7236:
5427:
4047:
527:
prove a particular sentence, called the Gödel sentence of the theory, which is a formalized statement of the claim that the theory is indeed consistent. Thus the consistency of a sufficiently strong, recursively enumerable, consistent theory of arithmetic can never be proven in that system itself.
3285:
4182:, p. 601. Tarski defines "proof" informally as "statements follow one another in a definite order according to certain principles ⊠and accompanied by considerations intended to establish their validity for all true premises â
3044:
522:
Moreover, Gödel's second incompleteness theorem shows that the consistency of sufficiently strong recursively enumerable theories of arithmetic can be tested in a particular way. Such a theory is consistent if and only if it does
399:, every satisfiable theory is consistent, but the converse does not hold. If there exists a deductive system for which these semantic and syntactic definitions are equivalent for any theory formulated in a particular deductive
4194:
defines the notion with respect to either an induction or as to paraphrase) a finite sequence of formulas such that each formula in the sequence is either an axiom or an "immediate consequence" of the preceding formulas; "A
3535:
1947:
1716:
1896:
1819:
1341:
4033:
2758:
3367:
1184:
3989:
3156:
3810:
3644:
341:
3103:
1515:
4130:
if no two asserted statements of this theory contradict each other, or in other words, if of any two contradictory sentences ⊠at least one cannot be proved," (p. 135) where Tarski defines
3448:
2169:
306:
2617:
2405:
2349:
2262:
1593:
824:
1848:
1745:
1477:
874:
749:
2862:
2129:
3773:
3674:
2045:
1771:
1665:
795:
1371:
1263:
4394:
4344:
4795:
4769:
2288:
2195:
2100:
2071:
1210:
210:
4703:
4637:
4506:
5615:
4848:
2668:
2570:
1563:
3935:
3397:
2314:
2221:
140:
2889:
679:
6290:
4932:
4908:
4868:
4817:
4723:
4657:
4606:
4546:
4480:
4414:
4364:
3910:
3870:
3605:
1996:
1636:
1415:
1140:
984:
960:
920:
769:
274:
117:
97:
5133:
2637:
653:
3313:
3151:
2963:
5153:
5113:
3721:
3701:
3562:
2913:
2785:
2510:
2460:
2016:
1972:
1616:
1539:
1232:
1120:
1096:
1074:
1054:
1026:
1004:
940:
896:
844:
719:
4972:
4952:
4888:
4743:
4677:
4586:
4566:
4526:
4458:
4438:
4314:
4294:
3890:
3850:
3585:
3468:
3125:
2968:
2937:
2828:
2805:
2692:
2530:
2480:
2436:
2369:
1435:
1395:
1286:
463:
Although consistency can be proved using model theory, it is often done in a purely syntactical way, without any need to reference some model of the logic. The
250:
230:
180:
160:
74:
6373:
5514:
528:
The same result is true for recursively enumerable theories that can describe a strong enough fragment of arithmetic—including set theories such as
3473:
1901:
1670:
5380:
6687:
4154:
is defined to be the smallest class of formulas that contains the axioms and is closed under the relation "immediate consequence", i.e., formula
5307:
6845:
5079:(this is equivalent since these two theories have been proved equiconsistent; that is, if one is consistent, the same is true for the other).
4238:
3703:
is in fact an equivalence relation. Then, it needs to be verified that (1), (2), and (3) are well defined. This falls out of the fact that
1853:
1776:
475:
if there is one) implies the consistency of the calculus: since there is no cut-free proof of falsity, there is no contradiction in general.
7340:
5633:
500:
6700:
6023:
1291:
354:(e.g., classical or intuitionistic propositional or first-order logics) every inconsistent theory is trivial. Consistency of a theory is a
7272:
5347:
6285:
6705:
6695:
6432:
5638:
7335:
6183:
5629:
2697:
7380:
6841:
5294:
5272:
5250:
5227:
608:
3318:
5016:
Post proves both consistency and completeness of the propositional calculus of PM, cf van
Heijenoort's commentary and Post's 1931
1145:
6938:
6682:
5507:
3280:{\displaystyle f^{{\mathfrak {T}}_{\Phi }}({\overline {t_{0}}}\ldots {\overline {t_{n-1}}}):={\overline {ft_{0}\ldots t_{n-1}}};}
7365:
6243:
5936:
5677:
5072:
3995:
3821:
532:(ZF). These set theories cannot prove their own Gödel sentenceâprovided that they are consistent, which is generally believed.
529:
491:. A theory is complete if, for every formula Ï in its language, at least one of Ï or ÂŹÏ is a logical consequence of the theory.
4797:). For first-order logic, the two kinds of entailment coincide by the completeness theorem for the proof calculus in question.
7416:
7199:
6901:
6664:
6659:
6484:
5905:
5589:
2831:
512:
4095:
3951:
3778:
3612:
460:, which showed that sufficiently strong proof theories cannot prove their consistency (provided that they are consistent).
7194:
6977:
6894:
6607:
6538:
6415:
5657:
4980:
We say that two theories S and T in L infinity omega are equivalent if they have the same models, i.e. if Mod(S) = Mod(T).
6265:
311:
7119:
6945:
6631:
5864:
347:
theory (i.e., one which proves every sentence in the language of the theory) is clearly inconsistent. Conversely, in an
77:
31:
7390:
6270:
5194:
Gödel, Kurt (1 December 1931). "Ăber formal unentscheidbare SĂ€tze der
Principia Mathematica und verwandter Systeme I".
6602:
6341:
5599:
5500:
5441:
5373:
3049:
1484:
548:
49:
7385:
6997:
6992:
4075:
7323:
6926:
6516:
5910:
5878:
5569:
3402:
2136:
279:
5643:
2575:
2374:
2321:
2229:
7216:
7165:
7062:
6560:
6521:
5998:
1568:
7308:
7057:
5672:
5164:
the common case in many applications to other areas of mathematics as well as the ordinary mode of reasoning of
800:
6987:
6526:
6378:
6361:
6084:
5564:
1827:
1724:
1456:
853:
728:
4679:. In any case, with infinitary languages, it's not always clear what would constitute proof. Some writers use
2836:
2105:
7328:
7265:
6889:
6866:
6827:
6713:
6654:
6300:
6220:
6064:
6008:
5621:
457:
3726:
7360:
7179:
6906:
6884:
6851:
6744:
6590:
6575:
6548:
6499:
6383:
6318:
6143:
6109:
6104:
5978:
5809:
5786:
5366:
5071:
A consistency proof often assumes the consistency of another theory. In most cases, this other theory is
4100:
3649:
3540:
2024:
1750:
1644:
774:
504:
468:
430:
410:
371:
348:
183:
1346:
1241:
7411:
7109:
6962:
6754:
6472:
6208:
6114:
5973:
5958:
5839:
5814:
4369:
4319:
2483:
1975:
516:
507:
theory of arithmetic cannot be both complete and consistent. Gödel's theorem applies to the theories of
494:
7235:
4774:
4748:
4070:
3723:
is an equivalence relation and also requires a proof that (1) and (2) are independent of the choice of
2267:
2174:
2079:
2050:
1189:
189:
4682:
4616:
4485:
7355:
7350:
7302:
7082:
7044:
6921:
6725:
6565:
6489:
6467:
6295:
6253:
6152:
6119:
5983:
5771:
5682:
5165:
4830:
4085:
4060:
2646:
2535:
2490:
1544:
1374:
488:
453:
405:
3375:
2293:
2200:
122:
7296:
7211:
7102:
7087:
7067:
7024:
6911:
6861:
6787:
6732:
6669:
6462:
6457:
6405:
6173:
6162:
5834:
5734:
5662:
5653:
5649:
5584:
5579:
5282:
5178:
3942:
699:
452:
was driven by the desire to provide finitary consistency proofs for all of mathematics as part of
433:
were proved by
Ackermann (1924), von Neumann (1927) and Herbrand (1931). Stronger logics, such as
7421:
7258:
7240:
7009:
6972:
6957:
6950:
6933:
6719:
6585:
6511:
6494:
6447:
6260:
6169:
6003:
5988:
5948:
5900:
5885:
5873:
5829:
5804:
5574:
5523:
629:
445:
434:
422:
400:
363:
6737:
6193:
4080:
3915:
2867:
658:
497:
is an axiom system for the natural numbers under addition. It is both consistent and complete.
7175:
6982:
6792:
6782:
6674:
6555:
6390:
6366:
6147:
6131:
6036:
6013:
5890:
5859:
5824:
5719:
5554:
5416:
5342:
5290:
5268:
5246:
5223:
4917:
4893:
4853:
4802:
4708:
4642:
4591:
4531:
4465:
4399:
4349:
4234:
4052:
3895:
3855:
3825:
3590:
2671:
2640:
2439:
1981:
1621:
1438:
1400:
1125:
969:
945:
905:
754:
259:
102:
82:
5118:
2622:
638:
7318:
7189:
7184:
7077:
7034:
6856:
6817:
6812:
6797:
6623:
6580:
6477:
6275:
6225:
5799:
5761:
5238:
5203:
4252:
4226:
4090:
3292:
3130:
2942:
633:
508:
484:
5138:
5091:
4248:
4142:
of any sentence; two sentences, of which the first is a negation of the second, are called
3706:
3686:
3547:
2898:
2763:
2495:
2445:
2001:
1957:
1601:
1524:
1217:
1105:
1081:
1059:
1039:
1011:
989:
925:
881:
829:
704:
7285:
7170:
7160:
7114:
7097:
7052:
7014:
6916:
6836:
6643:
6570:
6543:
6531:
6437:
6351:
6325:
6280:
6248:
6049:
5851:
5794:
5744:
5709:
5667:
5076:
4256:
4244:
4065:
3039:{\displaystyle R^{{\mathfrak {T}}_{\Phi }}{\overline {t_{0}}}\ldots {\overline {t_{n-1}}}}
543:
is interesting in set theory (and in other sufficiently expressive axiomatic systems). If
464:
42:
38:
7375:
7155:
7134:
7092:
7072:
6967:
6822:
6420:
6410:
6400:
6395:
6329:
6203:
6079:
5968:
5963:
5941:
5542:
5410:
5215:
4957:
4937:
4873:
4728:
4662:
4571:
4551:
4511:
4443:
4423:
4299:
4279:
3875:
3835:
3570:
3453:
3110:
2922:
2813:
2790:
2677:
2515:
2465:
2421:
2354:
1420:
1380:
1271:
235:
232:
under some (specified, possibly implicitly) formal deductive system. The set of axioms
215:
165:
145:
59:
7405:
7129:
6807:
6314:
6099:
6089:
6059:
6044:
5714:
5476:
5471:
5390:
5260:
5003:, a result not published until 1926, but he says nothing about Bernays proving their
426:
351:
53:
17:
7029:
6876:
6777:
6769:
6649:
6597:
6506:
6442:
6425:
6356:
6215:
6074:
5776:
5559:
5446:
1266:
487:, there is an intricate relationship between the consistency of the theory and its
472:
449:
414:
367:
3530:{\displaystyle {\mathfrak {I}}_{\Phi }:=({\mathfrak {T}}_{\Phi },\beta _{\Phi })}
7370:
7250:
7139:
7019:
6198:
6188:
6135:
5819:
5724:
5604:
5549:
5405:
388:
4225:. Logic, Epistemology, and the Unity of Science. Vol. 40. Cham: Springer.
