186:" between 1929 and 1931, and spoke about them to Polish audiences. However, as he emphasized in the paper, the undefinability theorem was the only result he did not obtain earlier. According to the footnote to the undefinability theorem (Twierdzenie I) of the 1933 monograph, the theorem and the sketch of the proof were added to the monograph only after the manuscript had been sent to the printer in 1931. Tarski reports there that, when he presented the content of his monograph to the Warsaw Academy of Science on March 21, 1931, he expressed at this place only some conjectures, based partly on his own investigations and partly on Gödel's short report on the incompleteness theorems "
25:
5292:
178:. Gödel also discovered the undefinability theorem in 1930, while proving his incompleteness theorems published in 1931, and well before the 1933 publication of Tarski's work (Murawski 1998). While Gödel never published anything bearing on his independent discovery of undefinability, he did describe it in a 1931 letter to
170:
are true in the standard model of arithmetic) must have expressive power exceeding that of the object language. The metalanguage includes primitive notions, axioms, and rules absent from the object language, so that there are theorems provable in the metalanguage not provable in the object language.
2779:
1961) is less than evident. Tarski's theorem, on the other hand, is not directly about mathematics but about the inherent limitations of any formal language sufficiently expressive to be of real interest. Such languages are necessarily capable of enough
151:. Moreover, any computable set of numbers can be defined by some arithmetical formula. For example, there are formulas in the language of arithmetic defining the set of codes for arithmetic sentences, and for provable arithmetic sentences.
2732:
The formal machinery of the proof given above is wholly elementary except for the diagonalization which the diagonal lemma requires. The proof of the diagonal lemma is likewise surprisingly simple; for example, it does not invoke
984:
Informally, the theorem says that the concept of truth of first-order arithmetic statements cannot be defined by a formula in first-order arithmetic. This implies a major limitation on the scope of "self-representation". It
2698:
2594:
2289:
955:
2449:
2380:
2190:
1942:
1320:
1172:
1027:
856:
766:
415:
3103:
1833:
1770:
2922:
2722:
2618:
2517:
2473:
2313:
2130:
1645:
1511:
1280:
1216:
979:
623:
379:
335:
284:
2106:
1543:
493:
1971:
1487:
1438:
516:
438:
2896:
The undefinability theorem does not prevent truth in one theory from being defined in a stronger theory. For example, the set of (codes for) formulas of first-order
2867:
1057:
796:
1801:
1738:
1691:
700:
650:
2020:
1621:
1592:
1369:
2233:
1856:
899:
673:
559:
311:
2966:
2946:
2887:
2821:
2755:
2638:
2537:
2493:
2400:
2340:
2210:
2150:
2060:
2040:
1991:
1876:
1711:
1563:
1458:
1409:
1389:
1340:
1256:
1236:
1192:
1132:
1109:
1081:
876:
816:
726:
599:
579:
536:
458:
355:
225:
3068:
1886:
Tarski proved a stronger theorem than the one stated above, using an entirely syntactical method. The resulting theorem applies to any formal language with
5331:
3671:
4346:
2761:, but the specifics of a coding method are not required. Hence Tarski's theorem is much easier to motivate and prove than the more celebrated
2775:. That the latter theorems have much to say about all of mathematics and more controversially, about a range of philosophical issues (e.g.,
4429:
3570:
204:
We will first state a simplified version of Tarski's theorem, then state and prove in the next section the theorem Tarski proved in 1933.
3438:
158:
concepts such as truth. It shows that no sufficiently rich interpreted language can represent its own semantics. A corollary is that any
1898:
holds. First-order arithmetic satisfies these preconditions, but the theorem applies to much more general formal systems, such as
4743:
4901:
3256:
3689:
3355:
2996:
2803:
concepts specific to the language. Hence the required functions include the "semantic valuation function" mapping a formula
2762:
120:
4756:
4079:
3161:
127:. Each expression of the formal language of arithmetic is assigned a distinct number. This procedure is known variously as
2784:
for the diagonal lemma to apply to them. The broader philosophical import of Tarski's theorem is more strikingly evident.
2772:
3287:
2981:
247:
extend over natural numbers, but not over sets or functions of natural numbers. The theory is strong enough to describe
4761:
4751:
4488:
4341:
3200:
2643:
3685:
3350:
2990:
4897:
3395:
3237:
68:
46:
4239:
39:
4994:
4738:
3563:
3083:
2771:(1991, 2001) has argued forcefully that Tarski's undefinability theorem deserves much of the attention garnered by
4299:
3992:
3431:
2969:
2542:
2237:
1662:, proved some years after Tarski (1933). A semantic proof of Tarski's theorem from Post's theorem is obtained by
903:
3733:
3380:
5255:
4957:
4720:
4715:
4540:
3961:
3645:
3507:
287:
5250:
5033:
4950:
4663:
4594:
4471:
3713:
1111:. To define a truth predicate for the metalanguage would require a still higher metametalanguage, and so on.
1091:. However, this formula would only be able to define a truth predicate for formulas in the original language
236:
5326:
5175:
5001:
4687:
4321:
3920:
3512:
2405:
193:
5316:
5053:
5048:
4658:
4397:
4326:
3655:
3556:
3522:
3517:
3502:
3405:
163:
2345:
2155:
1911:
1285:
1137:
992:
821:
731:
384:
337:
consists of the ordinary set of natural numbers and their addition and multiplication. Each sentence in
5336:
4982:
4572:
3966:
3934:
3625:
3532:
3424:
3400:
3338:
93:
5272:
5221:
5118:
4616:
4577:
4054:
3699:
3527:
3497:
3492:
1806:
1743:
1545:
when applied to its own Gödel number, yields a false statement). If we now consider the Gödel number
3728:
2903:
2703:
2599:
2498:
2454:
2294:
2111:
1626:
1492:
1261:
1197:
960:
604:
360:
316:
265:
5113:
5043:
4582:
4434:
4417:
4140:
3620:
3537:
3323:
3157:
2776:
2065:
33:
1516:
466:
4945:
4922:
4883:
4769:
4710:
4356:
4276:
4120:
4064:
3677:
3343:
3280:
2925:
2620:. But the diagonal lemma yields a counterexample to this equivalence, by giving a "liar" formula
1947:
1463:
1414:
1088:
5235:
4962:
4940:
4907:
4800:
4646:
4631:
4604:
4555:
4439:
4374:
4199:
4165:
4160:
4034:
3865:
3842:
3375:
1773:
1659:
498:
228:
124:
50:
423:
162:
capable of expressing the semantics of some object language (e.g. a predicate is definable in
107:, showing that truth in the standard model of the system cannot be defined within the system.
5321:
5165:
5018:
4810:
4528:
4264:
4170:
4029:
4014:
3895:
3870:
2829:
1032:
771:
100:. Informally, the theorem states that "arithmetical truth cannot be defined in arithmetic".
