496:. A theory is semidecidable if there is a well-defined method whose result, given an arbitrary formula, arrives as positive, if the formula is in the theory; otherwise, may never arrive at all; otherwise, arrives as negative. A logical system is semidecidable if there is a well-defined method for generating a sequence of theorems such that each theorem will eventually be generated. This is different from decidability because in a semidecidable system there may be no effective procedure for checking that a formula is
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631:
203:. Decidability for a theory concerns whether there is an effective procedure that decides whether the formula is a member of the theory or not, given an arbitrary formula in the signature of the theory. The problem of decidability arises naturally when a theory is defined as the set of logical consequences of a fixed set of axioms.
573:. Indeed, the proof that a logical system or theory is undecidable will use the formal definition of computability to show that an appropriate set is not a decidable set, and then invoke Church's thesis to show that the theory or logical system is not decidable by any effective method (Enderton 2001, pp. 206
540:
is decidable but incomplete, whereas the set of all true first-order statements about nonnegative integers in the language with + and Ă is complete but undecidable. Unfortunately, as a terminological ambiguity, the term "undecidable statement" is sometimes used as a synonym for
430:
in 1953. Remarkably, not only the general theory of groups is undecidable, but also several more specific theories, for example (as established by Mal'cev 1961) the theory of finite groups. Mal'cev also established that the theory of semigroups and the theory of
170:
has no theorems at all.) In such cases, alternative definitions of decidability of a logical system are often used, which ask for an effective method for determining something more general than just validity of formulas; for instance, validity of
503:
Every decidable theory or logical system is semidecidable, but in general the converse is not true; a theory is decidable if and only if both it and its complement are semi-decidable. For example, the set of logical validities
217:
first-order theory is decidable. An extension of a decidable theory may not be decidable. For example, there are undecidable theories in propositional logic, although the set of validities (the smallest theory) is decidable.
162:
with identity are decidable, however. This system is first-order logic restricted to those signatures that have no function symbols and whose relation symbols other than equality never take more than one argument.
82:, that is, it has been proven that no effective method for determining membership (returning a correct answer after finite, though possibly very long, time in all cases) can exist for them.
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78:) in a fixed logical system is decidable if there is an effective method for determining whether arbitrary formulas are included in the theory. Many important problems are
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1908:
210:) inconsistent theory is decidable, as every formula in the signature of the theory will be a logical consequence of, and thus a member of, the theory. Every
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of first-order logic is semi-decidable, but not decidable. In this case, it is because there is no effective method for determining for an arbitrary formula
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is known to be essentially undecidable, and thus every consistent theory that includes or interprets
Robinson arithmetic is also (essentially) undecidable.
147:
that includes equality and at least one other predicate with two or more arguments is not decidable. Logical systems extending first-order logic, such as
125:
A logical system is decidable if there is an effective method for determining whether arbitrary formulas are theorems of the logical system. For example,
607:(with limitations on rules and gamepieces) is decidable. However, there are positions (with finitely many pieces) that are forced wins, but not mate in
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225:. In fact, every consistent extension will be essentially undecidable. The theory of fields is undecidable but not essentially undecidable.
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The set of first-order logical validities in a signature with equality and one unary function, established by
Ehrenfeucht in 1959.
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of first-order axioms is semidecidable. Many of the examples of undecidable first-order theories given above are of this form.
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The first-order theory of the natural numbers with addition, multiplication, and equality, established by Tarski and
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no less than 2, or two unary function symbols, or one function symbol of arity no less than 2, established by
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The first-order theory of the natural numbers in the signature with equality and multiplication, also called
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The first-order theory of the rational numbers with addition, multiplication, and equality, established by
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The set of logical validities in any first-order signature with equality and either: a relation symbol of
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Some team games with imperfect information on a finite board (but with unlimited time) are undecidable.
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The first-order theory of the natural numbers in the signature with equality and addition, also called
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A consistent theory that has the property that every consistent extension is undecidable is said to be
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Mathoverflow.net/Decidability-of-chess-on-an-infinite-board
Decidability-of-chess-on-an-infinite-board
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Some logical systems are not adequately represented by the set of theorems alone. (For example,
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The set of first-order logical validities in the signature with only equality, established by
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establishes the equivalence of semantic and syntactic consequence. In other settings, such as
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is decidable. The same holds for all other finite two-player games with perfect information.
