
1256:
1252:
1229:
1225:
1204:
1184:
1162:
1158:
1148:
1134:
1112:
1108:
1087:
1065:
1061:
1052:
1048:
1045:and contains
1044:
1040:
1024:
1015:
1001:
995:
992:
972:
969:
966:
946:
926:
906:
898:
894:
890:
881:
878:
875:
873:
869:
859:
857:
850:
846:
842:
838:
828:
814:
811:
808:
785:
779:
773:
770:
747:
741:
738:
698:
692:
689:
669:
663:
660:
640:
637:
634:
623:
618:
616:
597:
577:
568:
566:
562:
558:
554:
550:
546:
542:
538:
534:
524:
521:
519:
514:
500:
497:
494:
487:
472:
469:
466:
443:
440:
437:
417:
414:
411:
402:
400:
396:
392:
388:
384:
380:
376:
360:
357:
354:
346:
342:
338:
334:
330:
314:
311:
308:
300:
296:
292:
288:
278:
276:
272:
271:
265:
257:
249:
245:
243:
242:
236:
231:
227:
226:
220:
218:
214:
210:
206:
202:
198:
194:
190:
183:Preliminaries
180:
178:
174:
170:
166:
161:
159:
155:
151:
146:
144:
136:
128:
124:
120:
116:
112:
107:
105:
101:
97:
93:
89:
82:
78:
74:
70:
66:
62:
59:
56:
52:
48:
44:
40:
36:
33:
30:The formula (
28:
22:
3796:Proof theory
3791:Model theory
3786:Metatheorems
3752:
3550:Ultraproduct
3397:Model theory
3362:Independence
3298:Formal proof
3290:Proof theory
3273:
3246:
3203:real numbers
3175:second-order
3086:Substitution
2963:Metalanguage
2904:conservative
2877:Axiom schema
2821:Constructive
2791:Morse–Kelley
2757:Set theories
2736:Aleph number
2729:inaccessible
2635:Grothendieck
2519:intersection
2406:Higher-order
2394:Second-order
2340:Truth tables
2297:Venn diagram
2149:
2080:Formal proof
1992:Independence
1967:Decidability
1962:Completeness
1946:
1912:
1814:
1790:
1753:
1747:
1733:
1694:
1664:(1): 42–48.
1661:
1657:
1651:
1618:
1612:
1606:
1584:
1542:
1537:
1531:
1511:
1505:
1490:
1449:
1446:
1442:
1431:
1417:
1395:
1389:
1385:
1381:
1379:
1149:
1050:
1016:
899:): A theory
892:
887:
879:
876:
865:
862:Consequences
848:
844:
840:
834:
626:
612:
610:has a model.
564:
560:
556:
552:
548:
544:
540:
530:
522:
518:group theory
515:
458:
398:
394:
386:
382:
374:
344:
340:
336:
332:
328:
298:
297:of a theory
294:
290:
284:
268:
266:
263:
255:
246:
239:
234:
223:
221:
208:
200:
186:
173:Ph.D. thesis
162:
154:proof theory
150:model theory
147:
142:
134:
122:
118:
114:
108:
87:
86:
72:
68:
64:
60:
54:
46:
42:
38:
34:
3660:Type theory
3608:undecidable
3540:Truth value
3427:equivalence
3106:non-logical
2719:Enumeration
2709:Isomorphism
2656:cardinality
2640:Von Neumann
2605:Ultrafilter
2570:Uncountable
2504:equivalence
2421:Quantifiers
2411:Fixed-point
2380:First-order
2260:Consistency
2245:Proposition
2222:Traditional
2193:Lindström's
2183:Compactness
2125:Type theory
2070:Cardinality
1982:Metatheorem
1940:of geometry
1925:Consistency
1843:Kurt Gödel.
1813:Chapter 5:
1787:Hans Hermes
1514:modal logic
533:consistency
169:Leon Henkin
100:provability
3775:Categories
3471:elementary
3164:arithmetic
3032:Quantifier
3010:functional
2882:Expression
2600:Transitive
2544:identities
2529:complement
2462:hereditary
2445:Set theory
1832:Kurt Gödel
1800:3540058192
1770:56.0046.04
1596:2112.06641
1576:References
1549:term model
1540:argument.
1039:consistent
347:, denoted
301:, denoted
209:conclusion
165:Kurt Gödel
77:structures
3742:Supertask
3645:Recursion
3603:decidable
3437:saturated
3415:of models
3338:deductive
3333:axiomatic
3253:Hilbert's
3240:Euclidean
3221:canonical
3144:axiomatic
3076:Signature
3005:Predicate
2894:Extension
2816:Ackermann
2741:Operation
2620:Universal
2610:Recursive
2585:Singleton
2580:Inhabited
2565:Countable
2555:Types of
2539:power set
2509:partition
2426:Predicate
2372:Predicate
2287:Syllogism
2277:Soundness
2250:Inference
2240:Tautology
2142:paradoxes
1957:Soundness
1893:Metalogic
1809:1431-4657
1778:123343522
1479:known as
1438:effective
1396:Also, if
1346:in which
1217:in which
1043:effective
1037:which is
999:¬
996:⊢
970:⊢
872:effective
812:⊢
789:⊥
786:⊢
777:¬
774:∪
745:¬
742:∪
719:⊥
696:¬
693:∪
667:¬
664:∪
638:⊨
622:countable
498:⊢
470:⊨
441:⊢
415:⊨
358:⊨
312:⊢
270:soundness
252:Statement
241:soundness
230:structure
179:in 1953.