5138:
5100:
4977:
4781:
4621:
4545:
4523:
4351:
4309:
4208:
4175:
4039:
3827:
3738:
3370:
3365:
3317:
2928:. Similarly, the set of true formulas of the standard model of second order arithmetic (or
2319:
1779:
1716:
1669:
1663:
678:
628:
2734:
1996:
1597:
1568:
1345:
8:
5267:
5158:
5143:
5123:
5080:
4967:
4917:
4843:
4788:
4725:
4518:
4513:
4461:
4229:
4218:
3890:
3790:
3718:
3709:
3705:
3640:
3635:
3311:
2792:
189:
Einige metamathematische
Resultate ĂŒber Entscheidungsdefinitheit und Widerspruchsfreiheit
2215:
1838:
881:
655:
541:
293:
123:, which he proved in part by showing how to represent the syntax of formal logic within
5296:
5065:
5028:
5013:
5006:
4989:
4793:
4775:
4641:
4567:
4550:
4503:
4316:
4225:
4059:
4044:
4004:
3956:
3941:
3929:
3885:
3860:
3630:
3579:
3273:
3183:
3031:
2951:
2931:
2872:
2806:
2740:
2623:
2522:
2478:
2385:
2325:
2195:
2135:
2045:
2025:
1976:
1861:
1696:
1548:
1443:
1394:
1374:
1325:
1241:
1221:
1177:
1117:
1094:
1066:
861:
801:
711:
584:
564:
521:
443:
340:
244:
210:
139:
of expressions are coded as sets of numbers. For various syntactic properties (such as
89:
4249:
128:
5291:
5231:
5038:
4848:
4838:
4730:
4611:
4446:
4422:
4203:
4187:
4092:
4069:
3946:
3915:
3880:
3775:
3610:
3476:
3461:
3252:
3233:
1944:
be any interpreted formal language which includes negation and has a Gödel numbering
1655:
240:
3187:
5245:
5240:
5133:
5090:
4912:
4873:
4868:
4853:
4679:
4636:
4533:
4331:
4281:
3855:
3817:
3333:
3212:
3173:
2897:
2768:
256:
179:
167:
5226:
5216:
5170:
5153:
5108:
5070:
4972:
4892:
4699:
4626:
4599:
4587:
4493:
4407:
4381:
4336:
4304:
4105:
3907:
3850:
3800:
3765:
3723:
3300:
3145:
3141:
2796:
1084:
3265:
3191:
2891:
No sufficiently powerful language is strongly-semantically-self-representational
2758:
461:
5211:
5190:
5148:
5128:
5023:
4878:
4476:
4466:
4456:
4451:
4385:
4259:
4135:
4024:
4019:
3997:
3598:
3390:
2781:
1895:
1891:
1648:
232:
148:
3216:
3178:
5310:
5185:
4863:
4370:
4155:
4145:
4115:
4100:
3770:
3137:
3055:
175:
104:
85:
3247:
Smullyan, R. (2001). "Gödel's
Incompleteness Theorems". In Goble, L. (ed.).
3220:
116:
5085:
4932:
4833:
4825:
4705:
4653:
4562:
4498:
4481:
4412:
4271:
4130:
3832:
3615:
3471:
1060:
159:
5195:
5075:
4254:
4244:
4191:
3875:
3795:
3780:
3660:
3605:
3466:
3385:
3328:
3030:
Joel David
Hamkins; Yang, Ruizhi (2013). "Satisfaction is not absolute".
2824:
3201:"Undefinability of truth. The problem of the priority: Tarski vs. Gödel"
2889:
to the object it denotes. Tarski's theorem then generalizes as follows:
4125:
3980:
3951:
3757:
154:
The undefinability theorem shows that this encoding cannot be done for
2318:
The proof of Tarski's undefinability theorem in this form is again by
5277:
5180:
4233:
4150:
4110:
4074:
4010:
3822:
3812:
3785:
3548:
3360:
3296:
1114:
To prove the theorem, we proceed by contradiction and assume that an
252:
248:
97:
5262:
5060:
4508:
4213:
3807:
2800:
1887:
155:
3036:
235:, including their addition and multiplication, axiomatized by the
4858:
3650:
2765:
about the metamathematical properties of first-order arithmetic.
182:. Tarski had obtained almost all results of his 1933 monograph "
135:
and, more generally, as arithmetization. In particular, various
184:
The
Concept of Truth in the Languages of the Deductive Sciences
16:
Theorem that arithmetical truth cannot be defined in arithmetic
103:
The theorem applies more generally to any sufficiently strong
4402:
3748:
3593:
3447:
174:
The undefinability theorem is conventionally attributed to
3062:(in Polish). NakĆadem Towarzystwa Naukowego Warszawskiego.
3016:
Cezary CieĆliĆski, "How Tarski
Defined the Undefinable,"
1899:
417:
is the "interpreted first-order language of arithmetic".
3416:
2986:
Pages displaying short descriptions of redirect targets
1693:
is arithmetically definable, there is a natural number
3029:
2869:
and the "semantic denotation function" mapping a term
3069:"Der Wahrheitsbegriff in den formalisierten Sprachen"
2954:
2934:
2906:
2875:
2832:
2809:
2743:
2706:
2646:
2626:
2602:
2545:
2525:
2501:
2481:
2457:
2408:
2388:
2348:
2328:
2297:
2240:
2218:
2198:
2158:
2138:
2114:
2068:
2048:
2028:
1999:
1979:
1950:
1914:
1864:
1841:
1809:
1782:
1746:
1719:
1699:
1672:
1629:
1600:
1571:
1551:
1519:
1495:
1466:
1446:
1417:
1397:
1377:
1348:
1328:
1288:
1264:
1244:
1224:
1200:
1180:
1140:
1120:
1097:
1069:
1035:
995:
963:
906:
884:
864:
824:
804:
774:
734:
714:
681:
658:
631:
607:
587:
567:
544:
524:
501:
469:
446:
426:
387:
363:
343:
319:
296:
268:
213:
3136:
1858:Thus the arithmetical hierarchy collapses at level
702:be defined by a formula of first-order arithmetic?
2960:
2940:
2916:
2881:
2861:
2815:
2749:
2716:
2692:
2632:
2612:
2588:
2531:
2511:
2487:
2467:
2443:
2394:
2374:
2334:
2307:
2283:
2227:
2204:
2184:
2144:
2124:
2100:
2054:
2034:
2014:
1985:
1965:
1936:
1870:
1850:
1827:
1795:
1764:
1732:
1705:
1685:
1639:
1615:
1586:
1557:
1537:
1505:
1481:
1452:
1432:
1403:
1383:
1363:
1334:
1314:
1274:
1250:
1230:
1210:
1186:
1166:
1126:
1103:
1075:
1051:
1021:
973:
949:
893:
870:
850:
810:
790:
760:
720:
694:
667:
644:
617:
593:
573:
553:
530:
510:
487:
452:
432:
409:
373:
349:
329:
305:
278:
219:
3295:
2993: â Fundamental theorem in mathematical logic
2693:{\displaystyle S\iff \lnot \mathrm {True} (g(S))}
1647:, we obtain a contradiction. (This is known as a
5308:
2999: â Limitative results in mathematical logic
675:The following theorem answers the question: Can
3104:"The Concept of Truth in Formalized Languages"
1973:satisfying the diagonal lemma, i.e. for every
1906:Tarski's undefinability theorem (general form)
88:in 1933, is an important limitative result in
3564:
3432:
3281:
3115:English translation of Tarski's 1936 article.
2968:) can be defined by a formula in first-order
2737:in any way. The proof does assume that every
1087:for first-order arithmetic can be defined in
652:the set of Gödel numbers of the sentences in
3127:
1174:exists which is true for the natural number
381:and then becomes either true or false. Thus
187:
3152:(4th ed.). Cambridge University Press.
3060:PojÄcie Prawdy w JÄzykach Nauk Dedukcyjnych
2789:strongly-semantically-self-representational
2589:{\displaystyle \mathrm {True} (g(A))\iff A}
2284:{\displaystyle \mathrm {True} (g(A))\iff A}
1063:whose expressive power goes beyond that of
950:{\displaystyle \mathrm {True} (g(A))\iff A}
5332:Theorems in the foundations of mathematics
3756:
3571:
3557:
3439:
3425:
3288:
3274:
3249:The Blackwell Guide to Philosophical Logic
2654:
2650:
2582:
2578:
2277:
2273:
2076:
2072:
943:
939:
251:integer functions such as exponentiation,
3177:
3035:
2984: â Measure of algorithmic complexity
69:Learn how and when to remove this message
3246:
3232:. Oxford: Oxford University Press, USA.
3227:
3198:
495:This is a natural number that "encodes"
166:for whether formulae in the language of
32:This article includes a list of general
3113:. Translated by J. H. Woodger. Hackett.