429:
in 1930, and consistency proofs for arithmetics restricted with respect to the
6069:
5924:
5895:
5701:
4230:
4042:
448:
that a particular theory is consistent. The early development of mathematical
383:
1942:{\displaystyle \operatorname {Con} \left(\Phi \cup \{\lnot \varphi \}\right)}
1711:{\displaystyle \operatorname {Con} \left(\Phi \cup \{\lnot \varphi \}\right)}
7345:
7281:
7221:
7124:
6177:
6094:
6054:
6018:
5954:
5766:
5756:
5729:
418:
396:
355:
7206:
7004:
6452:
6157:
5751:
1521:
Every satisfiable set of formulas is consistent, where a set of formulas
963:
359:
6802:
5594:
5451:
5207:
1891:{\displaystyle \operatorname {Con} \left(\Phi \cup \{\varphi \}\right)}
1814:{\displaystyle \operatorname {Con} \left(\Phi \cup \{\varphi \}\right)}
5492:
1336:{\displaystyle (\exists x\,\varphi \to \varphi {t \over x})\in \Phi }
5426:
5358:
535:
Because consistency of ZF is not provable in ZF, the weaker notion
6346:
5692:
5537:
5265:
Introduction to Logic and to the
Methodology of Deductive Sciences
5039:
The completeness of the axioms of the functional calculus of logic
1974:
be a maximally consistent set of formulas and suppose it contains
556:
375:
4771:
for our notion of entailment (a notation which clashes with our
7254:
5496:
5362:
5168:
in calculus and applications to physics, chemistry, engineering
2753:{\displaystyle T_{\Phi }:=\{\;{\overline {t}}\mid t\in T^{S}\}}
4223:
Paraconsistent logic: consistency, contradiction and negation
3362:{\displaystyle c^{{\mathfrak {T}}_{\Phi }}:={\overline {c}}.}
1179:{\displaystyle \operatorname {Con} (\Phi \cup \{\varphi \})}
5018:
Introduction to a general theory of elementary propositions
4028:{\displaystyle \varphi '\not \in T\lor \varphi \not \in T}
4745:
in some particular formal proof calculus, and they write
4146:" (p. 20). This definition requires a notion of "proof".
479:
Consistency and completeness in arithmetic and set theory
5287:
386:, although in contemporary mathematical logic the term
3984:{\displaystyle \{\varphi ,\varphi '\}\not \subseteq T}
3805:{\displaystyle {\mathfrak {I}}_{\Phi }\vDash \varphi }
3639:{\displaystyle {\mathfrak {I}}_{\Phi }\vDash \varphi }
5141:
5121:
5094:
4960:
4940:
4920:
4896:
4876:
4856:
4833:
4805:
4777:
4751:
4731:
4711:
4685:
4665:
4645:
4619:
4594:
4574:
4554:
4534:
4514:
4488:
4468:
4446:
4426:
4402:
4372:
4352:
4322:
4302:
4282:
4221:
Carnielli, Walter; Coniglio, Marcelo
Esteban (2016).
3998:
3954:
3918:
3898:
3878:
3858:
3838:
3781:
3729:
3709:
3689:
3652:
3615:
3593:
3573:
3550:
3476:
3456:
3405:
3378:
3321:
3295:
3159:
3133:
3113:
3052:
2971:
2945:
2925:
2901:
2870:
2839:
2816:
2793:
2766:
2700:
2680:
2649:
2625:
2578:
2538:
2518:
2498:
2468:
2448:
2424:
2377:
2357:
2324:
2296:
2270:
2232:
2203:
2177:
2139:
2108:
2082:
2053:
2027:
2004:
1984:
1960:
1904:
1856:
1830:
1779:
1753:
1727:
1673:
1647:
1624:
1604:
1571:
1547:
1527:
1487:
1459:
1423:
1403:
1383:
1349:
1294:
1274:
1244:
1220:
1192:
1148:
1128:
1108:
1084:
1062:
1042:
1014:
992:
972:
948:
928:
908:
884:
856:
832:
803:
777:
757:
731:
707:
661:
641:
314:
282:
262:
238:
218:
192:
168:
148:
125:
105:
85:
62:
4274:. New York: Cambridge University Press. p. 37.
7148:
7043:
6875:
6768:
6620:
6313:
6236:
6130:
6034:
5923:
5850:
5785:
5700:
5691:
5613:
5530:
5460:
5434:
5398:
5054:cf van Heijenoort's commentary and Herbrand's 1930
4983:(Please note the definition of Mod(T) on p. 30 ...)
1541:is satisfiable if and only if there exists a model
336:{\displaystyle \lnot \varphi \in \langle A\rangle }
5147:
5127:
5107:
4966:
4946:
4926:
4902:
4882:
4862:
4842:
4811:
4789:
4763:
4737:
4717:
4697:
4671:
4651:
4631:
4600:
4580:
4560:
4540:
4520:
4500:
4474:
4452:
4432:
4408:
4388:
4358:
4338:
4308:
4288:
4122:states it this way: "A deductive theory is called
4027:
3983:
3929:
3904:
3884:
3864:
3844:
3804:
3767:
3715:
3695:
3668:
3638:
3599:
3579:
3556:
3529:
3462:
3442:
3391:
3361:
3307:
3279:
3145:
3119:
3097:
3038:
2957:
2931:
2907:
2883:
2856:
2822:
2799:
2779:
2752:
2686:
2662:
2631:
2611:
2564:
2524:
2504:
2474:
2454:
2430:
2399:
2363:
2343:
2308:
2282:
2256:
2215:
2189:
2163:
2123:
2094:
2065:
2039:
2010:
1990:
1966:
1941:
1890:
1842:
1813:
1765:
1739:
1710:
1659:
1630:
1610:
1587:
1557:
1533:
1509:
1471:
1429:
1409:
1389:
1365:
1335:
1280:
1257:
1226:
1204:
1178:
1134:
1114:
1090:
1068:
1048:
1020:
998:
978:
954:
934:
914:
890:
868:
838:
818:
789:
763:
743:
713:
673:
647:
456:. Hilbert's program was strongly impacted by the
335:
300:
268:
244:
224:
204:
174:
154:
134:
111:
91:
68:
4995:, p. 265 states that Bernays determined the
4201:its last formula, and this formula is said to be
421:in 1921, while the completeness of (first order)
5088:This definition is independent of the choice of
5037:cf van Heijenoort's commentary and Gödel's 1930
3852:is one such that there exists a closed sentence
3683:There are several things to verify. First, that
3098:{\displaystyle \;Rt_{0}\ldots t_{n-1}\in \Phi ;}
2787:is the set of terms based on the set of symbols
1510:{\displaystyle \varphi ,\;\Phi \vdash \varphi .}
5059:
5042:
5021:
4992:
4179:
7266:
5508:
5374:
3443:{\displaystyle \beta _{\Phi }(x):={\bar {x}}}
2164:{\displaystyle (\varphi \lor \psi )\in \Phi }
301:{\displaystyle \varphi \in \langle A\rangle }
8:
3972:
3955:
2747:
2714:
2612:{\displaystyle \;t_{0}\equiv t_{1}\in \Phi }
2400:{\displaystyle \varphi {t \over x}\in \Phi }
2344:{\displaystyle \exists x\,\varphi \in \Phi }
2257:{\displaystyle (\varphi \to \psi )\in \Phi }
1931:
1922:
1880:
1874:
1803:
1797:
1700:
1691:
1170:
1164:
330:
324:
295:
289:
199:
193:
5289:. Cambridge, MA: Harvard University Press.
4954:-structure. Likewise, we say that a theory
4183:
4150:defines the notion this way: "The class of
1588:{\displaystyle {\mathfrak {I}}\vDash \Phi }
1036:if at least one formula in the language of
142:are elements of the set of consequences of
7273:
7259:
7251:
6334:
5929:
5697:
5515:
5501:
5493:
5381:
5367:
5359:
3812:can be verified by induction on formulas.
3653:
3053:
2717:
2579:
1494:
819:{\displaystyle \Phi \vdash \lnot \varphi }
212:the set of closed sentences provable from
5140:
5120:
5099:
5093:
4959:
4939:
4919:
4895:
4875:
4855:
4832:
4804:
4776:
4750:
4730:
4710:
4684:
4664:
4644:
4618:
4593:
4573:
4553:
4533:
4513:
4487:
4467:
4445:
4425:
4401:
4377:
4371:
4351:
4327:
4321:
4301:
4281:
3997:
3953:
3917:
3897:
3877:
3857:
3837:
3790:
3784:
3783:
3780:
3753:
3734:
3728:
3708:
3688:
3651:
3624:
3618:
3617:
3614:
3592:
3572:
3549:
3518:
3505:
3499:
3498:
3485:
3479:
3478:
3475:
3455:
3429:
3428:
3410:
3404:
3383:
3377:
3346:
3335:
3329:
3328:
3326:
3320:
3294:
3256:
3243:
3233:
3210:
3204:
3190:
3184:
3173:
3167:
3166:
3164:
3158:
3132:
3112:
3074:
3061:
3051:
3019:
3013:
2999:
2993:
2985:
2979:
2978:
2976:
2970:
2944:
2924:
2900:
2875:
2869:
2848:
2842:
2841:
2838:
2815:
2792:
2771:
2765:
2741:
2718:
2705:
2699:
2679:
2650:
2648:
2624:
2597:
2584:
2577:
2556:
2543:
2537:
2517:
2497:
2467:
2447:
2423:
2381:
2376:
2356:
2331:
2323:
2295:
2269:
2231:
2202:
2176:
2138:
2107:
2081:
2052:
2026:
2003:
1983:
1959:
1903:
1855:
1843:{\displaystyle \operatorname {Con} \Phi }
1829:
1778:
1752:
1740:{\displaystyle \operatorname {Con} \Phi }
1726:
1672:
1646:
1623:
1603:
1573:
1572:
1570:
1549:
1548:
1546:
1526:
1486:
1472:{\displaystyle \operatorname {Inc} \Phi }
1458:
1422:
1402:
1382:
1353:
1348:
1314:
1304:
1293:
1273:
1251:
1243:
1219:
1191:
1147:
1127:
1107:
1083:
1061:
1041:
1013:
991:
971:
947:
927:
907:
883:
869:{\displaystyle \operatorname {Inc} \Phi }
855:
831:
802:
776:
756:
744:{\displaystyle \operatorname {Con} \Phi }
730:
706:
660:
640:
313:
281:
261:
237:
217:
191:
167:
147:
124:
104:
84:
61:
5316:Ebbinghaus, H. D.; Flum, J.; Thomas, W.
5115:due to the substitutivity properties of
4134:as follows: "With the help of the word
4112:
2857:{\displaystyle {\mathfrak {T}}_{\Phi }}
2124:{\displaystyle \lnot \varphi \in \Phi }
52:is one that does not lead to a logical
5308:The Cambridge Dictionary of Philosophy
5025:
4208:
4191:
4187:
4119:
3941:theory is one such that the following
366:. A theory is satisfiable if it has a
5196:Monatshefte fĂŒr Mathematik und Physik
4175:
4147:
3768:{\displaystyle t_{0},\ldots ,t_{n-1}}
567:is said to be consistent relative to
378:in the theory are true. This is what
7:
5267:(Second ed.). New York: Dover.
3669:{\displaystyle \;\varphi \in \Phi .}
2040:{\displaystyle \Phi \vdash \varphi }
1766:{\displaystyle \Phi \vdash \varphi }
1660:{\displaystyle \Phi \vdash \varphi }
1122:is consistent and for every formula
790:{\displaystyle \Phi \vdash \varphi }
5348:Stanford Encyclopedia of Philosophy
3785:
3619:
3500:
3480:
3330:
3168:
2980:
2843:
1574:
1550:
1366:{\displaystyle \varphi {t \over x}}
1258:{\displaystyle \exists x\,\varphi }
689:(in some specified formal system).