3727:Logicism
3720:timeline
3696:Concrete
3555:Validity
3525:T-schema
3518:Kripke's
3513:Tarski's
3508:semantic
3498:Strength
3447:submodel
3442:spectrum
3410:function
3258:Tarski's
3247:Elements
3234:geometry
3190:Robinson
3111:variable
3096:function
3069:spectrum
3059:Sentence
3015:variable
2958:Language
2911:Relation
2872:Automata
2862:Alphabet
2846:language
2700:-jection
2678:codomain
2664:Function
2625:Universe
2595:Infinite
2499:Relation
2282:Validity
2272:Argument
2170:theorem,
1846:Archived
1789:(1973).
1744:Gödel, K
1731:(1929).
1729:Gödel, K
1708:Archived
1686:45705695
1643:28935946
1559:See also
1553:Isabelle
1532:Gödel's
1277:sentence
1147:itself.
893:complete
617:, says:
235:complete
217:computer
96:semantic
3669:Related
3466:Diagram
3364: (
3343:Hilbert
3328:Systems
3323:Theorem
3201:of the
3146:systems
2926:Formula
2921:Grammar
2837: (
2781:General
2494:Forcing
2479:Element
2399:Monadic
2174:paradox
2115:Theorem
2051:General
1678:2266326
1635:2267044
1306:within
653:, then
430:, then
389:with a
3432:finite
3195:Skolem
3148:
3123:Theory
3091:Symbol
3081:String
3064:atomic
2941:ground
2936:closed
2931:atomic
2887:ground
2850:syntax
2746:binary
2673:domain
2590:Finite
2355:finite
2213:Logics
2172:
2120:Theory
1807:
1797:
1776:
1768:
1684:
1676:
1641:
1633:
1545:Henkin
1538:ad hoc
1528:Proofs
1390:always
1150:Since
215:(by a
111:theory
3422:Model
3170:Peano
3027:Proof
2867:Arity
2796:Naive
2683:image
2615:Fuzzy
2575:Empty
2524:union
2469:Class
2110:Model
2100:Lemma
2058:Axiom
1834:"—by
1774:S2CID
1711:(PDF)
1704:(PDF)
1682:S2CID
1674:JSTOR
1639:S2CID
1631:JSTOR
1591:arXiv
1271:is a
895:(see
379:model
373:, if
339:is a
327:, if
293:is a
113:: If
3545:Type
3348:list
3152:list
3129:list
3118:Term
3052:rank
2946:open
2840:list
2652:Maps
2557:sets
2416:Free
2386:list
2136:list
2063:list
1895:and
1805:ISSN
1795:ISBN
205:tree
195:and
3232:of
3214:of
3162:of
2694:Sur
2668:Map
2475:Ur-
2457:Set
1830:: "
1766:JFM
1758:doi
1666:doi
1623:doi
1516:or
1382:any
1014:).
858:).
570:if
543:is
404:if
381:of
343:of
145:.)
102:in
49:))
3777::
3618:NP
3242::
3236::
3166::
2843:),
2698:Bi
2690:In
1803:.
1772:.
1764:.
1754:37
1680:.
1672:.
1662:18
1660:.
1637:.
1629:.
1619:14
1617:.
1524:.
1377:.
1049:("
1041:,
827:.
401:,
160:.
106:.
63:.
53:(∀
37:.
3698:/
3613:P
3368:)
3154:)
3150:(
3047:∀
3042:!
3037:∃
2998:=
2993:↔
2988:→
2983:∧
2978:∨
2973:¬
2696:/
2692:/
2666:/
2477:)
2473:(
2360:∞
2350:3
2138:)
2036:e
2029:t
2022:v
1885:e
1878:t
1871:v
1858:"
1838:.
1817:.
1811:.
1780:.
1760::
1688:.
1668::
1645:.
1625::
1599:.
1593::
1468:1
1460:0
1418:Q
1404:T
1386:Q
1359:T
1355:S
1334:T
1314:T
1292:T
1288:S
1275:1
1273:Π
1257:T
1253:S
1230:T
1226:S
1205:T
1185:T
1163:T
1159:S
1135:T
1113:T
1109:S
1088:T
1066:T
1062:S
1051:Q
1025:T
1002:S
993:T
973:S
967:T
947:T
927:S
907:T
855:2
853:Δ
849:T
845:T
841:T
815:s
809:T
783:)
780:s
771:T
768:(
748:s
739:T
699:s
690:T
670:s
661:T
641:s
635:T
598:T
578:T
565:T
561:T
557:s
553:s
549:s
541:T
501:s
495:T
473:s
467:T
456:.
444:s
438:T
418:s
412:T
399:T
395:s
387:T
383:T
375:s
361:s
355:T
345:T
337:s
333:T
329:s
315:s
309:T
299:T
291:s
143:T
139:u
135:T
131:u
123:T
119:T
115:T
73:y
71:,
69:x
67:(
65:R
61:y
58:∃
55:x
51:→
47:x
45:,
43:x
41:(
39:R
35:x
32:∀
23:.
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