2192:with the following property: for every
5309:
3578:
3101:
3066:
3054:
3552:
3420:
3269:
3156:
2444:{\displaystyle \mathrm {True} (g(A))}
1890:, and with sufficient capability for
1371:which is true for the natural number
1238:is the Gödel number of a sentence in
18:
2791:exactly when the language contains
1740:is definable by a formula at level
13:
3128:Bell, J. L.; Machover, M. (1977).
3121:
3048:
2909:
2709:
2668:
2665:
2662:
2659:
2655:
2605:
2556:
2553:
2550:
2547:
2504:
2460:
2419:
2416:
2413:
2410:
2402:is a sentence of arithmetic, then
2375:{\displaystyle \mathrm {True} (n)}
2359:
2356:
2353:
2350:
2300:
2251:
2248:
2245:
2242:
2185:{\displaystyle \mathrm {True} (n)}
2169:
2166:
2163:
2160:
2117:
1937:{\displaystyle (L,{\mathcal {N}})}
1926:
1811:
1748:
1632:
1498:
1315:{\displaystyle \mathrm {True} (n)}
1299:
1296:
1293:
1290:
1267:
1203:
1167:{\displaystyle \mathrm {True} (n)}
1151:
1148:
1145:
1142:
1022:{\displaystyle \mathrm {True} (n)}
1006:
1003:
1000:
997:
966:
917:
914:
911:
908:
851:{\displaystyle \mathrm {True} (n)}
835:
832:
829:
826:
761:{\displaystyle \mathrm {True} (n)}
745:
742:
739:
736:
610:
410:{\displaystyle (L,{\mathcal {N}})}
399:
366:
322:
271:
38:it lacks sufficient corresponding
14:
5348:
3111:Logic, Semantics, Metamathematics
1411:is the Gödel number of a formula
5290:
2982:Chaitin's incompleteness theorem
2724:. This is a contradiction. QED.
1878:, contradicting Post's theorem.
23:
3356:Gödel's incompleteness theorems
3230:Gödel's Incompleteness Theorems
3205:History and Philosophy of Logic
2997:Gödel's incompleteness theorems
2773:Gödel's incompleteness theorems
1881:
1828:{\displaystyle \Sigma _{k}^{0}}
1765:{\displaystyle \Sigma _{n}^{0}}
1594:, and ask whether the sentence
706:Tarski's undefinability theorem
82:Tarski's undefinability theorem
3130:A Course in Mathematical Logic
3023:
3010:
2917:{\displaystyle {\mathcal {N}}}
2852:
2847:
2839:
2834:
2717:{\displaystyle {\mathcal {N}}}
2687:
2684:
2678:
2672:
2651:
2613:{\displaystyle {\mathcal {N}}}
2579:
2575:
2572:
2566:
2560:
2512:{\displaystyle {\mathcal {N}}}
2468:{\displaystyle {\mathcal {N}}}
2438:
2435:
2429:
2423:
2369:
2363:
2308:{\displaystyle {\mathcal {N}}}
2274:
2270:
2267:
2261:
2255:
2179:
2173:
2125:{\displaystyle {\mathcal {N}}}
2095:
2092:
2086:
2080:
2073:
2009:
2003:
1960:
1954:
1931:
1915:
1654:The theorem is a corollary of
1640:{\displaystyle {\mathcal {N}}}
1610:
1604:
1581:
1575:
1529:
1523:
1506:{\displaystyle {\mathcal {N}}}
1476:
1470:
1427:
1421:
1358:
1352:
1309:
1303:
1275:{\displaystyle {\mathcal {N}}}
1211:{\displaystyle {\mathcal {N}}}
1161:
1155:
1016:
1010:
974:{\displaystyle {\mathcal {N}}}
940:
936:
933:
927:
921:
845:
839:
755:
749:
618:{\displaystyle {\mathcal {N}}}
479:
473:
404:
388:
374:{\displaystyle {\mathcal {N}}}
330:{\displaystyle {\mathcal {N}}}
279:{\displaystyle {\mathcal {N}}}
1:
5251:History of mathematical logic
3251:. Blackwell. pp. 72â89.
3228:Smullyan, Raymond M. (1992).
3003:
2948:-th order arithmetic for any
2924:is definable by a formula in
2727:
2101:{\displaystyle A\iff B(g(A))}
1489:is false when interpreted in
989:possible to define a formula
5176:Primitive recursive function
3351:Gödel's completeness theorem
2991:Gödel's completeness theorem
1538:{\displaystyle \varphi (x),}
561:not just about numbers. Let
488:{\displaystyle g(\varphi ).}
231:. This is the theory of the
199:
194:Austrian Academy of Sciences
7:
3162:"Mind, Machines, and Gödel"
2975:
2787:An interpreted language is
2382:as above existed, i.e., if
1966:{\displaystyle g(\varphi )}
1482:{\displaystyle \varphi (m)}
1433:{\displaystyle \varphi (x)}
538:can talk about formulas in
164:Zermelo-Fraenkel set theory
10:
5353:
4240:SchröderâBernstein theorem
3967:Monadic predicate calculus
3626:Foundations of mathematics
3339:Foundations of mathematics
518:In that way, the language
110:
94:foundations of mathematics
5286:
5273:Philosophy of mathematics
5222:Automated theorem proving
5204:
5099:
4931:
4824:
4676:
4393:
4369:
4347:Von NeumannâBernaysâGödel
4292:
4186:
4090:
3988:
3979:
3906:
3841:
3747:
3669:
3586:
3485:
3454:
3446:
3307:
3217:10.1080/01445349808837306
3179:10.1017/S0031819100057983
3109:. In Corcoran, J. (ed.).
3082:: 261â405. Archived from
1059:but only by drawing on a
511:{\displaystyle \varphi .}
188:
3381:LöwenheimâSkolem theorem
2022:(with one free variable
433:{\displaystyle \varphi }
237:first-order Peano axioms
147:, etc.), these sets are
4923:Self-verifying theories
4744:Tarski's axiomatization
3695:Tarski's undefinability
3690:incompleteness theorems
3406:Useâmention distinction
3150:Computability and Logic
2926:second order arithmetic
1089:second-order arithmetic
121:incompleteness theorems
84:, stated and proved by
53:more precise citations.
5297:Mathematics portal
4908:Proof of impossibility
4556:propositional variable
3866:Propositional calculus
3401:Typeâtoken distinction
2962:
2942:
2918:
2883:
2863:
2862:{\displaystyle ||A||,}
2817:
2751:
2718:
2694:
2634:
2614:
2590:
2533:
2513:
2489:
2469:
2445:
2396:
2376:
2336:
2309:
2285:
2229:
2206:
2186:
2146:
2126:
2102:
2056:
2042:) there is a sentence
2036:
2016:
1987:
1967:
1938:
1872:
1852:
1829:
1797:
1774:arithmetical hierarchy
1766:
1734:
1707:
1687:
1660:arithmetical hierarchy
1641:
1617:
1588:
1559:
1539:
1507:
1483:
1454:
1440:(with a free variable
1434:
1405:
1385:
1365:
1336:
1316:
1276:
1252:
1232:
1212:
1188:
1168:
1128:
1105:
1077:
1053:
1052:{\displaystyle T^{*},}
1023:
975:
951:
895:
872:
852:
812:
792:
791:{\displaystyle T^{*}.}
762:
722:
696:
669:
646:
619:
595:
575:
555:
532:
512:
489:
454:
434:
411:
375:
357:can be interpreted in
351:
331:
307:
280:
229:first-order arithmetic
221:
125:first-order arithmetic
5166:Kolmogorov complexity
5119:Computably enumerable
5019:Model complete theory
4811:Principia Mathematica
3871:Propositional formula
3700:BanachâTarski paradox
3199:Murawski, R. (1998).
3020:23.1 (2015): 139â149.