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1075:, Studies in Logic and the Foundations of Mathematics, Amsterdam: North-Holland,
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Set Theory for
Computing. From Decision Procedures to Logic Programming with Sets
963:, Studies in Logic and the Foundations of Mathematics, Amsterdam: North-Holland,
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is not decidable in general; in particular, the set of logical validities in any
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841:. Lecture Notes in Computer Science. Vol. 7318. Springer. pp. 78â88.
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is also essentially undecidable. This is closely related to the concept of a
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There are several basic results about decidability of theories. Every (non-
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Stack
Exchange Computer Science. "Is chess game movement TM decidable?"
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Whether a decision problem has an effective method to derive the answer
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A property of a theory or logical system weaker than decidability is
959:(1982), "Introduction to first-order logic", in Barwise, Jon (ed.),
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of the system, especially in the context of first-order logic where
110:. The logically valid formulas of a system are sometimes called the
888:
821:
1964:
1310:
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916:
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are undecidable. Robinson established in 1949 that the theory of
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Examples of decidable first-order theories include the theory of
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1114:
1071:(1982), "Fundamentals of model theory", in Barwise, Jon (ed.),
1001:, Oxford Logic Guides, vol. 35, Oxford University Press,
833:
Brumleve, Dan; Hamkins, Joel David; Schlicht, Philipp (2012).
456:
The first-order theory with equality and two function symbols
393:
Some undecidable theories include (Monk 1976, p. 279):
910:. Cambridge University Press. pp. 211â241 See p. 239.
904:"10. Undecidable Problems: A Sampler: §14.1 Abstract Games"
256:
Some decidable theories include (Monk 1976, p. 234):
66:
formulas (or theorems) can be effectively determined. A
199:
is a set of formulas, often assumed to be closed under
835:"The Mate-in-n Problem of Infinite Chess Is Decidable"
133:
method can be used to determine whether an arbitrary
520:. Similarly, the set of logical consequences of any
98:, which among other things determines the notion of
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997:Chagrov, Alexander; Zakharyaschev, Michael (1997),
977:Cantone, D.; Omodeo, E. G.; Policriti, A. (2013) ,
690:
445:(and therefore any consistent extension, such as
350:investigated in the 1980s through today.(Cantone
718:Tarski, A.; Mostovski, A.; Robinson, R. (1953),
569:. These are generally considered equivalent per
449:) is essentially undecidable, as established by
312:first-order theory of real-closed ordered fields
589:have been classified as to their decidability:
373:Methods used to establish decidability include
296:in 1940 (found in 1940 but announced in 1949).
2884:
1126:
8:
983:, Monographs in Computer Science, Springer,
62:are decidable if membership in their set of
50:(propositional logic) is decidable, whereas
889:"Lo.logic â Checkmate in $ \omega$ moves?"
746:"The Decision Problem for Standard Classes"
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532:Decidability should not be confused with
468:is interpretable in a consistent theory
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343:, established by SchwabhÀuser in 1959.
274:. The completeness was established by
248:are examples of undecidable theories.
839:Conference on Computability in Europe
320:Tarski's exponential function problem
7:
1048:A mathematical introduction to logic
348:decidable sublanguages of set theory
908:Interpreting Gödel: Critical Essays
336:, established by Szmielew in 1955.
25:
106:, which determines the notion of
46:for deriving the correct answer.
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329:, established by Tarski in 1949.
307:, established by Tarski in 1949.
86:Decidability of a logical system
2959:Gödel's incompleteness theorems
1029:Computability and Unsolvability
1073:Handbook of Mathematical Logic
961:Handbook of Mathematical Logic
906:. In Kennedy, Juliette (ed.).
528:Relationship with completeness
1:
2813:History of mathematical logic
549:Relationship to computability
536:. For example, the theory of
316:established by Tarski in 1949
2954:Gödel's completeness theorem
2738:Primitive recursive function
116:Gödel's completeness theorem
857:10.1007/978-3-642-30870-3_9
538:algebraically closed fields
301:algebraically closed fields
3056:
2942:Foundations of mathematics
1802:SchröderâBernstein theorem
1529:Monadic predicate calculus
1188:Foundations of mathematics
1046:Enderton, Herbert (2001),
811:Undecidable Chess Problem?