483:In theories of arithmetic, such as
5142:
4389:{\displaystyle L_{\infty \omega }}
4378:
4339:{\displaystyle L_{\infty \omega }}
4328:
3791:
3660:
3625:
3551:
3519:
3506:
3486:
3411:
3384:
3336:
3174:
3089:
2986:
2902:
2876:
2849:
2706:
2606:
2449:
2394:
2338:
2325:
2303:
2277:
2251:
2210:
2184:
2158:
2118:
2109:
2089:
2060:
2028:
1961:
1925:
1916:
1868:
1837:
1791:
1754:
1734:
1694:
1685:
1648:
1605:
1582:
1528:
1495:
1466:
1330:
1298:
1245:
1221:
1199:
1158:
1109:
1085:
1063:
1043:
1015:
993:
929:
885:
863:
833:
810:
804:
778:
738:
708:
503:show that any sufficiently strong
315:
126:
25:
4790:{\displaystyle A\models \varphi }
4764:{\displaystyle T\models \varphi }
3775:class representatives. Finally,
2462:be a maximally consistent set of
2283:{\displaystyle \varphi \in \Phi }
2190:{\displaystyle \varphi \in \Phi }
2095:{\displaystyle \varphi \in \Phi }
2066:{\displaystyle \varphi \in \Phi }
1238:if for every formula of the form
1205:{\displaystyle \varphi \in \Phi }
205:{\displaystyle \langle A\rangle }
7234:
5425:
5056:On the consistency of arithmetic
4698:{\displaystyle T\vdash \varphi }
4632:{\displaystyle T\vdash \varphi }
4501:{\displaystyle T\vdash \varphi }
4045:
655:means "provable from". That is,
7341:Gödel's incompleteness theorems
5220:Introduction to Metamathematics
5135:and the maximal consistency of
4843:{\displaystyle \vdash \varphi }
2663:{\displaystyle {\overline {t}}}
2565:{\displaystyle t_{0}\sim t_{1}}
2351:if and only if there is a term
1558:{\displaystyle {\mathfrak {I}}}
501:Gödel's incompleteness theorems
3524:
3494:
3434:
3422:
3416:
3392:{\displaystyle \beta _{\Phi }}
3227:
3181:
2309:{\displaystyle \psi \in \Phi }
2245:
2239:
2233:
2216:{\displaystyle \psi \in \Phi }
2152:
2140:
1451:The following are equivalent:
1324:
1308:
1295:
1173:
1155:
579:) if it can be proved that if
513:primitive recursive arithmetic
135:{\displaystyle \lnot \varphi }
1:
7195:History of mathematical logic
3372:Define a variable assignment
76:is consistent if there is no
27:Non-contradiction of a theory
7336:Gödel's completeness theorem
7120:Primitive recursive function
4197:proof is said to be a proof
3351:
3269:
3222:
3196:
3031:
3005:
2723:
2655:
628:In the following context of
32:Consistency (disambiguation)
5327:Elementary Lessons in Logic
5222:. New York: North-Holland.
5073:ZermeloâFraenkel set theory
4613:: we don't require that if
4096:Gentzen's consistency proof
530:ZermeloâFraenkel set theory
7438:
7324:Foundations of mathematics
6184:SchröderâBernstein theorem
5911:Monadic predicate calculus
5570:Foundations of mathematics
5343:"Inconsistent Mathematics"
5243:Elements of Symbolic Logic
409:. The completeness of the
186:(informally "axioms") and
29:
7292:
7230:
7217:Philosophy of mathematics
7166:Automated theorem proving
6337:
6291:Von NeumannâBernaysâGödel
5932:
5423:
5341:Mortensen, Chris (2017).
4639:then there is a proof of
4231:10.1007/978-3-319-33205-5
3930:{\displaystyle \varphi '}
3289:for each constant symbol
2884:{\displaystyle T_{\Phi }}
751:) if there is no formula
674:{\displaystyle a\vdash b}
256:when there is no formula
7366:LöwenheimâSkolem theorem
4927:{\displaystyle \varphi }
4903:{\displaystyle \varphi }
4890:-structure. We say that
4863:{\displaystyle \varphi }
4812:{\displaystyle \varphi }
4718:{\displaystyle \varphi }
4652:{\displaystyle \varphi }
4601:{\displaystyle \varphi }
4541:{\displaystyle \varphi }
4475:{\displaystyle \varphi }
4409:{\displaystyle \varphi }
4359:{\displaystyle \varphi }
4270:Hodges, Wilfrid (1997).
4076:Hilbert's second problem
3905:{\displaystyle \varphi }
3865:{\displaystyle \varphi }
3600:{\displaystyle \varphi }
1991:{\displaystyle \varphi }
1631:{\displaystyle \varphi }
1410:{\displaystyle \varphi }
1135:{\displaystyle \varphi }
979:{\displaystyle \varphi }
955:{\displaystyle \varphi }
915:{\displaystyle \varphi }
764:{\displaystyle \varphi }
721:in first-order logic is
370:, i.e., there exists an
269:{\displaystyle \varphi }
112:{\displaystyle \varphi }
92:{\displaystyle \varphi }
7391:Useâmention distinction
6867:Self-verifying theories
6688:Tarski's axiomatization
5639:Tarski's undefinability
5634:incompleteness theorems
5128:{\displaystyle \equiv }
5024:, pp. 264ff. Also
4144:contradictory sentences
2632:{\displaystyle \equiv }
2413:
648:{\displaystyle \vdash }
591:is consistent. If both
458:incompleteness theorems
7386:Typeâtoken distinction
7241:Mathematics portal
6852:Proof of impossibility
6500:propositional variable
5810:Propositional calculus
5325:Jevons, W. S. (1870).
5149:
5129:
5109:
4968:
4948:
4928:
4904:
4884:
4864:
4844:
4813:
4791:
4765:
4739:
4719:
4699:
4673:
4653:
4633:
4602:
4582:
4562:
4542:
4522:
4502:
4476:
4454:
4434:
4410:
4390:
4360:
4340:
4310:
4290:
4272:A Shorter Model Theory
4101:Proof by contradiction
4029:
3985:
3931:
3906:
3886:
3866:
3846:
3806:
3769:
3717:
3697:
3670:
3640:
3601:
3581:
3558:
3531:
3464:
3444:
3393:
3363:
3309:
3308:{\displaystyle c\in S}
3281:
3147:
3146:{\displaystyle f\in S}
3121:
3099:
3040:
2959:
2958:{\displaystyle R\in S}
2933:
2909:
2885:
2858:
2824:
2801:
2781:
2754:
2688:
2664:
2633:
2613:
2566:
2526:
2506:
2476:
2456:
2432:
2401:
2365:
2345:
2310:
2284:
2258:
2217:
2191:
2165:
2125:
2096:
2067:
2041:
2012:
1992:
1968:
1943:
1892:
1844:
1815:
1767:
1741:
1712:
1661:
1632:
1612:
1589:
1559:
1535:
1511:
1473:
1431:
1411:
1391:
1367:
1337:
1282:
1259:
1228:
1206:
1180:
1136:
1116:
1092:
1070:
1050:
1022:
1000:
980:
956:
936:
916:
892:
870:
840:
820:
791:
765:
745:
715:
675:
649:
505:recursively enumerable
431:induction axiom schema
411:propositional calculus
403:, the logic is called
337:
302:
270:
246:
226:
206:
176:
156:
136:
113:
93:
70:
7110:Kolmogorov complexity
7063:Computably enumerable
6963:Model complete theory
6755:Principia Mathematica
5815:Propositional formula
5644:BanachâTarski paradox
5234:10th impression 1991.
5150:
5148:{\displaystyle \Phi }
5130:
5110:
5108:{\displaystyle t_{i}}
5001:Principia Mathematica
4969:
4949:
4929:
4905:
4885:
4865:
4845:
4814:
4792:
4766:
4740:
4720:
4700:
4674:
4654:
4634:
4603:
4583:
4563:
4543:
4523:
4503:
4477:
4455:
4435:
4411:
4391:
4361:
4341:
4311:
4291:
4207:(formal) theorem" cf
4168:immediate consequence
4030:
3986:
3932:
3907:
3887:
3867:
3847:
3807:
3770:
3718:
3716:{\displaystyle \sim }
3698:
3696:{\displaystyle \sim }
3671:
3641:
3602:
3582:
3559:
3557:{\displaystyle \Phi }
3532:
3465:
3445:
3394:
3364:
3310:
3282:
3148:
3127:-ary function symbol
3122:
3100:
3041:
2960:
2939:-ary relation symbol
2934:
2910:
2908:{\displaystyle \Phi }
2886:
2859:
2825:
2802:
2782:
2780:{\displaystyle T^{S}}
2755:
2689:
2665:
2634:
2614:
2567:
2527:
2507:
2505:{\displaystyle \sim }
2482:-formulas containing
2477:
2457:
2455:{\displaystyle \Phi }
2433:
2402:
2366:
2346:
2311:
2285:
2259:
2218:
2192:
2166:
2126:
2097:
2068:
2042:
2013:
2011:{\displaystyle \psi }
1993:
1969:
1967:{\displaystyle \Phi }
1944:
1893:
1845:
1816:
1768:
1742:
1713:
1662:
1633:
1613:
1611:{\displaystyle \Phi }
1590:
1560:
1536:
1534:{\displaystyle \Phi }
1512:
1474:
1432:
1412:
1392:
1368:
1338:
1283:
1260:
1229:
1227:{\displaystyle \Phi }
1207:
1181:
1137:
1117:
1115:{\displaystyle \Phi }
1093:
1091:{\displaystyle \Phi }
1071:
1069:{\displaystyle \Phi }
1051:
1049:{\displaystyle \Phi }
1030:absolutely consistent
1023:
1021:{\displaystyle \Phi }
1001:
999:{\displaystyle \Phi }
981:
957:
937:
935:{\displaystyle \Phi }
917:
893:
891:{\displaystyle \Phi }
871:
841:
839:{\displaystyle \Phi }
821:
792:
766:
746:
716:
714:{\displaystyle \Phi }
676:
650:
517:Presburger arithmetic
495:Presburger arithmetic
467:(or equivalently the
382:meant in traditional
338:
303:
271:
247:
227:
207:
177:
157:
137:
114:
94:
71:
7309:ChurchâTuring thesis
7303:Entscheidungsproblem
7058:ChurchâTuring thesis
7045:Computability theory
6254:continuum hypothesis
5772:Square of opposition
5630:Gödel's completeness
5283:van Heijenoort, Jean
5166:informal mathematics
5139:
5119:
5092:
5075:with or without the
4958:
4938:
4918:
4894:
4874:
4854:
4831:
4803:
4775:
4749:
4729:
4709:
4683:
4663:
4643:
4617:
4592:
4572:
4552:
4548:. (In particular if
4532:
4512:
4508:, if every model of
4486:
4466:
4444:
4424:
4400:
4370:
4350:
4320:
4300:
4280:
4174:or substitution; cf
4086:Paraconsistent logic
4061:Cognitive dissonance
3996:
3952:
3943:logically equivalent
3916:
3896:
3876:
3856:
3836:
3779:
3727:
3707:
3687:
3650:
3613:
3591:
3571:
3548:
3474:
3454:
3403:
3376:
3319:
3293:
3157:
3131:
3111:
3050:
2969:
2943:
2923:
2899:
2868:
2837:
2814:
2791:
2764:
2698:
2678:
2674:of terms containing
2647:
2623:
2576:
2536:
2516:
2496:
2491:equivalence relation
2466:
2446:
2422:
2375:
2355:
2322:
2294:
2268:
2230:
2201:
2175:
2137:
2106:
2080:
2051:
2025:
2002:
1982:
1958:
1902:
1854:
1828:
1777:
1751:
1725:
1671:
1645:
1622:
1602:
1569:
1545:
1525:
1485:
1457:
1421:
1401:
1381:
1347:
1292:
1272:
1242:
1218:
1190:
1146:
1126:
1106:
1100:maximally consistent
1082:
1060:
1056:is not a theorem of
1040:
1012:
990:
970:
946:
926:
906:
882:
854:
830:
801:
775:
755:
729:
705:
659:
639:
599:are consistent with
539:relative consistency
437:, are not complete.