2963:
2943:
2919:
2884:
2864:
2818:
2752:
2719:
2695:
2635:
2615:
2591:
2534:
2514:
2490:
2470:
2446:
2397:
2377:
2337:
2310:
2286:
2230:
2207:
2187:
2147:
2127:
2103:
2057:
2037:
2017:
1988:
1968:
1939:
1873:
1853:
1830:
1798:
1796:{\displaystyle T^{*}}
1767:
1735:
1733:{\displaystyle T^{*}}
1708:
1688:
1686:{\displaystyle T^{*}}
1666:as follows. Assuming
1642:
1618:
1589:
1560:
1540:
1508:
1484:
1455:
1435:
1406:
1386:
1366:
1337:
1317:
1277:
1253:
1233:
1213:
1189:
1169:
1129:
1106:
1078:
1054:
1024:
976:
952:
896:
873:
853:
813:
798:That is, there is no
793:
763:
723:
697:
695:{\displaystyle T^{*}}
670:
647:
645:{\displaystyle T^{*}}
620:
596:
576:
556:
533:
513:
490:
455:
435:
412:
376:
352:
332:
308:
281:
222:
5114:ChurchâTuring thesis
5101:Computability theory
4310:continuum hypothesis
3828:Square of opposition
3686:Gödel's completeness
3324:ChurchâTuring thesis
3318:Entscheidungsproblem
2952:
2932:
2904:
2873:
2830:
2807:
2741:
2704:
2644:
2624:
2600:
2543:
2523:
2499:
2479:
2455:
2406:
2386:
2346:
2326:
2320:reductio ad absurdum
2295:
2238:
2216:
2196:
2156:
2136:
2112:
2066:
2046:
2026:
2015:{\displaystyle B(x)}
1997:
1977:
1948:
1912:
1862:
1839:
1807:
1780:
1744:
1717:
1697:
1670:
1664:reductio ad absurdum
1627:
1616:{\displaystyle S(g)}
1598:
1587:{\displaystyle S(m)}
1569:
1549:
1517:
1493:
1464:
1444:
1415:
1395:
1375:
1364:{\displaystyle S(m)}
1346:
1326:
1286:
1282:. We could then use
1262:
1242:
1222:
1198:
1178:
1138:
1118:
1095:
1067:
1033:
993:
961:
904:
882:
862:
858:such that for every
822:
802:
772:
732:
712:
679:
656:
629:
605:
585:
565:
542:
522:
499:
467:
444:
424:
385:
361:
341:
317:
294:
266:
211:
5327:Philosophy of logic
5268:Mathematical object
5159:P versus NP problem
5124:Computable function
4918:Reverse mathematics
4844:Logical consequence
4721:primitive recursive
4716:elementary function
4489:Free/bound variable
4342:TarskiâGrothendieck
3861:Logical connectives
3791:Logical equivalence
3641:Logical consequence
3102:Tarski, A. (1983).
3076:Studia Philosophica
3067:Tarski, A. (1936).
2735:recursive functions
2132:. Then there is no
1824:
1761:
1029:whose extension is
601:-sentences true in
581:denote the set of
249:recursively defined
227:be the language of
5317:Mathematical logic
5066:Transfer principle
5029:Semantics of logic
5014:Categorical theory
4990:Non-standard model
4504:Logical connective
3631:Information theory
3580:Mathematical logic
2958:
2938:
2914:
2879:
2859:
2813:
2747:
2714:
2690:
2630:
2610:
2586:
2529:
2509:
2485:
2465:
2441:
2392:
2372:
2332:
2322:. Suppose that an
2305:
2281:
2228:{\displaystyle A,}
2225:
2202:
2182:
2142:
2122:
2098:
2052:
2032:
2012:
1983:
1963:
1934:
1868:
1851:{\displaystyle k.}
1848:
1825:
1810:
1793:
1762:
1747:
1730:
1703:
1683:
1637:
1613:
1584:
1555:
1535:
1513:(i.e. the formula
1503:
1479:
1450:
1430:
1401:
1381:
1361:
1332:
1312:
1272:
1248:
1228:
1208:
1184:
1164:
1124:
1101:
1073:
1049:
1019:
971:
947:
894:{\displaystyle A,}
891:
868:
848:
808:
788:
758:
718:
692:
668:{\displaystyle T.}
665:
642:
615:
591:
571:
554:{\displaystyle L,}
551:
528:
508:
485:
450:
430:
407:
371:
347:
327:
306:{\displaystyle L,}
303:
276:
257:Fibonacci sequence
217:
90:mathematical logic
5337:Theories of truth
5304:
5303:
5236:Abstract category
5039:Theories of truth
4849:Rule of inference
4839:Natural deduction
4820:
4819:
4365:
4364:
4070:Cartesian product
3975:
3974:
3881:Many-valued logic
3856:Boolean functions
3739:Russell's paradox
3714:diagonal argument
3611:First-order logic
3546:
3545:
3414:
3413:
3258:978-0-631-20693-4
3089:on 9 January 2014
2961:{\displaystyle n}
2941:{\displaystyle n}
2900:that are true in
2882:{\displaystyle t}
2816:{\displaystyle A}
2799:defining all the
2763:theorems of Gödel
2750:{\displaystyle L}
2633:{\displaystyle S}
2532:{\displaystyle A}
2488:{\displaystyle A}
2395:{\displaystyle A}
2335:{\displaystyle L}
2205:{\displaystyle L}
2145:{\displaystyle L}
2055:{\displaystyle A}
2035:{\displaystyle x}
1986:{\displaystyle L}
1871:{\displaystyle n}
1706:{\displaystyle n}
1649:diagonal argument
1558:{\displaystyle g}
1453:{\displaystyle x}
1404:{\displaystyle m}
1384:{\displaystyle m}
1335:{\displaystyle L}
1251:{\displaystyle L}
1231:{\displaystyle n}
1187:{\displaystyle n}
1127:{\displaystyle L}
1104:{\displaystyle L}
1083:. For example, a
1076:{\displaystyle L}
871:{\displaystyle L}
811:{\displaystyle L}
721:{\displaystyle L}
594:{\displaystyle L}
574:{\displaystyle T}
531:{\displaystyle L}
453:{\displaystyle L}
350:{\displaystyle L}
220:{\displaystyle L}
79:
78:
71:
5344:
5295:
5294:
5246:History of logic
5241:Category of sets
5134:Decision problem
4913:Ordinal analysis
4854:Sequent calculus
4752:Boolean algebras
4692:
4691:
4666:
4637:logical/constant
4391:
4390:
4377:
4300:ZermeloâFraenkel
4051:Set operations:
3986:
3985:
3923:
3754:
3753:
3734:LöwenheimâSkolem
3621:Formal semantics
3573:
3566:
3559:
3550:
3549:
3441:
3434:
3427:
3418:
3417:
3334:Effective method
3312:Cantor's theorem
3290:
3283:
3276:
3267:
3266:
3262:
3243:
3224:
3219:. Archived from
3195:
3190:. Archived from
3181:
3153:
3133:
3132:. North-Holland.