650:Existential quantification
522:recursively enumerable set
422:The first-order theory of
339:The first-order theory of
332:The first-order theory of
325:The first-order theory of
299:The first-order theory of
288:The first-order theory of
160:monadic predicate calculus
129:is decidable, because the
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2848:
2835:Philosophy of mathematics
2784:Automated theorem proving
1955:
1909:Von NeumannâBernaysâGödel
1550:
1089:Monk, J. Donald (2012) ,
772:10.1017/S0022481200051513
553:As with the concept of a
389:Some undecidable theories
2984:LöwenheimâSkolem theorem
191:Decidability of a theory
155:, are also undecidable.
3009:Useâmention distinction
2485:Self-verifying theories
2306:Tarski's axiomatization
1257:Tarski's undefinability
1252:incompleteness theorems
744:Gurevich, Yuri (1976).
252:Some decidable theories
223:essentially undecidable
3004:Typeâtoken distinction
2859:Mathematics portal
2470:Proof of impossibility
2118:propositional variable
1428:Propositional calculus
902:Poonen, Bjorn (2014).
375:quantifier elimination
240:, while the theory of
215:recursively enumerable
2728:Kolmogorov complexity
2681:Computably enumerable
2581:Model complete theory
2373:Principia Mathematica
1433:Propositional formula
1262:BanachâTarski paradox
689:Monk, Donald (1976).
543:independent statement
272:Presburger arithmetic
238:Presburger arithmetic
135:propositional formula
2927:ChurchâTuring thesis
2921:Entscheidungsproblem
2676:ChurchâTuring thesis
2663:Computability theory
1872:continuum hypothesis
1390:Square of opposition
1248:Gödel's completeness
721:Undecidable Theories
645:Entscheidungsproblem
566:computable functions
482:computability theory
359:monadic second-order
177:consequence relation
137:is logically valid.
2830:Mathematical object
2721:P versus NP problem
2686:Computable function
2480:Reverse mathematics
2406:Logical consequence
2283:primitive recursive
2278:elementary function
2051:Free/bound variable
1904:TarskiâGrothendieck
1423:Logical connectives
1353:Logical equivalence
1203:Logical consequence
581:In context of games
443:Robinson arithmetic
341:hyperbolic geometry
246:Robinson arithmetic
227:Robinson arithmetic
201:logical consequence
127:propositional logic
96:syntactic component
76:logical consequence
42:if there exists an
2628:Transfer principle
2591:Semantics of logic
2576:Categorical theory
2552:Non-standard model
2066:Logical connective
1193:Information theory
1142:Mathematical logic
1091:Mathematical Logic
693:Mathematical Logic
478:many-one reduction
379:model completeness
327:Euclidean geometry
276:MojĆŒesz Presburger
234:real closed fields
158:The validities of
149:second-order logic
104:semantic component
94:comes with both a
70:(set of sentences
48:Zeroth-order logic
3040:Concepts in logic
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2601:Theories of truth
2411:Rule of inference
2401:Natural deduction
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1632:Cartesian product
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1443:Many-valued logic
1418:Boolean functions
1301:Russell's paradox
1276:diagonal argument
1173:First-order logic
1082:978-0-444-86388-1
1061:978-0-12-238452-3
1008:978-0-19-853779-3
970:978-0-444-86388-1
866:978-3-642-30870-3
637:Philosophy portal
560:effective methods
426:, established by
410:Andrzej Mostowski
292:, established by
283:Skolem arithmetic
262:Leopold Löwenheim
141:First-order logic
16:(Redirected from
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2937:Effective method
2915:Cantor's theorem
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2199:logical/constant
1953:
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1862:ZermeloâFraenkel
1613:Set operations:
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1183:Formal semantics
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488:Semidecidability
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451:Raphael Robinson
447:Peano arithmetic
290:Boolean algebras
187:} of the logic.