312:
280:
260:
236:
216:
190:
166:
146:
123:
103:
83:
60:
30:For other uses, see
18:Relative consistency
7212:Mathematical object
7103:P versus NP problem
7068:Computable function
6862:Reverse mathematics
6788:Logical consequence
6665:primitive recursive
6660:elementary function
6433:Free/bound variable
6286:TarskiâGrothendieck
5805:Logical connectives
5735:Logical equivalence
5585:Logical consequence
5245:. New York: Dover.
5060:van Heijenoort 1967
5043:van Heijenoort 1967
5022:van Heijenoort 1967
4993:van Heijenoort 1967
4568:has no models then
4203:(formally) provable
4180:van Heijenoort 1967
583:is consistent then
575:is consistent with
473:underlying calculus
397:sound formal system
7417:Hilbert's problems
7010:Transfer principle
6973:Semantics of logic
6958:Categorical theory
6934:Non-standard model
6448:Logical connective
5575:Information theory
5524:Mathematical logic
5318:Mathematical Logic
5208:10.1007/BF01700692
5145:
5125:
5105:
4978:if it has a model.
4964:
4944:
4924:
4900:
4880:
4860:
4840:
4809:
4787:
4761:
4735:
4725:is deducible from
4715:
4695:
4669:
4649:
4629:
4598:
4578:
4558:
4538:
4518:
4498:
4472:
4450:
4430:
4406:
4386:
4356:
4336:
4306:
4286:
4186:, p. 68]" cf
4071:Hilbert's problems
4025:
3981:
3927:
3902:
3882:
3862:
3842:
3802:
3765:
3713:
3693:
3666:
3636:
3597:
3577:
3554:
3527:
3460:
3450:for each variable
3440:
3389:
3359:
3305:
3277:
3143:
3117:
3095:
3036:
2955:
2929:
2905:
2891:, also called the
2881:
2854:
2820:
2797:
2777:
2750:
2684:
2660:
2629:
2609:
2562:
2522:
2502:
2472:
2452:
2428:
2397:
2361:
2341:
2306:
2280:
2254:
2213:
2187:
2161:
2121:
2092:
2063:
2037:
2008:
1988:
1964:
1939:
1888:
1840:
1811:
1763:
1737:
1708:
1657:
1628:
1608:
1585:
1555:
1531:
1507:
1469:
1427:
1407:
1387:
1363:
1333:
1278:
1255:
1224:
1202:
1176:
1132:
1112:
1088:
1066:
1046:
1018:
996:
976:
952:
932:
912:
902:if for no formula
888:
866:
836:
816:
787:
761:
741:
711:
671:
645:
630:mathematical logic
515:(PRA), but not to
446:mathematical proof
435:second-order logic
423:predicate calculus
384:Aristotelian logic
333:
298:
266:
242:
222:
202:
172:
152:
132:
109:
89:
66:
7399:
7398:
7248:
7247:
7180:Abstract category
6983:Theories of truth
6793:Rule of inference
6783:Natural deduction
6764:
6763:
6309:
6308:
6014:Cartesian product
5919:
5918:
5825:Many-valued logic
5800:Boolean functions
5683:Russell's paradox
5658:diagonal argument
5555:First-order logic
5490:
5489:
5239:Reichenbach, Hans
5062:, pp. 618ff.
5045:, pp. 582ff.
5028:, pp. 134ff.
4999:of the axioms of
4967:{\displaystyle T}
4947:{\displaystyle L}
4883:{\displaystyle L}
4870:is true in every
4738:{\displaystyle T}
4672:{\displaystyle T}
4581:{\displaystyle T}
4561:{\displaystyle T}
4521:{\displaystyle T}
4453:{\displaystyle T}
4433:{\displaystyle T}
4309:{\displaystyle T}
4289:{\displaystyle L}
4240:978-3-319-33203-1
4166:is defined as an
4152:provable formulas
4128:non-contradictory
4053:Philosophy portal
3912:and its negation
3885:{\displaystyle T}
3845:{\displaystyle T}
3826:first-order logic
3580:{\displaystyle S}
3463:{\displaystyle x}
3437:
3354:
3272:
3225:
3199:
3120:{\displaystyle n}
3034:
3008:
2932:{\displaystyle n}
2895:corresponding to
2823:{\displaystyle S}
2800:{\displaystyle S}
2726:
2687:{\displaystyle t}
2672:equivalence class
2658:
2525:{\displaystyle S}
2475:{\displaystyle S}
2431:{\displaystyle S}
2389:
2364:{\displaystyle t}
1439:First-order logic
1430:{\displaystyle t}
1390:{\displaystyle x}
1361:
1322:
1281:{\displaystyle t}
1236:contain witnesses
900:simply consistent
685:is provable from
619:First-order logic
555:is an additional
454:Hilbert's program
442:consistency proof
392:is used instead.
245:{\displaystyle A}
225:{\displaystyle A}
175:{\displaystyle A}
155:{\displaystyle T}
119:and its negation
69:{\displaystyle T}
16:(Redirected from
7429:
7319:Effective method
7297:Cantor's theorem
7275:
7268:
7261:
7252:
7239:
7238:
7190:History of logic
7185:Category of sets
7078:Decision problem
6857:Ordinal analysis
6798:Sequent calculus
6696:Boolean algebras
6636:
6635:
6610:
6581:logical/constant
6335:
6321:
6244:ZermeloâFraenkel
5995:Set operations:
5930:
5867:
5698:
5678:LöwenheimâSkolem
5565:Formal semantics
5517:
5510:
5503:
5494:
5429:
5383:
5376:
5369:
5360:
5352:
5330:
5321:
5312:
5300:
5278:
5256:
5233:
5211:
5181:
5179:De Morgan's laws
5175:
5169:
5162:
5156:
5154:
5152:
5151:
5146:
5134:
5132:
5131:
5126:
5114:
5112:
5111:
5106:
5104:
5103:
5086:
5080:
5069:
5063:
5052:
5046:
5035:
5029:
5014:
5008:
4990:
4984:
4982:
4973:
4971:
4970:
4965:
4953:
4951:
4950:
4945:
4934:is true in some
4933:
4931:
4930:
4925:
4909:
4907:
4906:
4901:
4889:
4887:
4886:
4881:
4869:
4867:
4866:
4861:
4849:
4847:
4846:
4841:
4818:
4816:
4815:
4810:
4796:
4794:
4793:
4788:
4770:
4768:
4767:
4762:
4744:
4742:
4741:
4736:
4724:
4722:
4721:
4716:
4704:
4702:
4701:
4696:
4678:
4676:
4675:
4670:
4658:
4656:
4655:
4650:
4638:
4636:
4635:
4630:
4607:
4605:
4604:
4599:
4587:
4585:
4584:
4579:
4567:
4565:
4564:
4559:
4547:
4545:
4544:
4539:
4527:
4525:
4524:
4519:
4507:
4505:
4504:
4499:
4481:
4479:
4478:
4473:
4459:
4457:
4456:
4451:
4439:
4437:
4436:
4431:
4415:
4413:
4412:
4407:
4395:
4393:
4392:
4387:
4385:
4384:
4365:
4363:
4362:
4357:
4345:
4343:
4342:
4337:
4335:
4334:
4315:
4313:
4312:
4307:
4296:be a signature,
4295:
4293:
4292:
4287:
4267:
4261:
4260:
4218:
4212:
4184:Reichenbach 1947
4117:
4055:
4050:
4049:
4048:
4034:
4032:
4031:
4026:
4006:
3990:
3988:
3987:
3982:
3971:
3945:conditions hold
3936:
3934:
3933:
3928:
3926:
3911:
3909:
3908:
3903:
3891:
3889:
3888:
3883:
3871:
3869:
3868:
3863:
3851:
3849:
3848:
3843:
3811:
3809:
3808:
3803:
3795:
3794:
3789:
3788:
3774:
3772:
3771:
3766:
3764:
3763:
3739:
3738:
3722:
3720:
3719:
3714:
3702:
3700:
3699:
3694:
3675:
3673:
3672:
3667:
3645:
3643:
3642:
3637:
3629:
3628:
3623:
3622:
3606:
3604:
3603:
3598:
3586:
3584:
3583:
3578:
3563:
3561:
3560:
3555:
3544:associated with
3536:
3534:
3533:
3528:
3523:
3522:
3510:
3509:
3504:
3503:
3490:
3489:
3484:
3483:
3469:
3467:
3466:
3461:
3449:
3447:
3446:
3441:
3439:
3438:
3430:
3415:
3414:
3398:
3396:
3395:
3390:
3388:
3387:
3368:
3366:
3365:
3360:
3355:
3347:
3342:
3341:
3340:
3339:
3334:
3333:
3314:
3312:
3311:
3306:
3286:
3284:
3283:
3278:
3273:
3268:
3267:
3266:
3248:
3247:
3234:
3226:
3221:
3220:
3205:
3200:
3195:
3194:
3185:
3180:
3179:
3178:
3177:
3172:
3171:
3152:
3150:
3149:
3144:
3126:
3124:
3123:
3118:
3104:
3102:
3101:
3096:
3085:
3084:
3066:
3065:
3045:
3043:
3042:
3037:
3035:
3030:
3029:
3014:
3009:
3004:
3003:
2994:
2992:
2991:
2990:
2989:
2984:
2983:
2964:
2962:
2961:
2956:
2938:
2936:
2935:
2930:
2914:
2912:
2911:
2906:
2890:
2888:
2887:
2882:
2880:
2879:
2863:
2861:
2860:
2855:
2853:
2852:
2847:
2846:
2829:
2827:
2826:
2821:
2806:
2804:
2803:
2798:
2786:
2784:
2783:
2778:
2776:
2775:
2759:
2757:
2756:
2751:
2746:
2745:
2727:
2719:
2710:
2709:
2693:
2691:
2690:
2685:
2669:
2667:
2666:
2661:
2659:
2651:
2638:
2636:
2635:
2630:
2618:
2616:
2615:
2610:
2602:
2601:
2589:
2588:
2571:
2569:
2568:
2563:
2561:
2560:
2548:
2547:
2531:
2529:
2528:
2523:
2511:
2509:
2508:
2503:
2481:
2479:
2478:
2473:
2461:
2459:
2458:
2453:
2437:
2435:
2434:
2429:
2414:Henkin's theorem
2406:
2404:
2403:
2398:
2390:
2382:
2370:
2368:
2367:
2362:
2350:
2348:
2347:
2342:
2315:
2313:
2312:
2307:
2289:
2287:
2286:
2281:
2263:
2261:
2260:
2255:
2222:
2220:
2219:
2214:
2196:
2194:
2193:
2188:
2170:
2168:
2167:
2162:
2130:
2128:
2127:
2122:
2101:
2099:
2098:
2093:
2072:
2070:
2069:
2064:
2046:
2044:
2043:
2038:
2017:
2015:
2014:
2009:
1997:
1995:
1994:
1989:
1973:
1971:
1970:
1965:
1948:
1946:
1945:
1940:
1938:
1934:
1897:
1895:
1894:
1889:
1887:
1883:
1849:
1847:
1846:
1841:
1820:
1818:
1817:
1812:
1810:
1806:
1772:
1770:
1769:
1764:
1746:
1744:
1743:
1738:
1717:
1715:
1714:
1709:
1707:
1703:
1666:
1664:
1663:
1658:
1637:
1635:
1634:
1629:
1617:
1615:
1614:
1609:
1594:
1592:
1591:
1586:
1578:
1577:
1564:
1562:
1561:
1556:
1554:
1553:
1540:
1538:
1537:
1532:
1516:
1514:
1513:
1508:
1478:
1476:
1475:
1470:
1436:
1434:
1433:
1428:
1416:
1414:
1413:
1408:
1396:
1394:
1393:
1388:
1372:
1370:
1369:
1364:
1362:
1354:
1342:
1340:
1339:
1334:
1323:
1315:
1287:
1285:
1284:
1279:
1264:
1262:
1261:
1256:
1233:
1231:
1230:
1225:
1211:
1209:
1208:
1203:
1185:
1183:
1182:
1177:
1141:
1139:
1138:
1133:
1121:
1119:
1118:
1113:
1097:
1095:
1094:
1089:
1075:
1073:
1072:
1067:
1055:
1053:
1052:
1047:
1027:
1025:
1024:
1019:
1005:
1003:
1002:
997:
986:are theorems of
985:
983:
982:
977:
961:
959:
958:
953:
941:
939:
938:
933:
921:
919:
918:
913:
897:
895:
894:
889:
875:
873:
872:
867:
845:
843:
842:
837:
825:
823:
822:
817:
796:
794:
793:
788:
770:
768:
767:
762:
750:
748:
747:
742:
720:
718:
717:
712:
680:
678:
677:
672:
654:
652:
651:
646:
634:turnstile symbol
571:(or simply that
541:
540:
509:Peano arithmetic
485:Peano arithmetic
374:under which all
342:
340:
339:
334:
307:
305:
304:
299:
275:
273:
272:
267:
251:
249:
248:
243:
231:
229:
228:
223:
211:
209:
208:
203:
184:closed sentences
181:
179:
178:
173:
161:
159:
158:
153:
141:
139:
138:
133:
118:
116:
115:
110:
98:
96:
95:
90:
75:
73:
72:
67:
21:
7437:
7436:
7432:
7431:
7430:
7428:
7427:
7426:
7402:
7401:
7400:
7395:
7288:
7286:metamathematics
7279:
7249:
7244:
7233:
7226:
7171:Category theory
7161:Algebraic logic
7144:
7115:Lambda calculus
7053:Church encoding
7039:
7015:Truth predicate
6871:
6837:Complete theory
6760:
6629:
6625:
6621:
6616:
6608:
6328: and
6324:
6319:
6305:
6281:New Foundations
6249:axiom of choice
6232:
6194:Gödel numbering
6134: and
6126:
6030:
5915:
5865:
5846:
5795:Boolean algebra
5781:
5745:Equiconsistency
5710:Classical logic
5687:
5668:Halting problem
5656: and
5632: and
5620: and
5619:
5614:Theorems (
5609:
5526:
5521:
5491:
5486:
5456:
5430:
5421:
5394:
5387:
5356:
5340:
5337:
5324:
5315:
5305:"Consistency".
5304:
5297:
5281:
5275:
5259:
5253:
5237:
5230:
5216:Kleene, Stephen
5214:
5193:
5190:
5185:
5184:
5176:
5172:
5163:
5159:
5137:
5136:
5117:
5116:
5095:
5090:
5089:
5087:
5083:
5077:axiom of choice
5070:
5066:
5053:
5049:
5036:
5032:
5015:
5011:
4991:
4987:
4979:
4956:
4955:
4936:
4935:
4916:
4915:
4892:
4891:
4872:
4871:
4852:
4851:
4829:
4828:
4825:logical theorem
4801:
4800:
4798:
4773:
4772:
4747:
4746:
4727:
4726:
4707:
4706:
4681:
4680:
4661:
4660:
4641:
4640:
4615:
4614:
4609:
4590:
4589:
4570:
4569:
4550:
4549:
4530:
4529:
4510:
4509:
4484:
4483:
4464:
4463:
4442:
4441:
4422:
4421:
4398:
4397:
4373:
4368:
4367:
4348:
4347:
4323:
4318:
4317:
4298:
4297:
4278:
4277:
4269:
4268:
4264:
4241:
4220:
4219:
4215:
4118:
4114:
4109:
4081:Jan Ćukasiewicz
4066:Equiconsistency
4051:
4046:
4044:
4041:
3999:
3994:
3993:
3964:
3950:
3949:
3919:
3914:
3913:
3894:
3893:
3874:
3873:
3854:
3853:
3834:
3833:
3824:with classical
3818:
3782:
3777:
3776:
3749:
3730:
3725:
3724:
3705:
3704:
3685:
3684:
3681:
3679:Sketch of proof
3676:
3648:
3647:
3646:if and only if
3616:
3611:
3610:
3589:
3588:
3569:
3568:
3546:
3545:
3514:
3497:
3477:
3472:
3471:
3452:
3451:
3406:
3401:
3400:
3379:
3374:
3373:
3327:
3322:
3317:
3316:
3291:
3290:
3252:
3239:
3235:
3206:
3186:
3165:
3160:
3155:
3154:
3129:
3128:
3109:
3108:
3070:
3057:
3048:
3047:
3015:
2995:
2977:
2972:
2967:
2966:
2941:
2940:
2921:
2920:
2897:
2896:
2871:
2866:
2865:
2840:
2835:
2834:
2812:
2811:
2789:
2788:
2767:
2762:
2761:
2737:
2701:
2696:
2695:
2676:
2675:
2645:
2644:
2621:
2620:
2593:
2580:
2574:
2573:
2552:
2539:
2534:
2533:
2514:
2513:
2494:
2493:
2464:
2463:
2444:
2443:
2420:
2419:
2416:
2373:
2372:
2353:
2352:
2320:
2319:
2292:
2291:
2266:
2265:
2228:
2227:
2199:
2198:
2173:
2172:
2171:if and only if
2135:
2134:
2104:
2103:
2078:
2077:
2049:
2048:
2023:
2022:
2000:
1999:
1980:
1979:
1956:
1955:
1915:
1911:
1900:
1899:
1867:
1863:
1852:
1851:
1826:
1825:
1790:
1786:
1775:
1774:
1749:
1748:
1723:
1722:
1684:
1680:
1669:
1668:
1643:
1642:
1620:
1619:
1600:
1599:
1567:
1566:
1543:
1542:
1523:
1522:
1483:
1482:
1455:
1454:
1448:
1419:
1418:
1399:
1398:
1379:
1378:
1345:
1344:
1290:
1289:
1270:
1269:
1265:there exists a
1240:
1239:
1216:
1215:
1188:
1187:
1144:
1143:
1124:
1123:
1104:
1103:
1080:
1079:
1058:
1057:
1038:
1037:
1034:Post consistent
1010:
1009:
988:
987:
968:
967:
944:
943:
924:
923:
904:
903:
880:
879:
852:
851:
828:
827:
799:
798:
773:
772:
753:
752:
727:
726:
703:
702:
695:
657:
656:
637:
636:
626:
621:
538:
537:
481:
465:cut-elimination
362:counterpart is
310:
309:
278:
277:
258:
257:
234:
233:
214:
213:
188:
187:
164:
163:
144:
143:
121:
120:
101:
100:
99:such that both
81:
80:
58:
57:
43:deductive logic
35:
28:
23:
22:
15:
12:
11:
5:
7435:
7433:
7425:
7424:
7419:
7414:
7404:
7403:
7397:
7396:
7394:
7393:
7388:
7383:
7378:
7376:Satisfiability
7373:
7368:
7363:
7361:Interpretation
7358:
7353:
7348:
7343:
7338:
7333:
7332:
7331:
7321:
7316:
7311:
7306:
7299:
7293:
7290:
7289:
7280:
7278:
7277:
7270:
7263:
7255:
7246:
7245:
7231:
7228:
7227:
7225:
7224:
7219:
7214:
7209:
7204:
7203:
7202:
7192:
7187:
7182:
7173:
7168:
7163:
7158:
7156:Abstract logic
7152:
7150:
7146:
7145:
7143:
7142:
7137:
7135:Turing machine
7132:
7127:
7122:
7117:
7112:
7107:
7106:
7105:
7100:
7095:
7090:
7085:
7075:
7073:Computable set
7070:
7065:
7060:
7055:
7049:
7047:
7041:
7040:
7038:
7037:
7032:
7027:
7022:
7017:
7012:
7007:
7002:
7001:
7000:
6995:
6990:
6980:
6975:
6970:
6968:Satisfiability
6965:
6960:
6955:
6954:
6953:
6943:
6942:
6941:
6931:
6930:
6929:
6924:
6919:
6914:
6909:
6899:
6898:
6897:
6892:
6885:Interpretation
6881:
6879:
6873:
6872:
6870:
6869:
6864:
6859:
6854:
6849:
6839:
6834:
6833:
6832:
6831:
6830:
6820:
6815:
6805:
6800:
6795:
6790:
6785:
6780:
6774:
6772:
6766:
6765:
6762:
6761:
6759:
6758:
6750:
6749:
6748:
6747:
6742:
6741:
6740:
6735:
6730:
6710:
6709:
6708:
6706:minimal axioms
6703:
6692:
6691:
6690:
6679:
6678:
6677:
6672:
6667:
6662:
6657:
6652:
6639:
6637:
6618:
6617:
6615:
6614:
6613:
6612:
6600:
6595:
6594:
6593:
6588:
6583:
6578:
6568:
6563:
6558:
6553:
6552:
6551:
6546:
6536:
6535:
6534:
6529:
6524:
6519:
6509:
6504:
6503:
6502:
6497:
6492:
6482:
6481:
6480:
6475:
6470:
6465:
6460:
6455:
6445:
6440:
6435:
6430:
6429:
6428:
6423:
6418:
6413:
6403:
6398:
6396:Formation rule
6393:
6388:
6387:
6386:
6381:
6371:
6370:
6369:
6359:
6354:
6349:
6344:
6338:
6332:
6315:Formal systems
6311:
6310:
6307:
6306:
6304:
6303:
6298:
6293:
6288:
6283:
6278:
6273:
6268:
6263:
6258:
6257:
6256:
6251:
6240:
6238:
6234:
6233:
6231:
6230:
6229:
6228:
6218:
6213:
6212:
6211:
6204:Large cardinal
6201:
6196:
6191:
6186:
6181:
6167:
6166:
6165:
6160:
6155:
6140:
6138:
6128:
6127:
6125:
6124:
6123:
6122:
6117:
6112:
6102:
6097:
6092:
6087:
6082:
6077:
6072:
6067:
6062:
6057:
6052:
6047:
6041:
6039:
6032:
6031:
6029:
6028:
6027:
6026:
6021:
6016:
6011:
6006:
6001:
5993:
5992:
5991:
5986:
5976:
5971:
5969:Extensionality
5966:
5964:Ordinal number
5961:
5951:
5946:
5945:
5944:
5933:
5927:
5921:
5920:
5917:
5916:
5914:
5913:
5908:
5903:
5898:
5893:
5888:
5883:
5882:
5881:
5871:
5870:
5869:
5856:
5854:
5848:
5847:
5845:
5844:
5843:
5842:
5837:
5832:
5822:
5817:
5812:
5807:
5802:
5797:
5791:
5789:
5783:
5782:
5780:
5779:
5774:
5769:
5764:
5759:
5754:
5749:
5748:
5747:
5737:
5732:
5727:
5722:
5717:
5712:
5706:
5704:
5695:
5689:
5688:
5686:
5685:
5680:
5675:
5670:
5665:
5660:
5648:Cantor's
5646:
5641:
5636:
5626:
5624:
5611:
5610:
5608:
5607:
5602:
5597:
5592:
5587:
5582:
5577:
5572:
5567:
5562:
5557:
5552:
5547:
5546:
5545:
5534:
5532:
5528:
5527:
5522:
5520:
5519:
5512:
5505:
5497:
5488:
5487:
5485:
5484:
5479:
5474:
5464:
5462:
5461:Negation
5458:
5457:
5455:
5454:
5449:
5444:
5438:
5436:
5432:
5431:
5424:
5422:
5420:
5419:
5413:
5411:truth function
5408:
5402:
5400:
5396:
5395:
5388:
5386:
5385:
5378:
5371:
5363:
5354:
5353:
5336:
5335:External links
5333:
5332:
5331:
5322:
5313:
5302:
5295:
5279:
5273:
5261:Tarski, Alfred
5257:
5251:
5235:
5228:
5212:
5202:(1): 173â198.