3114:
3108:
3098:
3096:
3094:
3088:
3073:
3063:
3042:
3041:
3039:
3027:
3021:
3014:
2987:
2967:
2965:
2964:
2959:
2947:
2945:
2944:
2939:
2923:
2921:
2920:
2915:
2913:
2912:
2898:Peano arithmetic
2888:
2886:
2885:
2880:
2868:
2866:
2865:
2860:
2855:
2850:
2842:
2837:
2822:
2820:
2819:
2814:
2797:function symbols
2756:
2754:
2753:
2748:
2723:
2721:
2720:
2715:
2713:
2712:
2699:
2697:
2696:
2691:
2671:
2639:
2637:
2636:
2631:
2619:
2617:
2616:
2611:
2609:
2608:
2595:
2593:
2592:
2587:
2559:
2538:
2536:
2535:
2530:
2519:. Hence for all
2518:
2516:
2515:
2510:
2508:
2507:
2494:
2492:
2491:
2486:
2474:
2472:
2471:
2466:
2464:
2463:
2450:
2448:
2447:
2442:
2422:
2401:
2399:
2398:
2393:
2381:
2379:
2378:
2373:
2362:
2341:
2339:
2338:
2333:
2314:
2312:
2311:
2306:
2304:
2303:
2290:
2288:
2287:
2282:
2254:
2234:
2232:
2231:
2226:
2211:
2209:
2208:
2203:
2191:
2189:
2188:
2183:
2172:
2151:
2149:
2148:
2143:
2131:
2129:
2128:
2123:
2121:
2120:
2107:
2105:
2104:
2099:
2061:
2059:
2058:
2053:
2041:
2039:
2038:
2033:
2021:
2019:
2018:
2013:
1992:
1990:
1989:
1984:
1972:
1970:
1969:
1964:
1943:
1941:
1940:
1935:
1930:
1929:
1877:
1875:
1874:
1869:
1857:
1855:
1854:
1849:
1834:
1832:
1831:
1826:
1823:
1818:
1802:
1800:
1799:
1794:
1792:
1791:
1771:
1769:
1768:
1763:
1760:
1755:
1739:
1737:
1736:
1731:
1729:
1728:
1712:
1710:
1709:
1704:
1692:
1690:
1689:
1684:
1682:
1681:
1646:
1644:
1643:
1638:
1636:
1635:
1622:
1620:
1619:
1614:
1593:
1591:
1590:
1585:
1564:
1562:
1561:
1556:
1544:
1542:
1541:
1536:
1512:
1510:
1509:
1504:
1502:
1501:
1488:
1486:
1485:
1480:
1459:
1457:
1456:
1451:
1439:
1437:
1436:
1431:
1410:
1408:
1407:
1402:
1390:
1388:
1387:
1382:
1370:
1368:
1367:
1362:
1341:
1339:
1338:
1333:
1322:to define a new
1321:
1319:
1318:
1313:
1302:
1281:
1279:
1278:
1273:
1271:
1270:
1258:that is true in
1257:
1255:
1254:
1249:
1237:
1235:
1234:
1229:
1217:
1215:
1214:
1209:
1207:
1206:
1193:
1191:
1190:
1185:
1173:
1171:
1170:
1165:
1154:
1133:
1131:
1130:
1125:
1110:
1108:
1107:
1102:
1082:
1080:
1079:
1074:
1058:
1056:
1055:
1050:
1045:
1044:
1028:
1026:
1025:
1020:
1009:
980:
978:
977:
972:
970:
969:
956:
954:
953:
948:
920:
900:
898:
897:
892:
877:
875:
874:
869:
857:
855:
854:
849:
838:
817:
815:
814:
809:
797:
795:
794:
789:
784:
783:
767:
765:
764:
759:
748:
727:
725:
724:
719:
701:
699:
698:
693:
691:
690:
674:
672:
671:
666:
651:
649:
648:
643:
641:
640:
624:
622:
621:
616:
614:
613:
600:
598:
597:
592:
580:
578:
577:
572:
560:
558:
557:
552:
537:
535:
534:
529:
517:
515:
514:
509:
494:
492:
491:
486:
459:
457:
456:
451:
439:
437:
436:
431:
416:
414:
413:
408:
403:
402:
380:
378:
377:
372:
370:
369:
356:
354:
353:
348:
336:
334:
333:
328:
326:
325:
312:
310:
309:
304:
286:be the standard
285:
283:
282:
277:
275:
274:
226:
224:
223:
218:
196:, Vienna, 1930.
191:
190:
180:John von Neumann
168:Peano arithmetic
145:being a sentence
96:, and in formal
74:
67:
63:
60:
54:
49:this article by
40:inline citations
27:
26:
19:
5352:
5351:
5347:
5346:
5345:
5343:
5342:
5341:
5307:
5306:
5305:
5300:
5289:
5282:
5227:Category theory
5217:Algebraic logic
5200:
5171:Lambda calculus
5109:Church encoding
5095:
5071:Truth predicate
4927:
4893:Complete theory
4816:
4685:
4681:
4677:
4672:
4664:
4384: and
4380:
4375:
4361:
4337:New Foundations
4305:axiom of choice
4288:
4250:Gödel numbering
4190: and
4182:
4086:
3971:
3921:
3902:
3851:Boolean algebra
3837:
3801:Equiconsistency
3766:Classical logic
3743:
3724:Halting problem
3712: and
3688: and
3676: and
3675:
3670:Theorems (
3665:
3582:
3577:
3547:
3542:
3481:
3450:
3445:
3415:
3410:
3303:
3301:metamathematics
3294:
3259:
3240:
3172:(137): 112â27.
3124:
3122:Further reading
3106:
3092:
3090:
3086:
3071:
3051:
3049:Primary sources
3046:
3045:
3028:
3024:
3018:European Review
3015:
3011:
3006:
2985:
2978:
2953:
2950:
2949:
2933:
2930:
2929:
2908:
2907:
2905:
2902:
2901:
2874:
2871:
2870:
2851:
2846:
2838:
2833:
2831:
2828:
2827:
2808:
2805:
2804:
2757:-formula has a
2742:
2739:
2738:
2730:
2708:
2707:
2705:
2702:
2701:
2658:
2645:
2642:
2641:
2625:
2622:
2621:
2604:
2603:
2601:
2598:
2597:
2546:
2544:
2541:
2540:
2524:
2521:
2520:
2503:
2502:
2500:
2497:
2496:
2480:
2477:
2476:
2475:if and only if
2459:
2458:
2456:
2453:
2452:
2409:
2407:
2404:
2403:
2387:
2384:
2383:
2349:
2347:
2344:
2343:
2327:
2324:
2323:
2299:
2298:
2296:
2293:
2292:
2241:
2239:
2236:
2235:
2217:
2214:
2213:
2197:
2194:
2193:
2159:
2157:
2154:
2153:
2137:
2134:
2133:
2116:
2115:
2113:
2110:
2109:
2067:
2064:
2063:
2047:
2044:
2043:
2027:
2024:
2023:
1998:
1995:
1994:
1978:
1975:
1974:
1949:
1946:
1945:
1925:
1924:
1913:
1910:
1909:
1884:
1863:
1860:
1859:
1840:
1837:
1836:
1819:
1814:
1808:
1805:
1804:
1787:
1783:
1781:
1778:
1777:
1756:
1751:
1745:
1742:
1741:
1724:
1720:
1718:
1715:
1714:
1698:
1695:
1694:
1677:
1673:
1671:
1668:
1667:
1631:
1630:
1628:
1625:
1624:
1599:
1596:
1595:
1570:
1567:
1566:
1565:of the formula
1550:
1547:
1546:
1518:
1515:
1514:
1497:
1496:
1494:
1491:
1490:
1465:
1462:
1461:
1445:
1442:
1441:
1416:
1413:
1412:
1396:
1393:
1392:
1391:if and only if
1376:
1373:
1372:
1347:
1344:
1343:
1327:
1324:
1323:
1289:
1287:
1284:
1283:
1266:
1265:
1263:
1260:
1259:
1243:
1240:
1239:
1223:
1220:
1219:
1218:if and only if
1202:
1201:
1199:
1196:
1195:
1179:
1176:
1175:
1141:
1139:
1136:
1135:
1119:
1116:
1115:
1096:
1093:
1092:
1085:truth predicate
1068:
1065:
1064:
1040:
1036:
1034:
1031:
1030:
996:
994:
991:
990:
965:
964:
962:
959:
958:
907:
905:
902:
901:
883:
880:
879:
863:
860:
859:
825:
823:
820:
819:
803:
800:
799:
779:
775:
773:
770:
769:
735:
733:
730:
729:
713:
710:
709:
708:: There is no
686:
682:
680:
677:
676:
657:
654:
653:
636:
632:
630:
627:
626:
609:
608:
606:
603:
602:
586:
583:
582:
566:
563:
562:
543:
540:
539:
523:
520:
519:
500:
497:
496:
468:
465:
464:
445:
442:
441:
425:
422:
421:
398:
397:
386:
383:
382:
365:
364:
362:
359:
358:
342:
339:
338:
321:
320:
318:
315:
314:
295:
292:
291:
270:
269:
267:
264:
263:
233:natural numbers
212:
209:
208:
202:
141:being a formula
129:Gödel numbering
113:
75:
64:
58:
55:
45:Please help to
44:
28:
24:
17:
12:
11:
5:
5350:
5340:
5339:
5334:
5329:
5324:
5319:
5302:
5301:
5287:
5284:
5283:
5281:
5280:
5275:
5270:
5265:
5260:
5259:
5258:
5248:
5243:
5238:
5229:
5224:
5219:
5214:
5212:Abstract logic
5208:
5206:
5202:
5201:
5199:
5198:
5193:
5191:Turing machine
5188:
5183:
5178:
5173:
5168:
5163:
5162:
5161:
5156:
5151:
5146:
5141:
5131:
5129:Computable set
5126:
5121:
5116:
5111:
5105:
5103:
5097:
5096:
5094:
5093:
5088:
5083:
5078:
5073:
5068:
5063:
5058:
5057:
5056:
5051:
5046:
5036:
5031:
5026:
5024:Satisfiability
5021:
5016:
5011:
5010:
5009:
4999:
4998:
4997:
4987:
4986:
4985:
4980:
4975:
4970:
4965:
4955:
4954:
4953:
4948:
4941:Interpretation
4937:
4935:
4929:
4928:
4926:
4925:
4920:
4915:
4910:
4905:
4895:
4890:
4889:
4888:
4887:
4886:
4876:
4871:
4861:
4856:
4851:
4846:
4841:
4836:
4830:
4828:
4822:
4821:
4818:
4817:
4815:
4814:
4806:
4805:
4804:
4803:
4798:
4797:
4796:
4791:
4786:
4766:
4765:
4764:
4762:minimal axioms
4759:
4748:
4747:
4746:
4735:
4734:
4733:
4728:
4723:
4718:
4713:
4708:
4695:
4693:
4674:
4673:
4671:
4670:
4669:
4668:
4656:
4651:
4650:
4649:
4644:
4639:
4634:
4624:
4619:
4614:
4609:
4608:
4607:
4602:
4592:
4591:
4590:
4585:
4580:
4575:
4565:
4560:
4559:
4558:
4553:
4548:
4538:
4537:
4536:
4531:
4526:
4521:
4516:
4511:
4501:
4496:
4491:
4486:
4485:
4484:
4479:
4474:
4469:
4459:
4454:
4452:Formation rule
4449:
4444:
4443:
4442:
4437:
4427:
4426:
4425:
4415:
4410:
4405:
4400:
4394:
4388:
4371:Formal systems
4367:
4366:
4363:
4362:
4360:
4359:
4354:
4349:
4344:
4339:
4334:
4329:
4324:
4319:
4314:
4313:
4312:
4307:
4296:
4294:
4290:
4289:
4287:
4286:
4285:
4284:
4274:
4269:
4268:
4267:
4260:Large cardinal
4257:
4252:
4247:
4242:
4237:
4223:
4222:
4221:
4216:
4211:
4196:
4194:
4184:
4183:
4181:
4180:
4179:
4178:
4173:
4168:
4158:
4153:
4148:
4143:
4138:
4133:
4128:
4123:
4118:
4113:
4108:
4103:
4097:
4095:
4088:
4087:
4085:
4084:
4083:
4082:
4077:
4072:
4067:
4062:
4057:
4049:
4048:
4047:
4042:
4032:
4027:
4025:Extensionality
4022:
4020:Ordinal number
4017:
4007:
4002:
4001:
4000:
3989:
3983:
3977:
3976:
3973:
3972:
3970:
3969:
3964:
3959:
3954:
3949:
3944:
3939:
3938:
3937:
3927:
3926:
3925:
3912:
3910:
3904:
3903:
3901:
3900:
3899:
3898:
3893:
3888:
3878:
3873:
3868:
3863:
3858:
3853:
3847:
3845:
3839:
3838:
3836:
3835:
3830:
3825:
3820:
3815:
3810:
3805:
3804:
3803:
3793:
3788:
3783:
3778:
3773:
3768:
3762:
3760:
3751:
3745:
3744:
3742:
3741:
3736:
3731:
3726:
3721:
3716:
3704:Cantor's
3702:
3697:
3692:
3682:
3680:
3667:
3666:
3664:
3663:
3658:
3653:
3648:
3643:
3638:
3633:
3628:
3623:
3618:
3613:
3608:
3603:
3602:
3601:
3590:
3588:
3584:
3583:
3576:
3575:
3568:
3561:
3553:
3544:
3543:
3541:
3540:
3535:
3530:
3525:
3520:
3515:
3510:
3508:Correspondence
3505:
3503:Constructivist
3500:
3495:
3489:
3487:
3483:
3482:
3480:
3479:
3474:
3469:
3464:
3458:
3456:
3452:
3451:
3444:
3443:
3436:
3429:
3421:
3412:
3411:
3409:
3408:
3403:
3398:
3393:
3391:Satisfiability
3388:
3383:
3378:
3376:Interpretation
3373:
3368:
3363:
3358:
3353:
3348:
3347:
3346:
3336:
3331:
3326:
3321:
3314:
3308:
3305:
3304:
3293:
3292:
3285:
3278:
3270:
3264:
3263:
3257:
3244:
3238:
3225:
3223:on 2011-06-08.
3211:(3): 153â160.
3196:
3194:on 2007-08-19.