108:logical validity
44:effective method
36:decision problem
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18:Semidecidability
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1399:
1363:Equiconsistency
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1232:Theorems (
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383:ĆoĆ-Vaught test
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2618:
2613:
2608:
2598:
2593:
2588:
2586:Satisfiability
2583:
2578:
2573:
2572:
2571:
2561:
2560:
2559:
2549:
2548:
2547:
2542:
2537:
2532:
2527:
2517:
2516:
2515:
2510:
2503:Interpretation
2499:
2497:
2491:
2490:
2488:
2487:
2482:
2477:
2472:
2467:
2457:
2452:
2451:
2450:
2449:
2448:
2438:
2433:
2423:
2418:
2413:
2408:
2403:
2398:
2392:
2390:
2384:
2383:
2380:
2379:
2377:
2376:
2368:
2367:
2366:
2365:
2360:
2359:
2358:
2353:
2348:
2328:
2327:
2326:
2324:minimal axioms
2321:
2310:
2309:
2308:
2297:
2296:
2295:
2290:
2285:
2280:
2275:
2270:
2257:
2255:
2236:
2235:
2233:
2232:
2231:
2230:
2218:
2213:
2212:
2211:
2206:
2201:
2196:
2186:
2181:
2176:
2171:
2170:
2169:
2164:
2154:
2153:
2152:
2147:
2142:
2137:
2127:
2122:
2121:
2120:
2115:
2110:
2100:
2099:
2098:
2093:
2088:
2083:
2078:
2073:
2063:
2058:
2053:
2048:
2047:
2046:
2041:
2036:
2031:
2021:
2016:
2014:Formation rule
2011:
2006:
2005:
2004:
1999:
1989:
1988:
1987:
1977:
1972:
1967:
1962:
1956:
1950:
1933:Formal systems
1929:
1928:
1925:
1924:
1922:
1921:
1916:
1911:
1906:
1901:
1896:
1891:
1886:
1881:
1876:
1875:
1874:
1869:
1858:
1856:
1852:
1851:
1849:
1848:
1847:
1846:
1836:
1831:
1830:
1829:
1822:Large cardinal
1819:
1814:
1809:
1804:
1799:
1785:
1784:
1783:
1778:
1773:
1758:
1756:
1746:
1745:
1743:
1742:
1741:
1740:
1735:
1730:
1720:
1715:
1710:
1705:
1700:
1695:
1690:
1685:
1680:
1675:
1670:
1665:
1659:
1657:
1650:
1649:
1647:
1646:
1645:
1644:
1639:
1634:
1629:
1624:
1619:
1611:
1610:
1609:
1604:
1594:
1589:
1587:Extensionality
1584:
1582:Ordinal number
1579:
1569:
1564:
1563:
1562:
1551:
1545:
1539:
1538:
1535:
1534:
1532:
1531:
1526:
1521:
1516:
1511:
1506:
1501:
1500:
1499:
1489:
1488:
1487:
1474:
1472:
1466:
1465:
1463:
1462:
1461:
1460:
1455:
1450:
1440:
1435:
1430:
1425:
1420:
1415:
1409:
1407:
1401:
1400:
1398:
1397:
1392:
1387:
1382:
1377:
1372:
1367:
1366:
1365:
1355:
1350:
1345:
1340:
1335:
1330:
1324:
1322:
1313:
1307:
1306:
1304:
1303:
1298:
1293:
1288:
1283:
1278:
1266:Cantor's
1264:
1259:
1254:
1244:
1242:
1229:
1228:
1226:
1225:
1220:
1215:
1210:
1205:
1200:
1195:
1190:
1185:
1180:
1175:
1170:
1165:
1164:
1163:
1152:
1150:
1146:
1145:
1140:
1138:
1137:
1130:
1123:
1115:
1109:
1108:
1103:
1086:
1081:
1069:Keisler, H. J.
1065:
1060:
1052:Academic Press
1043:
1038:
1020:
1007:
994:
989:
974:
969:
951:
948:
945:
944:
936:
894:
880:
865:
825:
814:
803:
792:
756:(2): 460â464.