5189:
5186:
5183:
5182:
5170:
5157:
5144:
5124:
5102:
5098:
5081:
5064:
5047:
5030:
5009:
4985:
4963:
4943:
4923:
4899:
4879:
4859:
4839:
4836:
4808:
4786:
4783:
4780:
4760:
4757:
4754:
4734:
4714:
4694:
4691:
4688:
4668:
4648:
4628:
4625:
4622:
4597:
4577:
4557:
4537:
4528:is a model of
4517:
4497:
4494:
4491:
4471:
4449:
4429:
4405:
4396:. We say that
4383:
4380:
4376:
4366:a sentence in
4355:
4333:
4330:
4326:
4305:
4285:
4262:
4239:
4213:
4138:one forms the
4111:
4110:
4108:
4105:
4104:
4103:
4098:
4093:
4088:
4083:
4078:
4073:
4068:
4063:
4057:
4056:
4040:
4037:
4036:
4035:
4024:
4021:
4018:
4015:
4012:
4009:
4005:
4002:
3991:
3980:
3977:
3974:
3970:
3967:
3963:
3960:
3957:
3925:
3922:
3901:
3892:contains both
3881:
3861:
3841:
3822:ZFC set theory
3817:
3814:
3801:
3798:
3793:
3787:
3762:
3759:
3756:
3752:
3748:
3745:
3742:
3737:
3733:
3712:
3692:
3680:
3677:
3665:
3662:
3659:
3656:
3635:
3632:
3627:
3621:
3609:
3596:
3576:
3567:Then for each
3553:
3541:interpretation
3526:
3521:
3517:
3513:
3508:
3502:
3496:
3493:
3488:
3482:
3459:
3436:
3433:
3427:
3424:
3421:
3418:
3413:
3409:
3386:
3382:
3370:
3369:
3358:
3353:
3350:
3345:
3338:
3332:
3325:
3304:
3301:
3298:
3287:
3276:
3271:
3265:
3262:
3259:
3255:
3251:
3246:
3242:
3238:
3232:
3229:
3224:
3219:
3216:
3213:
3209:
3203:
3198:
3193:
3189:
3183:
3176:
3170:
3163:
3142:
3139:
3136:
3116:
3105:
3094:
3091:
3088:
3083:
3080:
3077:
3073:
3069:
3064:
3060:
3056:
3033:
3028:
3025:
3022:
3018:
3012:
3007:
3002:
2998:
2988:
2982:
2975:
2954:
2951:
2948:
2928:
2904:
2893:term-structure
2878:
2874:
2851:
2845:
2819:
2796:
2774:
2770:
2749:
2744:
2740:
2736:
2733:
2730:
2725:
2722:
2716:
2713:
2708:
2704:
2683:
2657:
2654:
2628:
2608:
2605:
2600:
2596:
2592:
2587:
2583:
2559:
2555:
2551:
2546:
2542:
2521:
2512:on the set of
2501:
2471:
2451:
2440:set of symbols
2427:
2415:
2412:
2411:
2410:
2409:
2408:
2396:
2393:
2388:
2385:
2380:
2360:
2340:
2337:
2334:
2330:
2327:
2317:
2305:
2302:
2299:
2279:
2276:
2273:
2253:
2250:
2247:
2244:
2241:
2238:
2235:
2224:
2212:
2209:
2206:
2186:
2183:
2180:
2160:
2157:
2154:
2151:
2148:
2145:
2142:
2132:
2120:
2117:
2114:
2111:
2091:
2088:
2085:
2074:
2062:
2059:
2056:
2036:
2033:
2030:
2007:
1987:
1963:
1952:
1951:
1950:
1937:
1933:
1930:
1927:
1924:
1921:
1918:
1914:
1910:
1907:
1886:
1882:
1879:
1876:
1873:
1870:
1866:
1862:
1859:
1839:
1836:
1833:
1822:
1809:
1805:
1802:
1799:
1796:
1793:
1789:
1785:
1782:
1762:
1759:
1756:
1736:
1733:
1730:
1719:
1706:
1702:
1699:
1696:
1693:
1690:
1687:
1683:
1679:
1676:
1656:
1653:
1650:
1627:
1607:
1596:
1584:
1581:
1576:
1552:
1530:
1519:
1518:
1517:
1506:
1503:
1500:
1497:
1493:
1490:
1479:
1468:
1465:
1462:
1447:
1444:
1443:
1442:
1426:
1406:
1386:
1360:
1357:
1352:
1332:
1329:
1326:
1321:
1318:
1313:
1310:
1307:
1303:
1300:
1297:
1277:
1254:
1250:
1247:
1223:
1213:
1201:
1198:
1195:
1175:
1172:
1169:
1166:
1163:
1160:
1157:
1154:
1151:
1131:
1111:
1098:is said to be
1087:
1077:
1065:
1045:
1028:is said to be
1017:
1007:
995:
975:
951:
931:
911:
898:is said to be
887:
877:
865:
862:
859:
835:
815:
812:
809:
806:
786:
783:
780:
760:
740:
737:
734:
710:
694:
691:
670:
667:
664:
644:
625:
622:
620:
617:
607:is said to be
480:
477:
425:was proved by
413:was proved by
372:interpretation
364:satisfiability
358:notion, whose
332:
329:
326:
323:
320:
317:
297:
294:
291:
288:
285:
265:
241:
221:
201:
198:
195:
171:
151:
131:
128:
108:
88:
65:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
7434:
7423:
7420:
7418:
7415:
7413:
7410:
7409:
7407:
7392:
7389:
7387:
7384:
7382:
7379:
7377:
7374:
7372:
7369:
7367:
7364:
7362:
7359:
7357:
7354:
7352:
7349:
7347:
7344:
7342:
7339:
7337:
7334:
7330:
7327:
7326:
7325:
7322:
7320:
7317:
7315:
7312:
7310:
7307:
7305:
7304:
7300:
7298:
7295:
7294:
7291:
7287:
7283:
7276:
7271:
7269:
7264:
7262:
7257:
7256:
7253:
7243:
7242:
7237:
7229:
7223:
7220:
7218:
7215:
7213:
7210:
7208:
7205:
7201:
7198:
7197:
7196:
7193:
7191:
7188:
7186:
7183:
7181:
7177:
7174:
7172:
7169:
7167:
7164:
7162:
7159:
7157:
7154:
7153:
7151:
7147:
7141:
7138:
7136:
7133:
7131:
7130:Recursive set
7128:
7126:
7123:
7121:
7118:
7116:
7113:
7111:
7108:
7104:
7101:
7099:
7096:
7094:
7091:
7089:
7086:
7084:
7081:
7080:
7079:
7076:
7074:
7071:
7069:
7066:
7064:
7061:
7059:
7056:
7054:
7051:
7050:
7048:
7046:
7042:
7036:
7033:
7031:
7028:
7026:
7023:
7021:
7018:
7016:
7013:
7011:
7008:
7006:
7003:
6999:
6996:
6994:
6991:
6989:
6986:
6985:
6984:
6981:
6979:
6976:
6974:
6971:
6969:
6966:
6964:
6961:
6959:
6956:
6952:
6949:
6948:
6947:
6944:
6940:
6939:of arithmetic
6937:
6936:
6935:
6932:
6928:
6925:
6923:
6920:
6918:
6915:
6913:
6910:
6908:
6905:
6904:
6903:
6900:
6896:
6893:
6891:
6888:
6887:
6886:
6883:
6882:
6880:
6878:
6874:
6868:
6865:
6863:
6860:
6858:
6855:
6853:
6850:
6847:
6846:from ZFC
6843:
6840:
6838:
6835:
6829:
6826:
6825:
6824:
6821:
6819:
6816:
6814:
6811:
6810:
6809:
6806:
6804:
6801:
6799:
6796:
6794:
6791:
6789:
6786:
6784:
6781:
6779:
6776:
6775:
6773:
6771:
6767:
6757:
6756:
6752:
6751:
6746:
6745:non-Euclidean
6743:
6739:
6736:
6734:
6731:
6729:
6728:
6724:
6723:
6721:
6718:
6717:
6715:
6711:
6707:
6704:
6702:
6699:
6698:
6697:
6693:
6689:
6686:
6685:
6684:
6680:
6676:
6673:
6671:
6668:
6666:
6663:
6661:
6658:
6656:
6653:
6651:
6648:
6647:
6645:
6641:
6640:
6638:
6633:
6627:
6622:Example
6619:
6611:
6606:
6605:
6604:
6601:
6599:
6596:
6592:
6589:
6587:
6584:
6582:
6579:
6577:
6574:
6573:
6572:
6569:
6567:
6564:
6562:
6559:
6557:
6554:
6550:
6547:
6545:
6542:
6541:
6540:
6537:
6533:
6530:
6528:
6525:
6523:
6520:
6518:
6515:
6514:
6513:
6510:
6508:
6505:
6501:
6498:
6496:
6493:
6491:
6488:
6487:
6486:
6483:
6479:
6476:
6474:
6471:
6469:
6466:
6464:
6461:
6459:
6456:
6454:
6451:
6450:
6449:
6446:
6444:
6441:
6439:
6436:
6434:
6431:
6427:
6424:
6422:
6419:
6417:
6414:
6412:
6409:
6408:
6407:
6404:
6402:
6399:
6397:
6394:
6392:
6389:
6385:
6382:
6380:
6379:by definition
6377:
6376:
6375:
6372:
6368:
6365:
6364:
6363:
6360:
6358:
6355:
6353:
6350:
6348:
6345:
6343:
6340:
6339:
6336:
6333:
6331:
6327:
6322:
6316:
6312:
6302:
6299:
6297:
6294:
6292:
6289:
6287:
6284:
6282:
6279:
6277:
6274:
6272:
6269:
6267:
6266:KripkeâPlatek
6264:
6262:
6259:
6255:
6252:
6250:
6247:
6246:
6245:
6242:
6241:
6239:
6235:
6227:
6224:
6223:
6222:
6219:
6217:
6214:
6210:
6207:
6206:
6205:
6202:
6200:
6197:
6195:
6192:
6190:
6187:
6185:
6182:
6179:
6175:
6171:
6168:
6164:
6161:
6159:
6156:
6154:
6151:
6150:
6149:
6145:
6142:
6141:
6139:
6137:
6133:
6129:
6121:
6118:
6116:
6113:
6111:
6110:constructible
6108:
6107:
6106:
6103:
6101:
6098:
6096:
6093:
6091:
6088:
6086:
6083:
6081:
6078:
6076:
6073:
6071:
6068:
6066:
6063:
6061:
6058:
6056:
6053:
6051:
6048:
6046:
6043:
6042:
6040:
6038:
6033:
6025:
6022:
6020:
6017:
6015:
6012:
6010:
6007:
6005:
6002:
6000:
5997:
5996:
5994:
5990:
5987:
5985:
5982:
5981:
5980:
5977:
5975:
5972:
5970:
5967:
5965:
5962:
5960:
5956:
5952:
5950:
5947:
5943:
5940:
5939:
5938:
5935:
5934:
5931:
5928:
5926:
5922:
5912:
5909:
5907:
5904:
5902:
5899:
5897:
5894:
5892:
5889:
5887:
5884:
5880:
5877:
5876:
5875:
5872:
5868:
5863:
5862:
5861:
5858:
5857:
5855:
5853:
5849:
5841:
5838:
5836:
5833:
5831:
5828:
5827:
5826:
5823:
5821:
5818:
5816:
5813:
5811:
5808:
5806:
5803:
5801:
5798:
5796:
5793:
5792:
5790:
5788:
5787:Propositional
5784:
5778:
5775:
5773:
5770:
5768:
5765:
5763:
5760:
5758:
5755:
5753:
5750:
5746:
5743:
5742:
5741:
5738:
5736:
5733:
5731:
5728:
5726:
5723:
5721:
5718:
5716:
5715:Logical truth
5713:
5711:
5708:
5707:
5705:
5703:
5699:
5696:
5694:
5690:
5684:
5681:
5679:
5676:
5674:
5671:
5669:
5666:
5664:
5661:
5659:
5655:
5651:
5647:
5645:
5642:
5640:
5637:
5635:
5631:
5628:
5627:
5625:
5623:
5617:
5612:
5606:
5603:
5601:
5598:
5596:
5593:
5591:
5588:
5586:
5583:
5581:
5578:
5576:
5573:
5571:
5568:
5566:
5563:
5561:
5558:
5556:
5553:
5551:
5548:
5544:
5541:
5540:
5539:
5536:
5535:
5533:
5529:
5525:
5518:
5513:
5511:
5506:
5504:
5499:
5498:
5495:
5483:
5482:inconsistency
5480:
5478:
5477:contradiction
5475:
5473:
5469:
5466:
5465:
5463:
5459:
5453:
5450:
5448:
5445:
5443:
5440:
5439:
5437:
5433:
5428:
5418:
5415:⊨
5414:
5412:
5409:
5407:
5404:
5403:
5401:
5397:
5392:
5391:Logical truth
5384:
5379:
5377:
5372:
5370:
5365:
5364:
5361:
5357:
5350:
5349:
5344:
5339:
5338:
5334:
5328:
5323:
5319:
5314:
5310:
5309:
5303:
5298:
5296:0-674-32449-8
5292:
5288:
5284:
5280:
5276:
5274:0-486-28462-X
5270:
5266:
5262:
5258:
5254:
5252:0-486-24004-5
5248:
5244:
5240:
5236:
5231:
5229:0-7204-2103-9
5225:
5221:
5217:
5213:
5209:
5205:
5201:
5197:
5192:
5191:
5187:
5180:
5177:according to
5174:
5171:
5167:
5161:
5158:
5122:
5100:
5096:
5085:
5082:
5078:
5074:
5068:
5065:
5061:
5057:
5051:
5048:
5044:
5040:
5034:
5031:
5027:
5023:
5019:
5013:
5010:
5006:
5002:
4998:
4994:
4989:
4986:
4981:
4977:
4961:
4941:
4921:
4913:
4897:
4877:
4857:
4837:
4834:
4827:, in symbols
4826:
4822:
4806:
4784:
4781:
4778:
4758:
4755:
4752:
4732:
4712:
4705:to mean that
4692:
4689:
4686:
4666:
4646:
4626:
4623:
4620:
4612:
4595:
4575:
4555:
4535:
4515:
4495:
4492:
4489:
4482:, in symbols
4469:
4462:
4447:
4427:
4419:
4403:
4381:
4374:
4353:
4331:
4324:
4303:
4283:
4273:
4266:
4263:
4258:
4254:
4250:
4246:
4242:
4236:
4232:
4228:
4224:
4217:
4214:
4211:, p. 83.
4210:
4206:
4202:
4198:
4193:
4190:, p. 3.
4189:
4185:
4181:
4177:
4173:
4169:
4165:
4161:
4157:
4153:
4149:
4145:
4141:
4137:
4133:
4132:contradictory
4129:
4125:
4121:
4116:
4113:
4106:
4102:
4099:
4097:
4094:
4092:
4091:Ï-consistency
4089:
4087:
4084:
4082:
4079:
4077:
4074:
4072:
4069:
4067:
4064:
4062:
4059:
4058:
4054:
4043:
4038:
4022:
4019:
4016:
4013:
4010:
4007:
4003:
4000:
3992:
3978:
3975:
3968:
3965:
3961:
3958:
3948:
3947:
3946:
3944:
3940:
3923:
3920:
3899:
3879:
3859:
3839:
3831:
3827:
3823:
3815:
3813:
3799:
3796:
3760:
3757:
3754:
3750:
3746:
3743:
3740:
3735:
3731:
3710:
3690:
3678:
3663:
3657:
3654:
3633:
3630:
3608:
3594:
3574:
3565:
3543:
3542:
3515:
3511:
3491:
3457:
3431:
3425:
3419:
3407:
3380:
3356:
3348:
3343:
3323:
3302:
3299:
3296:
3288:
3274:
3263:
3260:
3257:
3253:
3249:
3244:
3240:
3236:
3230:
3217:
3214:
3211:
3207:
3201:
3191:
3187:
3161:
3140:
3137:
3134:
3114:
3106:
3092:
3086:
3081:
3078:
3075:
3071:
3067:
3062:
3058:
3054:
3026:
3023:
3020:
3016:
3010:
3000:
2996:
2973:
2952:
2949:
2946:
2926:
2918:
2917:
2916:
2894:
2872:
2833:
2817:
2808:
2794:
2772:
2768:
2742:
2738:
2734:
2731:
2728:
2720:
2711:
2702:
2681:
2673:
2652:
2642:
2626:
2603:
2598:
2594:
2590:
2585:
2581:
2557:
2553:
2549:
2544:
2540:
2519:
2499:
2492:
2487:
2485:
2469:
2441:
2425:
2391:
2386:
2383:
2378:
2358:
2335:
2332:
2328:
2318:
2300:
2297:
2274:
2271:
2248:
2242:
2236:
2225:
2207:
2204:
2181:
2178:
2155:
2149:
2146:
2143:
2133:
2115:
2112:
2086:
2083:
2075:
2057:
2054:
2034:
2031:
2020:
2019:
2005:
1985:
1977:
1953:
1935:
1928:
1919:
1912:
1908:
1905:
1884:
1877:
1871:
1864:
1860:
1857:
1834:
1831:
1823:
1807:
1800:
1794:
1787:
1783:
1780:
1760:
1757:
1731:
1728:
1720:
1704:
1697:
1688:
1681:
1677:
1674:
1654:
1651:
1640:
1639:
1625:
1597:
1579:
1520:
1504:
1501:
1498:
1491:
1488:
1480:
1463:
1460:
1453:
1452:
1450:
1449:
1446:Basic results
1445:
1440:
1424:
1404:
1384:
1376:
1358:
1355:
1350:
1327:
1319:
1316:
1311:
1305:
1301:
1275:
1268:
1252:
1248:
1237:
1214:
1196:
1193:
1167:
1161:
1152:
1149:
1129:
1101:
1078:
1035:
1031:
1008:
973:
965:
949:
909:
901:
878:
860:
857:
849:
826:. Otherwise
813:
807:
784:
781:
758:
735:
732:
724:
701:
697:
696:
692:
690:
688:
684:
668:
665:
662:
642:
635:
631:
623:
618:
616:
614:
610:
606:
602:
598:
594:
590:
586:
582:
578:
574:
570:
566:
562:
558:
554:
550:
546:
542:
533:
531:
526:
520:
518:
514:
510:
506:
502:
498:
496:
492:
490:
486:
478:
476:
474:
470:
469:normalization
466:
461:
459:
455:
451:
447:
443:
438:
436:
432:
428:
424:
420:
416:
412:
408:
407:
402:
398:
393:
391:
390:
385:
381:
377:
373:
369:
365:
361:
357:
353:
352:formal system
350:
346:
327:
321:
318:
292:
286:
283:
263:
255:
239:
219:
196:
185:
169:
149:
129:
106:
86:
79:
63:
55:
54:contradiction
51:
48:
44:
40:
33:
19:
7412:Proof theory
7381:Independence
7356:Decidability
7351:Completeness
7313:
7301:
7232:
7030:Ultraproduct
6877:Model theory
6842:Independence
6778:Formal proof
6770:Proof theory
6753:
6726:
6683:real numbers
6655:second-order
6566:Substitution
6443:Metalanguage
6384:conservative
6357:Axiom schema
6301:Constructive
6271:MorseâKelley
6237:Set theories
6216:Aleph number
6209:inaccessible
6115:Grothendieck
5999:intersection
5886:Higher-order
5874:Second-order
5820:Truth tables
5777:Venn diagram
5739:
5560:Formal proof
5481:
5467:
5447:formal proof
5355:
5346:
5326:
5317:
5306:
5286:
5264:
5242:
5219:
5199:
5195:
5173:
5160:
5084:
5067:
5055:
5050:
5038:
5033:
5017:
5012:
5004:
5000:
4997:independence
4996:
4988:
4975:
4911:
4824:
4820:
4799:We say that
4610:
4460:
4417:
4316:a theory in
4275:
4271:
4265:
4222:
4216:
4204:
4200:
4196:
4172:modus ponens
4171:
4170:in terms of
4167:
4163:
4159:
4155:
4151:
4143:
4139:
4135:
4131:
4127:
4123:
4115:
3938:
3830:inconsistent
3829:
3819:
3816:Model theory
3682:
3566:
3538:
3371:
2892:
2809:
2488:
2417:
1375:substitution
1373:denotes the
1235:
1099:
1033:
1029:
899:
848:inconsistent
847:
722:
686:
682:
627:
612:
604:
600:
596:
592:
588:
584:
580:
576:
572:
568:
564:
560:
552:
544:
536:
534:
524:
521:
499:
493:
489:completeness
482:
462:
450:proof theory
441:
439:
417:in 1918 and
415:Paul Bernays
404:
394:
387:
379:
344:
253:
182:be a set of
46:
36:
7371:Metatheorem
7329:of geometry
7314:Consistency
7140:Type theory
7088:undecidable
7020:Truth value
6907:equivalence
6586:non-logical
6199:Enumeration
6189:Isomorphism
6136:cardinality
6120:Von Neumann
6085:Ultrafilter
6050:Uncountable
5984:equivalence
5901:Quantifiers
5891:Fixed-point
5860:First-order
5740:Consistency
5725:Proposition
5702:Traditional
5673:Lindström's
5663:Compactness
5605:Type theory
5550:Cardinality
5406:truth value
5399:Functional:
5026:Tarski 1946
5005:consistency
4418:consequence
4209:Kleene 1952
4192:Kleene 1952
4188:Tarski 1946
4120:Tarski 1946
2810:Define the
2670:denote the
1978:. For all
1437:; see also
