2832:
3258:
998:
446:
are interpreted in each other, then by combining the interpretations in two possible ways, one obtains an interpretation of each of the two structures in itself. This observation permits one to define an equivalence relation among structures, reminiscent of the
308:
249:
160:
328:
220:
184:
1211:
1886:
584:
the rational numbers. To see that it is an interpretation (without parameters), one needs to check the following preimages of definable sets in
394:. Similarly, an interpretation with parameters may be referred to as simply an interpretation, and an interpretation without parameters as a
1969:
1110:
390:
often refers to definability with parameters; if this convention is used, definability without parameters is expressed by the term
2283:
103:, rather than being used in the sense defined here. These two notions of "interpretation" are related but nevertheless distinct.
3282:
2441:
1079:
1052:
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945:
1229:
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1619:
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2301:
2291:
2028:
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1234:
1225:
3192:
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2534:
2278:
1103:
1071:
1014:
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1839:
1532:
1273:
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2080:
1501:
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100:
35:
2790:
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2011:
1253:
2715:
2541:
2227:
1861:
1460:
3272:
2593:
2588:
2198:
1937:
1866:
1195:
1096:
1012:
Ahlbrandt, Gisela; Ziegler, Martin (1986), "Quasi finitely axiomatizable totally categorical theories",
3223:
2522:
2112:
1506:
1474:
1165:
1044:
341:(with or without parameters), it is sufficient to check the preimages of the following definable sets:
3202:
3127:
2909:
2812:
2761:
2658:
2156:
2117:
1594:
1239:
274:) by a first-order formula with parameters (or without parameters, respectively). Since the value of
267:
1268:
968:
Hodges, Wilfrid (2009), "Functional
Modelling and Mathematical Models", in Meijers, Anthonie (ed.),
72:
Many model-theoretic properties are preserved under interpretability. For example, if the theory of
3100:
2653:
2583:
2122:
1974:
1957:
1680:
1160:
3184:
2485:
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2423:
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2250:
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1217:
20:
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8:
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2001:
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555:
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96:
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1986:
1962:
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951:
941:
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190:
46:
3167:
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2413:
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931:
3013:
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2610:
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2239:
2166:
2139:
2127:
2033:
1947:
1921:
1876:
1844:
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1447:
1390:
1340:
1305:
1263:
562:
3267:
2960:
2939:
2854:
2751:
2730:
2688:
2668:
2563:
2418:
2016:
2006:
1996:
1991:
1925:
1799:
1675:
1564:
1559:
1537:
1138:
1036:
3297:
3117:
3058:
2725:
2403:
1910:
1695:
1685:
1655:
1640:
1310:
955:
259:
77:
502:, respectively (the composite interpretations being viewed as operations on
353:
3107:
2929:
2625:
2472:
2373:
2365:
2245:
2193:
2102:
2038:
2021:
1952:
1811:
1670:
1372:
1155:
383:
333:
To verify that the preimage of every definable (without parameters) set in
27:
935:
3142:
3137:
3090:
2735:
2615:
1794:
1784:
1731:
1415:
1335:
1320:
1200:
1145:
3085:
3053:
3018:
1665:
1520:
1491:
1297:
187:
821:
the preimage of the graph of multiplication is defined by the formula
3147:
3008:
2919:
2817:
2720:
1773:
1690:
1650:
1614:
1550:
1362:
1352:
1325:
1088:
3068:
2802:
2600:
2048:
1753:
1347:
202:
49:) is a technical notion that approximates the idea of representing
3132:
3063:
2398:
1190:
573:
972:, Handbook of the Philosophy of Science, vol. 9, Elsevier,
705:
the preimage of the graph of addition is defined by the formula
2970:
430:
then one can naturally construct a composite interpretation of
58:
3162:
2877:
1942:
1288:
1133:
580:)). In fact, this particular interpretation is often used to
3122:
2846:
682:
the preimages of 0 and 1 are defined by the formulas Ï(
316:
284:
230:
208:
172:
136:
993:
940:(2nd ed.), Mineola, N.Y.: Dover Publications,
1063:
322:
302:
243:
214:
178:
154:
970:Philosophy of technology and engineering sciences
3295:
1011:
576:(to be precise, the interpretation is (2,
934:(2006), "11.2 Formal Language and Semantics",
2862:
1104:
937:Topoi : the categorial analysis of logic
486:such that the composite interpretations of
99:, the term "interpretation" may refer to a
2869:
2855:
1296:
1111:
1097:
1027:
930:
330:itself is also called an interpretation.