3154:
3134:
3123:
3120:
3119:
3118:
3117:
3116:
3064:
3050:
3047:
3044:
3043:
3022:
3008:
3007:
3005:
3002:
3001:
3000:
2994:
2988:
2977:
2974:
2957:
2937:
2911:
2878:
2858:
2854:
2849:
2845:
2841:
2836:
2812:
2782:self-reference
2746:
2729:
2726:
2711:
2689:
2686:
2683:
2680:
2677:
2674:
2670:
2667:
2664:
2661:
2657:
2653:
2649:
2629:
2607:
2585:
2581:
2577:
2574:
2571:
2568:
2565:
2562:
2558:
2555:
2552:
2549:
2539:, the formula
2528:
2506:
2484:
2462:
2440:
2437:
2434:
2431:
2428:
2425:
2421:
2418:
2415:
2412:
2391:
2371:
2368:
2365:
2361:
2358:
2355:
2352:
2331:
2302:
2280:
2276:
2272:
2269:
2266:
2263:
2260:
2257:
2253:
2250:
2247:
2244:
2224:
2221:
2201:
2181:
2178:
2175:
2171:
2168:
2165:
2162:
2141:
2119:
2097:
2094:
2091:
2088:
2085:
2082:
2079:
2075:
2071:
2051:
2031:
2011:
2008:
2005:
2002:
1982:
1962:
1959:
1956:
1953:
1933:
1928:
1923:
1920:
1917:
1896:diagonal lemma
1892:self-reference
1883:
1880:
1867:
1847:
1844:
1835:-hard for all
1822:
1817:
1813:
1790:
1786:
1759:
1754:
1750:
1727:
1723:
1702:
1680:
1676:
1656:Post's theorem
1634:
1612:
1609:
1606:
1603:
1583:
1580:
1577:
1574:
1554:
1534:
1531:
1528:
1525:
1522:
1500:
1478:
1475:
1472:
1469:
1449:
1429:
1426:
1423:
1420:
1400:
1380:
1360:
1357:
1354:
1351:
1331:
1311:
1308:
1305:
1301:
1298:
1295:
1292:
1269:
1247:
1227:
1205:
1183:
1163:
1160:
1157:
1153:
1150:
1147:
1144:
1123:
1100:
1072:
1048:
1043:
1039:
1018:
1015:
1012:
1008:
1005:
1002:
999:
968:
946:
942:
938:
935:
932:
929:
926:
923:
919:
916:
913:
910:
890:
887:
867:
847:
844:
841:
837:
834:
831:
828:
807:
787:
782:
778:
757:
754:
751:
747:
744:
741:
738:
717:
689:
685:
664:
661:
639:
635:
612:
590:
570:
550:
547:
527:
507:
504:
484:
481:
478:
475:
472:
449:
429:
406:
401:
396:
393:
390:
368:
346:
324:
302:
299:
273:
243:" theory: the
216:
201:
198:
119:published the
112:
109:
77:
76:
31:
29:
22:
15:
9:
6:
4:
3:
2:
5349:
5338:
5335:
5333:
5330:
5328:
5325:
5323:
5320:
5318:
5315:
5314:
5312:
5299:
5298:
5293:
5285:
5279:
5276:
5274:
5271:
5269:
5266:
5264:
5261:
5257:
5254:
5253:
5252:
5249:
5247:
5244:
5242:
5239:
5237:
5233:
5230:
5228:
5225:
5223:
5220:
5218:
5215:
5213:
5210:
5209:
5207:
5203:
5197:
5194:
5192:
5189:
5187:
5186:Recursive set
5184:
5182:
5179:
5177:
5174:
5172:
5169:
5167:
5164:
5160:
5157:
5155:
5152:
5150:
5147:
5145:
5142:
5140:
5137:
5136:
5135:
5132:
5130:
5127:
5125:
5122:
5120:
5117:
5115:
5112:
5110:
5107:
5106:
5104:
5102:
5098:
5092:
5089:
5087:
5084:
5082:
5079:
5077:
5074:
5072:
5069:
5067:
5064:
5062:
5059:
5055:
5052:
5050:
5047:
5045:
5042:
5041:
5040:
5037:
5035:
5032:
5030:
5027:
5025:
5022:
5020:
5017:
5015:
5012:
5008:
5005:
5004:
5003:
5000:
4996:
4995:of arithmetic
4993:
4992:
4991:
4988:
4984:
4981:
4979:
4976:
4974:
4971:
4969:
4966:
4964:
4961:
4960:
4959:
4956:
4952:
4949:
4947:
4944:
4943:
4942:
4939:
4938:
4936:
4934:
4930:
4924:
4921:
4919:
4916:
4914:
4911:
4909:
4906:
4903:
4902:from ZFC
4899:
4896:
4894:
4891:
4885:
4882:
4881:
4880:
4877:
4875:
4872:
4870:
4867:
4866:
4865:
4862:
4860:
4857:
4855:
4852:
4850:
4847:
4845:
4842:
4840:
4837:
4835:
4832:
4831:
4829:
4827:
4823:
4813:
4812:
4808:
4807:
4802:
4801:non-Euclidean
4799:
4795:
4792:
4790:
4787:
4785:
4784:
4780:
4779:
4777:
4774:
4773:
4771:
4767:
4763:
4760:
4758:
4755:
4754:
4753:
4749:
4745:
4742:
4741:
4740:
4736:
4732:
4729:
4727:
4724:
4722:
4719:
4717:
4714:
4712:
4709:
4707:
4704:
4703:
4701:
4697:
4696:
4694:
4689:
4683:
4678:Example
4675:
4667:
4662:
4661:
4660:
4657:
4655:
4652:
4648:
4645:
4643:
4640:
4638:
4635:
4633:
4630:
4629:
4628:
4625:
4623:
4620:
4618:
4615:
4613:
4610:
4606:
4603:
4601:
4598:
4597:
4596:
4593:
4589:
4586:
4584:
4581:
4579:
4576:
4574:
4571:
4570:
4569:
4566:
4564:
4561:
4557:
4554:
4552:
4549:
4547:
4544:
4543:
4542:
4539:
4535:
4532:
4530:
4527:
4525:
4522:
4520:
4517:
4515:
4512:
4510:
4507:
4506:
4505:
4502:
4500:
4497:
4495:
4492:
4490:
4487:
4483:
4480:
4478:
4475:
4473:
4470:
4468:
4465:
4464:
4463:
4460:
4458:
4455:
4453:
4450:
4448:
4445:
4441:
4438:
4436:
4435:by definition
4433:
4432:
4431:
4428:
4424:
4421:
4420:
4419:
4416:
4414:
4411:
4409:
4406:
4404:
4401:
4399:
4396:
4395:
4392:
4389:
4387:
4383:
4378:
4372:
4368:
4358:
4355:
4353:
4350:
4348:
4345:
4343:
4340:
4338:
4335:
4333:
4330:
4328:
4325:
4323:
4322:KripkeâPlatek
4320:
4318:
4315:
4311:
4308:
4306:
4303:
4302:
4301:
4298:
4297:
4295:
4291:
4283:
4280:
4279:
4278:
4275:
4273:
4270:
4266:
4263:
4262:
4261:
4258:
4256:
4253:
4251:
4248:
4246:
4243:
4241:
4238:
4235:
4231:
4227:
4224:
4220:
4217:
4215:
4212:
4210:
4207:
4206:
4205:
4201:
4198:
4197:
4195:
4193:
4189:
4185:
4177:
4174:
4172:
4169:
4167:
4166:constructible
4164:
4163:
4162:
4159:
4157:
4154:
4152:
4149:
4147:
4144:
4142:
4139:
4137:
4134:
4132:
4129:
4127:
4124:
4122:
4119:
4117:
4114:
4112:
4109:
4107:
4104:
4102:
4099:
4098:
4096:
4094:
4089:
4081:
4078:
4076:
4073:
4071:
4068:
4066:
4063:
4061:
4058:
4056:
4053:
4052:
4050:
4046:
4043:
4041:
4038:
4037:
4036:
4033:
4031:
4028:
4026:
4023:
4021:
4018:
4016:
4012:
4008:
4006:
4003:
3999:
3996:
3995:
3994:
3991:
3990:
3987:
3984:
3982:
3978:
3968:
3965:
3963:
3960:
3958:
3955:
3953:
3950:
3948:
3945:
3943:
3940:
3936:
3933:
3932:
3931:
3928:
3924:
3919:
3918:
3917:
3914:
3913:
3911:
3909:
3905:
3897:
3894:
3892:
3889:
3887:
3884:
3883:
3882:
3879:
3877:
3874:
3872:
3869:
3867:
3864:
3862:
3859:
3857:
3854:
3852:
3849:
3848:
3846:
3844:
3843:Propositional
3840:
3834:
3831:
3829:
3826:
3824:
3821:
3819:
3816:
3814:
3811:
3809:
3806:
3802:
3799:
3798:
3797:
3794:
3792:
3789:
3787:
3784:
3782:
3779:
3777:
3774:
3772:
3771:Logical truth
3769:
3767:
3764:
3763:
3761:
3759:
3755:
3752:
3750:
3746:
3740:
3737:
3735:
3732:
3730:
3727:
3725:
3722:
3720:
3717:
3715:
3711:
3707:
3703:
3701:
3698:
3696:
3693:
3691:
3687:
3684:
3683:
3681:
3679:
3673:
3668:
3662:
3659:
3657:
3654:
3652:
3649:
3647:
3644:
3642:
3639:
3637:
3634:
3632:
3629:
3627:
3624:
3622:
3619:
3617:
3614:
3612:
3609:
3607:
3604:
3600:
3597:
3596:
3595:
3592:
3591:
3589:
3585:
3581:
3574:
3569:
3567:
3562:
3560:
3555:
3554:
3551:
3539:
3536:
3534:
3531:
3529:
3526:
3524:
3521:
3519:
3516:
3514:
3511:
3509:
3506:
3504:
3501:
3499:
3496:
3494:
3491:
3490:
3488:
3484:
3478:
3475:
3473:
3470:
3468:
3465:
3463:
3460:
3459:
3457:
3453:
3449:
3442:
3437:
3435:
3430:
3428:
3423:
3422:
3419:
3407:
3404:
3402:
3399:
3397:
3394:
3392:
3389:
3387:
3384:
3382:
3379:
3377:
3374:
3372:
3369:
3367:
3364:
3362:
3359:
3357:
3354:
3352:
3349:
3345:
3342:
3341:
3340:
3337:
3335:
3332:
3330:
3327:
3325:
3322:
3320:
3319:
3315:
3313:
3310:
3309:
3306:
3302:
3298:
3291:
3286:
3284:
3279:
3277:
3272:
3271:
3268:
3260:
3254:
3250:
3245:
3241:
3239:0-19-504672-2
3235:
3231:
3226:
3222:
3218:
3214:
3210:
3206:
3202:
3197:
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3175:
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3167:
3163:
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3078:(in German).