1234:is said to
609:independent
389:satisfiable
56:. A theory
7406:Categories
6951:elementary
6644:arithmetic
6512:Quantifier
6490:functional
6362:Expression
6080:Transitive
6024:identities
6009:complement
5942:hereditary
5925:Set theory
5188:References
4976:consistent
4912:consistent
4823:, or is a
4440:, or that
4257:1355.03001
4176:Gödel 1931
4148:Gödel 1931
4124:consistent
3939:consistent
3872:such that
2694:; and let
2532:-terms by
2489:Define an
2371:such that
1565:such that
1288:such that
771:such that
723:consistent
693:Definition
427:Kurt Gödel
380:consistent
276:such that
254:consistent
47:consistent
7422:Metalogic
7346:Soundness
7282:Metalogic
7222:Supertask
7125:Recursion
7083:decidable
6917:saturated
6895:of models
6818:deductive
6813:axiomatic
6733:Hilbert's
6720:Euclidean
6701:canonical
6624:axiomatic
6556:Signature
6485:Predicate
6374:Extension
6296:Ackermann
6221:Operation
6100:Universal
6090:Recursive
6065:Singleton
6060:Inhabited
6045:Countable
6035:Types of
6019:power set
5989:partition
5906:Predicate
5852:Predicate
5767:Syllogism
5757:Soundness
5730:Inference
5720:Tautology
5622:paradoxes
5417:tautology
5143:Φ
5123:≡
4922:φ
4898:φ
4858:φ
4838:φ
4835:⊢
4807:φ
4785:φ
4782:⊨
4759:φ
4756:⊨
4713:φ
4693:φ
4690:⊢
4647:φ
4627:φ
4624:⊢
4596:φ
4536:φ
4496:φ
4493:⊢
4470:φ
4404:φ
4382:ω
4379:∞
4354:φ
4332:ω
4329:∞
4017:φ
4014:∨
4001:φ
3966:φ
3959:φ
3921:φ
3900:φ
3860:φ
3800:φ
3797:⊨
3792:Φ
3758:−
3744:…
3711:∼
3691:∼
3661:Φ
3658:∈
3655:φ
3634:φ
3631:⊨
3626:Φ
3595:φ
3587:-formula
3552:Φ
3520:Φ
3516:β
3507:Φ
3487:Φ
3435:¯
3412:Φ
3408:β
3385:Φ
3381:β
3352:¯
3337:Φ
3315:, define
3300:∈
3270:¯
3261:−
3250:…
3223:¯
3215:−
3202:…
3197:¯
3175:Φ
3153:, define
3138:∈
3107:for each
3090:Φ
3087:∈
3079:−
3068:…
3032:¯
3024:−
3011:…
3006:¯
2987:Φ
2965:, define
2950:∈
2919:for each
2903:Φ
2877:Φ
2850:Φ
2832:structure
2735:∈
2729:∣
2724:¯
2707:Φ
2656:¯
2627:≡
2607:Φ
2604:∈
2591:≡
2550:∼
2500:∼
2484:witnesses
2450:Φ
2395:Φ
2392:∈
2379:φ
2339:Φ
2336:∈
2333:φ
2326:∃
2304:Φ
2301:∈
2298:ψ
2278:Φ
2275:∈
2272:φ
2252:Φ
2249:∈
2243:ψ
2240:→
2237:φ
2211:Φ
2208:∈
2205:ψ
2185:Φ
2182:∈
2179:φ
2159:Φ
2156:∈
2150:ψ
2147:∨
2144:φ
2119:Φ
2116:∈
2113:φ
2110:¬
2090:Φ
2087:∈
2084:φ
2061:Φ
2058:∈
2055:φ
2035:φ
2032:⊢
2029:Φ
2006:ψ
1986:φ
1976:witnesses
1962:Φ
1929:φ
1926:¬
1920:∪
1917:Φ
1909:
1878:φ
1872:∪
1869:Φ
1861:
1838:Φ
1835:
1801:φ
1795:∪
1792:Φ
1784:
1761:φ
1758:⊢
1755:Φ
1735:Φ
1732:
1698:φ
1695:¬
1689:∪
1686:Φ
1678:
1655:φ
1652:⊢
1649:Φ
1626:φ
1606:Φ
1583:Φ
1580:⊨
1529:Φ
1502:φ
1499:⊢
1496:Φ
1489:φ
1467:Φ
1464:
1405:φ
1351:φ
1331:Φ
1328:∈
1312:φ
1309:→
1306:φ
1299:∃
1253:φ
1246:∃
1222:Φ
1200:Φ
1197:∈
1194:φ
1168:φ
1162:∪
1159:Φ
1153:
1130:φ
1110:Φ
1086:Φ
1064:Φ
1044:Φ
1016:Φ
994:Φ
974:φ
950:φ
930:Φ
910:φ
886:Φ
864:Φ
861:
850:(written
834:Φ
814:φ
811:¬
808:⊢
805:Φ
785:φ
782:⊢
779:Φ
759:φ
739:Φ
736:
725:(written
709:Φ
698:A set of
666:⊢
643:⊢
511:(PA) and
419:Emil Post
356:syntactic
349:explosive
331:⟩
325:⟨
322:∈
319:φ
316:¬
296:⟩
290:⟨
287:∈
284:φ
264:φ
200:⟩
194:⟨
130:φ
127:¬
107:φ
87:φ
39:classical
7207:Logicism
7200:timeline
7176:Concrete
7035:Validity
7005:T-schema
6998:Kripke's
6993:Tarski's
6988:semantic
6978:Strength
6927:submodel
6922:spectrum
6890:function
6738:Tarski's
6727:Elements
6714:geometry
6670:Robinson
6591:variable
6576:function
6549:spectrum
6539:Sentence
6495:variable
6438:Language
6391:Relation
6352:Automata
6342:Alphabet
6326:language
6180:-jection
6158:codomain
6144:Function
6105:Universe
6075:Infinite
5979:Relation
5762:Validity
5752:Argument
5650:theorem,
5285:(1967).
5263:(1946).
5241:(1947).
5218:(1952).
4588:entails
4205:or be a
4140:negation
4039:See also
4020:∉
4008:∉
4004:′
3976:⊈
3969:′
3924:′
2641:equality
2639:denotes
2619:, where
1598:For all
1481:For all
1377:of each
1343:, where
1186:implies
964:negation
962:and the
700:formulas
624:Notation
406:complete
360:semantic
7149:Related
6946:Diagram
6844: (
6823:Hilbert
6808:Systems
6803:Theorem
6681:of the
6626:systems
6406:Formula
6401:Grammar
6317: (
6261:General
5974:Forcing
5959:Element
5879:Monadic
5654:paradox
5595:Theorem
5531:General
5452:theorem
5435:Formal:
5393: â€
4611:Warning
4461:entails
4249:3822731
3832:theory
3537:be the
2290:, then
2076:either
2047:, then
1850:, then
1773:, then
1667:, then
1641:if not
942:, both
681:reads:
603:, then
471:of the
345:trivial
78:formula
6912:finite
6675:Skolem
6628:
6603:Theory
6571:Symbol
6561:String
6544:atomic
6421:ground
6416:closed
6411:atomic
6367:ground
6330:syntax
6226:binary
6153:domain
6070:Finite
5835:finite
5693:Logics
5652:
5600:Theory
5470:
5442:theory
5301:(pbk.)
5293:
5271:
5249:
5226:
4255:
4247:
4237:
3470:. Let
2915:, by:
2760:where
2643:. Let
2442:. Let
632:, the
549:theory
376:axioms
162:. Let
50:theory
6902:Model
6650:Peano
6507:Proof
6347:Arity
6276:Naive
6163:image
6095:Fuzzy
6055:Empty
6004:union
5949:Class
5590:Model
5580:Lemma
5538:Axiom
5472:false
4850:, if
4821:valid
4659:from
4416:is a
4107:Notes
3828:, an
3539:term
2864:over
2438:be a
1417:by a
595:and ÂŹ
557:axiom
547:is a
444:is a
401:logic
395:In a
368:model
7284:and
7025:Type
6828:list
6632:list
6609:list
6598:Term
6532:rank
6426:open
6320:list
6132:Maps
6037:sets
5896:Free
5866:list
5616:list
5543:list
5291:ISBN
5269:ISBN
5247:ISBN
5224:ISBN
4346:and
4276:Let
4235:ISBN
4162:and
3937:. A
2418:Let
2264:and
1998:and
1954:Let
1747:and
1618:and
1267:term
797:and
551:and
343:. A
308:and
45:, a
6712:of
6694:of
6642:of
6174:Sur
6148:Map
5955:Ur-
5937:Set
5204:doi
5058:in
5041:in
5020:in
4974:is
4914:if
4910:is
4819:is
4420:of
4253:Zbl
4227:doi
4158:of
4136:not
4126:or
3820:In
3399:by
3046:if
2572:if
2226:if
2197:or
2102:or
2021:if
1906:Con
1898:or
1858:Con
1832:Con
1824:if
1781:Con
1729:Con
1721:if
1675:Con
1461:Inc
1397:in
1150:Con
1102:if
1032:or
966:of
922:of
858:Inc
846:is
733:Con
611:of
525:not
252:is
37:In
7408::
7098:NP
6722::
6716::
6646::
6323:),
6178:Bi
6170:In
5345:.
5200:38
5198:.
4608:.)
4251:.
4245:MR
4243:.
4233:.
4199:of
4178:,
3607::
3564:.
3492::=
3426::=
3344::=
3231::=
2807:.
2712::=
2486:.
2018::
1638::
1142:,
876:).
615:.
587:+
563:+
559:,
519:.
440:A
41:,
7274:e
7267:t
7260:v
7178:/
7093:P
6848:)
6634:)
6630:(
6527:â
6522:!
6517:â
6478:=
6473:â
6468:â
6463:â§
6458:âš
6453:ÂŹ
6176:/
6172:/
6146:/
5957:)
5953:(
5840:â
5830:3
5618:)
5516:e
5509:t
5502:v
5468:â„
5389:â
5382:e
5375:t
5368:v
5351:.
5329:.
5320:.
5311:.
5299:.
5277:.
5255:.
5232:.
5210:.
5206::
5155:.
5101:i
5097:t
5007:.
4962:T
4942:L
4878:L
4779:A
4753:T
4733:T
4687:T
4667:T
4621:T
4576:T
4556:T
4516:T
4490:T
4448:T
4428:T
4375:L
4325:L
4304:T
4284:L
4259:.
4229::
4164:b
4160:a
4156:c
4023:T
4011:T
3979:T
3973:}
3962:,
3956:{
3880:T
3840:T
3786:I
3761:1
3755:n
3751:t
3747:,
3741:,
3736:0
3732:t
3664:.
3620:I
3575:S
3525:)
3512:,
3501:T
3495:(
3481:I
3458:x
3432:x
3423:)
3420:x
3417:(
3357:.
3349:c
3331:T
3324:c
3303:S
3297:c
3275:;
3264:1
3258:n
3254:t
3245:0
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240:A
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150:T
64:T
34:.
20:)
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