61:or definitional expansion of a structure
3296:
1118:
1061:
1035:
967:
401:
374:of every function in the signature of
2850:
1092:
470:if there exists an interpretation of
310:is often clear from context, the map
554:0 provides an interpretation of the
270:without parameters is definable (in
3044:Analytic and synthetic propositions
2915:Formal semantics (natural language)
363:every relation in the signature of
13:
989:
14:
3320:
3256:
2830:
1015:Annals of Pure and Applied Logic
996:
611:the preimage of the diagonal of
961:
924:
297:
285:
149:
137:
1:
2791:History of mathematical logic
917:
106:
2716:Primitive recursive function
1029:10.1016/0168-0072(86)90037-0
596:is defined by the formula Ï(
95:Note that in other areas of
7:
3309:Interpretation (philosophy)
494:in itself are definable in
10:
3325:
1780:SchröderâBernstein theorem
1507:Monadic predicate calculus
1166:Foundations of mathematics
1045:Cambridge University Press
615:is defined by the formula
513:
130:, respectively) is a pair
45:(typically of a different
18:
3251:
3211:
3183:
3176:
3128:Necessity and sufficiency
3031:
2996:
2948:
2902:
2884:
2876:
2826:
2813:Philosophy of mathematics
2762:Automated theorem proving
2744:
2639:
2471:
2364:
2216:
1933:
1909:
1887:Von NeumannâBernaysâGödel
1832:
1726:
1630:
1528:
1519:
1446:
1381:
1287:
1209:
1126:
478:and an interpretation of
65:has an interpretation in
1066:A Course in Model Theory
251:-preimage) of every set
166:is a natural number and
19:Not to be confused with
2463:Self-verifying theories
2284:Tarski's axiomatization
1235:Tarski's undefinability
1230:incompleteness theorems
21:Interpretation function
16:Concept in model theory
2837:Mathematics portal
2448:Proof of impossibility
2096:propositional variable
1406:Propositional calculus
1062:Poizat, Bruno (2000),
1041:A shorter model theory
414:are three structures,
324:
304:
278:for an interpretation
245:
216:
180:
156:
3263:Philosophy portal
2706:Kolmogorov complexity
2659:Computably enumerable
2559:Model complete theory
2351:Principia Mathematica
1411:Propositional formula
1240:BanachâTarski paradox
325:
305:
303:{\displaystyle (n,f)}
246:
244:{\displaystyle f^{k}}
217:
181:
157:
155:{\displaystyle (n,f)}
88:, then the theory of
57:. For example, every
41:in another structure
2654:ChurchâTuring thesis
2641:Computability theory
1850:continuum hypothesis
1368:Square of opposition
1226:Gödel's completeness
449:homotopy equivalence
314:
282:
228:
224:(more precisely the
206:
170:
134:
84:is interpretable in
2925:Philosophy of logic
2808:Mathematical object
2699:P versus NP problem
2664:Computable function
2458:Reverse mathematics
2384:Logical consequence
2261:primitive recursive
2256:elementary function
2029:Free/bound variable
1882:TarskiâGrothendieck
1401:Logical connectives
1331:Logical equivalence
1181:Logical consequence
694: = 0 and
604:) given by ÂŹ (
402:Bi-interpretability
268:first-order formula
3224:Rules of inference
3193:Mathematical logic
2935:Semantics of logic
2606:Transfer principle
2569:Semantics of logic
2554:Categorical theory
2530:Non-standard model
2044:Logical connective
1171:Information theory
1120:Mathematical logic
1085:(Section 9.4)
1058:(Section 4.3)
526: ×
453:topological spaces
438:If two structures
426:is interpreted in
418:is interpreted in