736:
730:
710:
703:
678:
665:
664:
662:
659:
657:
654:
653:
652:
647:
641:
640:
624:
621:
620:
619:
616:
605:infinite chess
597:
582:
579:
550:
547:
529:
526:
489:
486:
458:
457:
454:
440:
420:
417:Julia Robinson
413:
406:
390:
387:
371:
370:
355:
344:
337:
334:Abelian groups
330:
323:
308:
305:characteristic
297:
286:
279:
268:
265:
253:
250:
208:paraconsistent
192:
189:
168:Kleene's logic
92:logical system
87:
84:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3052:
3041:
3038:
3036:
3033:
3031:
3028:
3027:
3025:
3010:
3007:
3005:
3002:
3000:
2997:
2995:
2992:
2990:
2987:
2985:
2982:
2980:
2977:
2975:
2972:
2970:
2967:
2965:
2962:
2960:
2957:
2955:
2952:
2948:
2945:
2944:
2943:
2940:
2938:
2935:
2933:
2930:
2928:
2925:
2923:
2922:
2918:
2916:
2913:
2912:
2909:
2905:
2901:
2894:
2889:
2887:
2882:
2880:
2875:
2874:
2871:
2861:
2860:
2855:
2847:
2841:
2838:
2836:
2833:
2831:
2828:
2826:
2823:
2819:
2816:
2815:
2814:
2811:
2809:
2806:
2804:
2801:
2799:
2795:
2792:
2790:
2787:
2785:
2782:
2780:
2777:
2775:
2772:
2771:
2769:
2765:
2759:
2756:
2754:
2751:
2749:
2748:Recursive set
2746:
2744:
2741:
2739:
2736:
2734:
2731:
2729:
2726:
2722:
2719:
2717:
2714:
2712:
2709:
2707:
2704:
2702:
2699:
2698:
2697:
2694:
2692:
2689:
2687:
2684:
2682:
2679:
2677:
2674:
2672:
2669:
2668:
2666:
2664:
2660:
2654:
2651:
2649:
2646:
2644:
2641:
2639:
2636:
2634:
2631:
2629:
2626:
2624:
2621:
2617:
2614:
2612:
2609:
2607:
2604:
2603:
2602:
2599:
2597:
2594:
2592:
2589:
2587:
2584:
2582:
2579:
2577:
2574:
2570:
2567:
2566:
2565:
2562:
2558:
2557:of arithmetic
2555:
2554:
2553:
2550:
2546:
2543:
2541:
2538:
2536:
2533:
2531:
2528:
2526:
2523:
2522:
2521:
2518:
2514:
2511:
2509:
2506:
2505:
2504:
2501:
2500:
2498:
2496:
2492:
2486:
2483:
2481:
2478:
2476:
2473:
2471:
2468:
2465:
2464:from ZFC
2461:
2458:
2456:
2453:
2447:
2444:
2443:
2442:
2439:
2437:
2434:
2432:
2429:
2428:
2427:
2424:
2422:
2419:
2417:
2414:
2412:
2409:
2407:
2404:
2402:
2399:
2397:
2394:
2393:
2391:
2389:
2385:
2375:
2374:
2370:
2369:
2364:
2363:non-Euclidean
2361:
2357:
2354:
2352:
2349:
2347:
2346:
2342:
2341:
2339:
2336:
2335:
2333:
2329:
2325:
2322:
2320:
2317:
2316:
2315:
2311:
2307:
2304:
2303:
2302:
2298:
2294:
2291:
2289:
2286:
2284:
2281:
2279:
2276:
2274:
2271:
2269:
2266:
2265:
2263:
2259:
2258:
2256:
2251:
2245:
2240:Example
2237:
2229:
2224:
2223:
2222:
2219:
2217:
2214:
2210:
2207:
2205:
2202:
2200:
2197:
2195:
2192:
2191:
2190:
2187:
2185:
2182:
2180:
2177:
2175:
2172:
2168:
2165:
2163:
2160:
2159:
2158:
2155:
2151:
2148:
2146:
2143:
2141:
2138:
2136:
2133:
2132:
2131:
2128:
2126:
2123:
2119:
2116:
2114:
2111:
2109:
2106:
2105:
2104:
2101:
2097:
2094:
2092:
2089:
2087:
2084:
2082:
2079:
2077:
2074:
2072:
2069:
2068:
2067:
2064:
2062:
2059:
2057:
2054:
2052:
2049:
2045:
2042:
2040:
2037:
2035:
2032:
2030:
2027:
2026:
2025:
2022:
2020:
2017:
2015:
2012:
2010:
2007:
2003:
2000:
1998:
1997:by definition
1995:
1994:
1993:
1990:
1986:
1983:
1982:
1981:
1978:
1976:
1973:
1971:
1968:
1966:
1963:
1961:
1958:
1957:
1954:
1951:
1949:
1945:
1940:
1934:
1930:
1920:
1917:
1915:
1912:
1910:
1907:
1905:
1902:
1900:
1897:
1895:
1892:
1890:
1887:
1885:
1884:KripkeâPlatek
1882:
1880:
1877:
1873:
1870:
1868:
1865:
1864:
1863:
1860:
1859:
1857:
1853:
1845:
1842:
1841:
1840:
1837:
1835:
1832:
1828:
1825:
1824:
1823:
1820:
1818:
1815:
1813:
1810:
1808:
1805:
1803:
1800:
1797:
1793:
1789:
1786:
1782:
1779:
1777:
1774:
1772:
1769:
1768:
1767:
1763:
1760:
1759:
1757:
1755:
1751:
1747:
1739:
1736:
1734:
1731:
1729:
1728:constructible
1726:
1725:
1724:
1721:
1719:
1716:
1714:
1711:
1709:
1706:
1704:
1701:
1699:
1696:
1694:
1691:
1689:
1686:
1684:
1681:
1679:
1676:
1674:
1671:
1669:
1666:
1664:
1661:
1660:
1658:
1656:
1651:
1643:
1640:
1638:
1635:
1633:
1630:
1628:
1625:
1623:
1620:
1618:
1615:
1614:
1612:
1608:
1605:
1603:
1600:
1599:
1598:
1595:
1593:
1590:
1588:
1585:
1583:
1580:
1578:
1574:
1570:
1568:
1565:
1561:
1558:
1557:
1556:
1553:
1552:
1549:
1546:
1544:
1540:
1530:
1527:
1525:
1522:
1520:
1517:
1515:
1512:
1510:
1507:
1505:
1502:
1498:
1495:
1494:
1493:
1490:
1486:
1481:
1480:
1479:
1476:
1475:
1473:
1471:
1467:
1459:
1456:
1454:
1451:
1449:
1446:
1445:
1444:
1441:
1439:
1436:
1434:
1431:
1429:
1426:
1424:
1421:
1419:
1416:
1414:
1411:
1410:
1408:
1406:
1405:Propositional
1402:
1396:
1393:
1391:
1388:
1386:
1383:
1381:
1378:
1376:
1373:
1371:
1368:
1364:
1361:
1360:
1359:
1356:
1354:
1351:
1349:
1346:
1344:
1341:
1339:
1336:
1334:
1333:Logical truth
1331:
1329:
1326:
1325:
1323:
1321:
1317:
1314:
1312:
1308:
1302:
1299:
1297:
1294:
1292:
1289:
1287:
1284:
1282:
1279:
1277:
1273:
1269:
1265:
1263:
1260:
1258:
1255:
1253:
1249:
1246:
1245:
1243:
1241:
1235:
1230:
1224:
1221:
1219:
1216:
1214:
1211:
1209:
1206:
1204:
1201:
1199:
1196:
1194:
1191:
1189:
1186:
1184:
1181:
1179:
1176:
1174:
1171:
1169:
1166:
1162:
1159:
1158:
1157:
1154:
1153:
1151:
1147:
1143:
1136:
1131:
1129:
1124:
1122:
1117:
1116:
1113:
1106:
1104:9781468494525
1100:
1096:
1092:
1087:
1084:
1078:
1074:
1070:
1066:
1063:
1057:
1053:
1049:
1044:
1041:
1039:9780486151069
1035:
1031:
1030:
1025:
1024:Davis, Martin
1021:
1018:
1014:
1010:
1004:
1000:
995:
992:
990:9781475734522
986:
982:
981:
975:
972:
966:
962:
958:
954:
953:
949:
939:
937:9781107002661
933:
928:
923:
918:
913:
909:
905:
898:
895:
890:
884:
881:
876:
872:
868:
862:
858:
854:
849:
844:
840:
836:
829:
826:
823:
818:
815:
812:
807:
804:
801:
796:
793:
781:
777:
773:
769:
764:
759:
755:
751:
747:
740:
737:
733:
731:9780444533784
727:
723:
722:
714:
711:
706:
704:9780387901701
700:
695:
694:
685:
683:
679:
675:
670:
667:
660:
655:
651:
648:
646:
643:
642:
638:
627:
622:
617:
614:
610:
606:
602:
598:
595:
592:
591:
590:
588:
580:
578:
576:
572:
568:
567:
562:
561:
556:
555:decidable set
548:
546:
544:
539:
535:
527:
525:
523:
519:
515:
511:
507:
501:
499:
495:
487:
485:
483:
479:
475:
471:
467:
463:
455:
452:
448:
444:
441:
438:
434:
429:
428:Alfred Tarski
425:
421:
418:
414:
411:
407:
404:
400:
396:
395:
394:
388:
386:
384:
380:
376:
368:
364:
360:
356:
353:
349:
345:
342:
338:
335:
331:
328:
324:
321:
317:
313:
309:
306:
302:
298:
295:
294:Alfred Tarski
291:
287:
284:
280:
277:
273:
269:
266:
263:
259:
258:
257:
251:
249:
247:
243:
239:
235:
230:
228:
224:
219:
216:
213:
209:
204:
202:
198:
190:
188:
186:
182:
178:
174:
169:
164:
161:
156:
154:
150:
146:
142:
138:
136:
132:
128:
123:
121:
117:
113:
109:
105:
101:
97:
93:
85:
83:
81:
77:
73:
69:
65:
61:
57:
53:
49:
45:
41:
37:
33:
19:
3030:Proof theory
2999:Independence
2974:Decidability
2973:
2969:Completeness
2919:
2850:
2700:
2648:Ultraproduct
2495:Model theory
2460:Independence
2396:Formal proof
2388:Proof theory
2371:
2344:
2301:real numbers
2273:second-order
2184:Substitution
2061:Metalanguage
2002:conservative
1975:Axiom schema
1919:Constructive
1889:MorseâKelley
1855:Set theories
1834:Aleph number
1827:inaccessible
1733:Grothendieck
1617:intersection
1504:Higher-order
1492:Second-order
1438:Truth tables
1395:Venn diagram
1178:Formal proof
1090:
1072:
1047:
1028:
998:
979:
960:
957:Barwise, Jon
950:Bibliography
907:
897:
883:
838:
828:
817:
806:
795:
783:. Retrieved
753:
750:J. Symb. Log
749:
739:
720:
713:
697:. Springer.
692:
674:Trakhtenbrot
669:
612:
608:
600:
584:
574:
564:
558:
552:
534:completeness
531:
517:
513:
509:
505:
502:
497:
493:
491:
473:
469:
465:
459:
403:Trakhtenbrot
392:
372:
351:
255:
231:
222:
220:
205:
194:
184:
180:
165:
157:
139:
124:
120:linear logic
111:
89:
56:higher-order
39:
29:
2989:Metatheorem
2947:of geometry
2932:Consistency
2758:Type theory
2706:undecidable
2638:Truth value
2525:equivalence
2204:non-logical
1817:Enumeration
1807:Isomorphism
1754:cardinality
1738:Von Neumann
1703:Ultrafilter
1668:Uncountable
1602:equivalence
1519:Quantifiers
1509:Fixed-point
1478:First-order
1358:Consistency
1343:Proposition
1320:Traditional
1291:Lindström's
1281:Compactness
1223:Type theory
1168:Cardinality
999:Modal logic
500:a theorem.
303:of a given
153:type theory
131:truth-table
100:provability
80:undecidable
52:first-order
3024:Categories
2569:elementary
2262:arithmetic
2130:Quantifier
2108:functional
1980:Expression
1698:Transitive
1642:identities
1627:complement
1560:hereditary
1543:Set theory
656:References
516:is not in
381:, and the
361:theory of
318:(see also
3035:Metalogic
2964:Soundness
2900:Metalogic
2840:Supertask
2743:Recursion
2701:decidable
2535:saturated
2513:of models
2436:deductive
2431:axiomatic
2351:Hilbert's
2338:Euclidean
2319:canonical
2242:axiomatic
2174:Signature
2103:Predicate
1992:Extension
1914:Ackermann
1839:Operation
1718:Universal
1708:Recursive
1683:Singleton
1678:Inhabited
1663:Countable
1653:Types of
1637:power set
1607:partition
1524:Predicate
1470:Predicate
1385:Syllogism
1375:Soundness
1348:Inference
1338:Tautology
1240:paradoxes
1032:, Dover,
1026:(2013) ,
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758:CiteSeerX
346:Specific
175:, or the
145:signature
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2794:Concrete
2653:Validity
2623:T-schema
2616:Kripke's
2611:Tarski's
2606:semantic
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2540:spectrum
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2345:Elements
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