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2329:
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2258:
2222:
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2139:
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2077:
2069:
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2000:
1980:
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1907:
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780:
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768:that defines
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687:
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662:
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637:
633:
588:
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548:
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420:Each formula
418:
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260:
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250:
246:
242:
239:. This is a "
238:
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197:
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185:
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176:Alfred Tarski
172:
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118:
108:
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105:formal system
101:
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87:
86:Alfred Tarski
83:
73:
70:
62:
52:
48:
42:
41:
35:
30:
21:
20:
5322:Metatheorems
5288:
5086:Ultraproduct
4933:Model theory
4898:Independence
4834:Formal proof
4826:Proof theory
4809:
4782:
4739:real numbers
4711:second-order
4622:Substitution
4499:Metalanguage
4440:conservative
4413:Axiom schema
4357:Constructive
4327:MorseâKelley
4293:Set theories
4272:Aleph number
4265:inaccessible
4171:Grothendieck
4055:intersection
3942:Higher-order
3930:Second-order
3876:Truth tables
3833:Venn diagram
3694:
3616:Formal proof
3513:Deflationary
3472:Truth-bearer
3467:Propositions
3396:Independence
3371:Decidability
3366:Completeness
3316:
3248:
3229:
3221:the original
3208:
3204:
3192:the original
3169:
3165:
3158:Lucas, J. R.
3149:
3129:
3110:
3091:. Retrieved
3084:the original
3079:
3075:
3059:
3025:
3017:
3012:
2895:
2890:
2788:
2786:
2767:
2759:Gödel number
2731:
2317:
1905:
1904:
1885:
1882:General form
1653:
1460:) such that
1113:
1061:metalanguage
986:
983:
705:
704:
462:Gödel number
419:
261:
206:
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183:
173:
160:metalanguage
153:
144:
140:
136:
132:
114:
102:
81:
80:
65:
56:
37:
5196:Type theory
5144:undecidable
5076:Truth value
4963:equivalence
4642:non-logical
4255:Enumeration
4245:Isomorphism
4192:cardinality
4176:Von Neumann
4141:Ultrafilter
4106:Uncountable
4040:equivalence
3957:Quantifiers
3947:Fixed-point
3916:First-order
3796:Consistency
3781:Proposition
3758:Traditional
3729:Lindström's
3719:Compactness
3661:Type theory
3606:Cardinality
3477:Truth-maker
3386:Metatheorem
3344:of geometry
3329:Consistency
3146:Jeffrey, R.
3142:Burgess, J.
2825:truth value
2291:is true in
1776:. However,
1623:is true in
245:quantifiers
241:first-order
59:August 2023
51:introducing
5311:Categories
5007:elementary
4700:arithmetic
4568:Quantifier
4546:functional
4418:Expression
4136:Transitive
4080:identities
4065:complement
3998:hereditary
3981:Set theory
3533:Redundancy
3166:Philosophy
3138:Boolos, G.
3056:Tarski, A.
3004:References
2793:predicates
2728:Discussion
2640:such that
2212:-sentence
2062:such that
1713:such that
1658:about the
878:-sentence
253:factorials
149:computable
117:Kurt Gödel
34:references
5278:Supertask
5181:Recursion
5139:decidable
4973:saturated
4951:of models
4874:deductive
4869:axiomatic
4789:Hilbert's
4776:Euclidean
4757:canonical
4680:axiomatic
4612:Signature
4541:Predicate
4430:Extension
4352:Ackermann
4277:Operation
4156:Universal
4146:Recursive
4121:Singleton
4116:Inhabited
4101:Countable
4091:Types of
4075:power set
4045:partition
3962:Predicate
3908:Predicate
3823:Syllogism
3813:Soundness
3786:Inference
3776:Tautology
3678:paradoxes
3528:Pragmatic
3523:Pluralist
3518:Epistemic
3498:Consensus
3493:Coherence
3462:Statement
3361:Soundness
3297:Metalogic
3037:1312.0670
2700:holds in
2656:¬
2652:⟺
2596:holds in
2580:⟺
2495:holds in
2451:holds in
2342:-formula
2275:⟺
2152:-formula
2108:holds in
2074:⟺
1993:-formula
1958:φ
1894:that the
1812:Σ
1789:∗
1749:Σ
1726:∗
1679:∗
1521:φ
1468:φ
1419:φ
1342:-formula
1134:-formula
1042:∗
957:holds in
941:⟺
818:-formula
781:∗
728:-formula
688:∗
638:∗
503:φ
477:φ
428:φ
288:structure
200:Statement
115:In 1931,
98:semantics
5263:Logicism
5256:timeline
5232:Concrete
5091:Validity
5061:T-schema
5054:Kripke's
5049:Tarski's
5044:semantic
5034:Strength
4983:submodel
4978:spectrum
4946:function
4794:Tarski's
4783:Elements
4770:geometry
4726:Robinson
4647:variable
4632:function
4605:spectrum
4595:Sentence
4551:variable
4494:Language
4447:Relation
4408:Automata
4398:Alphabet
4382:language
4236:-jection
4214:codomain
4200:Function
4161:Universe
4131:Infinite
4035:Relation
3818:Validity
3808:Argument
3706:theorem,
3538:Semantic
3486:Theories
3188:55408480
3160:(1961).
3148:(2002).
3058:(1933).
2976:See also
2801:semantic
2769:Smullyan
1888:negation
156:semantic
5205:Related
5002:Diagram
4900: (
4879:Hilbert
4864:Systems
4859:Theorem
4737:of the
4682:systems
4462:Formula
4457:Grammar
4373: (
4317:General
4030:Forcing
4015:Element
3935:Monadic
3710:paradox
3651:Theorem
3587:General
3455:General
3093:26 June
2823:to its
1772:of the
255:or the
111:History
47:improve
4968:finite
4731:Skolem
4684:
4659:Theory
4627:Symbol
4617:String
4600:atomic
4477:ground
4472:closed
4467:atomic
4423:ground
4386:syntax
4282:binary
4209:domain
4126:Finite
3891:finite
3749:Logics
3708:
3656:Theory
3255:
3236:
3186:
1908:: Let
625:, and
460:has a
133:coding
92:, the
36:, but
4958:Model
4706:Peano
4563:Proof
4403:Arity
4332:Naive
4219:image
4151:Fuzzy
4111:Empty
4060:union
4005:Class
3646:Model
3636:Lemma
3594:Axiom
3448:Truth
3184:S2CID
3107:(PDF)
3087:(PDF)
3072:(PDF)
3032:arXiv
2777:Lucas
313:i.e.
5081:Type
4884:list
4688:list
4665:list
4654:Term
4588:rank
4482:open
4376:list
4188:Maps
4093:sets
3952:Free
3922:list
3672:list
3599:list
3299:and
3253:ISBN
3234:ISBN
3095:2013
2795:and
290:for
262:Let
207:Let
192:" ,
137:sets
4768:of
4750:of
4698:of
4230:Sur
4204:Map
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3993:Set
3213:doi
3174:doi
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1900:ZFC
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440:in
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