320:
300:
241:
212:
176:
152:
128:without parameters
97:mathematical logic
3291:
3290:
3247:
3246:
3081:Deductive closure
3027:
3026:
2966:Critical thinking
2844:
2843:
2776:Abstract category
2579:Theories of truth
2389:Rule of inference
2379:Natural deduction
2360:
2359:
1905:
1904:
1610:Cartesian product
1515:
1514:
1421:Many-valued logic
1396:Boolean functions
1279:Russell's paradox
1254:diagonal argument
1151:First-order logic
1081:978-0-387-98655-5
1054:978-0-521-58713-6
1004:Philosophy portal
979:978-0-444-51667-1
947:978-0-486-31796-0
932:Goldblatt, Robert
490:in itself and of
323:{\displaystyle f}
215:{\displaystyle f}
193:from a subset of
179:{\displaystyle f}
3316:
3261:
3260:
3259:
3181:
3180:
2946:
2945:
2910:Computer science
2871:
2864:
2857:
2848:
2847:
2835:
2834:
2786:History of logic
2781:Category of sets
2674:Decision problem
2453:Ordinal analysis
2394:Sequent calculus
2292:Boolean algebras
2232:
2231:
2206:
2177:logical/constant
1931:
1930:
1917:
1840:ZermeloâFraenkel
1591:Set operations:
1526:
1525:
1463:
1294:
1293:
1274:LöwenheimâSkolem
1161:Formal semantics
1113:
1106:
1099:
1090:
1089:
1084:
1069:
1057:
1032:
1031:
1006:
1001:
1000:
999:
983:
982:
965:
959:
958:
928:
912:
889:
866:
817:
794:
750:
678:
662:
646:
608: = 0);
592:the preimage of
563:rational numbers
518:The partial map
468:bi-interpretable
396:0-interpretation
337:is definable in
329:
327:
326:
321:
309:
307:
306:
301:
250:
248:
247:
242:
240:
239:
221:
219:
218:
213:
185:
183:
182:
177:
161:
159:
158:
153:
92:is also stable.
3324:
3323:
3319:
3318:
3317:
3315:
3314:
3313:
3294:
3293:
3292:
3287:
3257:
3255:
3243:
3207:
3198:Boolean algebra
3172:
3023:
3014:Metamathematics
2992:
2944:
2898:
2880:
2875:
2845:
2840:
2829:
2822:
2767:Category theory
2757:Algebraic logic
2740:
2711:Lambda calculus
2649:Church encoding
2635:
2611:Truth predicate
2467:
2433:Complete theory
2356:
2225:
2221:
2217:
2212:
2204:
1924: and
1920:
1915:
1901:
1877:New Foundations
1845:axiom of choice
1828:
1790:Gödel numbering
1730: and
1722:
1626:
1511:
1461:
1442:
1391:Boolean algebra
1377:
1341:Equiconsistency
1306:Classical logic
1283:
1264:Halting problem
1252: and
1228: and
1216: and
1215:
1210:Theorems (
1205:
1122:
1117:
1082:
1055:
1037:Hodges, Wilfrid
1002:
997:
995:
992:
990:Further reading
987:
986:
980:
966:
962:
948:
929:
925:
920:
911:
904:
897:
891:
888:
881:
874:
868:
864:
857:
850:
843:
836:
829:
822:
816:
809:
802:
796:
793:
786:
779:
772:
765:
758:
752:
748:
741:
734:
727:
720:
713:
706:
677:
670:
664:
661:
654:
648:
644:
637:
630:
623:
616:
516:
458:Two structures
404:
315:
312:
311:
283:
280:
279:
235:
231:
229:
226:
225:
207:
204:
203:
171:
168:
167:
135:
132:
131:
124:with parameters
119:in a structure
115:of a structure
109:
24:
17:
12:
11:
5:
3322:
3312:
3311:
3306:
3289:
3288:
3286:
3285:
3280:
3270:
3265:
3252:
3249:
3248:
3245:
3244:
3242:
3241:
3236:
3231:
3226:
3221:
3215:
3213:
3209:
3208:
3206:
3205:
3200:
3195:
3189:
3187:
3178:
3174:
3173:
3171:
3170:
3165:
3160:
3155:
3150:
3145:
3140:
3135:
3130:
3125:
3120:
3115:
3110:
3105:
3104:
3103:
3093:
3088:
3083:
3078:
3073:
3072:
3071:
3066:
3056:
3051:
3046:
3041:
3035:
3033:
3029:
3028:
3025:
3024:
3022:
3021:
3016:
3011:
3006:
3000:
2998:
2994:
2993:
2991:
2990:
2985:
2980:
2975:
2974:
2973:
2968:
2958:
2952:
2950:
2943:
2942:
2937:
2932:
2927:
2922:
2917:
2912:
2906:
2904:
2900:
2899:
2897:
2896:
2891:
2885:
2882:
2881:
2874:
2873:
2866:
2859:
2851:
2842:
2841:
2827:
2824:
2823:
2821:
2820:
2815:
2810:
2805:
2800:
2799:
2798:
2788:
2783:
2778:
2769:
2764:
2759:
2754:
2752:Abstract logic
2748:
2746:
2742:
2741:
2739:
2738:
2733:
2731:Turing machine
2728:
2723:
2718:
2713:
2708:
2703:
2702:
2701:
2696:
2691:
2686:
2681:
2671:
2669:Computable set
2666:
2661:
2656:
2651:
2645:
2643:
2637:
2636:
2634:
2633:
2628:
2623:
2618:
2613:
2608:
2603:
2598:
2597:
2596:
2591:
2586:
2576:
2571:
2566:
2564:Satisfiability
2561:
2556:
2551:
2550:
2549:
2539:
2538:
2537:
2527:
2526:
2525:
2520:
2515:
2510:
2505:
2495:
2494:
2493:
2488:
2481:Interpretation
2477:
2475:
2469:
2468:
2466:
2465:
2460:
2455:
2450:
2445:
2435:
2430:
2429:
2428:
2427:
2426:
2416:
2411:
2401:
2396:
2391:
2386:
2381:
2376:
2370:
2368:
2362:
2361:
2358:
2357:
2355:
2354:
2346:
2345:
2344:
2343:
2338:
2337:
2336:
2331:
2326:
2306:
2305:
2304:
2302:minimal axioms
2299:
2288:
2287:
2286:
2275:
2274:
2273:
2268:
2263:
2258:
2253:
2248:
2235:
2233:
2214:
2213:
2211:
2210:
2209:
2208:
2196:
2191:
2190:
2189:
2184:
2179:
2174:
2164:
2159:
2154:
2149:
2148:
2147:
2142:
2132:
2131:
2130:
2125:
2120:
2115:
2105:
2100:
2099:
2098:
2093:
2088:
2078:
2077:
2076:
2071:
2066:
2061:
2056:
2051:
2041:
2036:
2031:
2026:
2025:
2024:
2019:
2014:
2009:
1999:
1994:
1992:Formation rule
1989:
1984:
1983:
1982:
1977:
1967:
1966:
1965:
1955:
1950:
1945:
1940:
1934:
1928:
1911:Formal systems
1907:
1906:
1903:
1902:
1900:
1899:
1894:
1889:
1884:
1879:
1874:
1869:
1864:
1859:
1854:
1853:
1852:
1847:
1836:
1834:
1830:
1829:
1827:
1826:
1825:
1824:
1814:
1809:
1808:
1807:
1800:Large cardinal
1797:
1792:
1787:
1782:
1777:
1763:
1762:
1761:
1756:
1751:
1736:
1734:
1724:
1723:
1721:
1720:
1719:
1718:
1713:
1708:
1698:
1693:
1688:
1683:
1678:
1673:
1668:
1663:
1658:
1653:
1648:
1643:
1637:
1635:
1628:
1627:
1625:
1624:
1623:
1622:
1617:
1612:
1607:
1602:
1597:
1589:
1588:
1587:
1582:
1572:
1567:
1565:Extensionality
1562:
1560:Ordinal number
1557:
1547:
1542:
1541:
1540:
1529:
1523:
1517:
1516:
1513:
1512:
1510:
1509:
1504:
1499:
1494:
1489:
1484:
1479:
1478:
1477:
1467:
1466:
1465:
1452:
1450:
1444:
1443:
1441:
1440:
1439:
1438:
1433:
1428:
1418:
1413:
1408:
1403:
1398:
1393:
1387:
1385:
1379:
1378:
1376:
1375:
1370:
1365:
1360:
1355:
1350:
1345:
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1343:
1333:
1328:
1323:
1318:
1313:
1308:
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1300:
1291:
1285:
1284:
1282:
1281:
1276:
1271:
1266:
1261:
1256:
1244:Cantor's
1242:
1237:
1232:
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1141:
1130:
1128:
1124:
1123:
1116:
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1108:
1101:
1093:
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1080:
1059:
1053:
1033:
1008:
1007:
991:
988:
985:
984:
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946:
922:
921:
919:
916:
915:
914:
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895:
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862:
855:
848:
841:
834:
827:
819:
814:
807:
800:
791:
784:
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763:
756:
746:
739:
732:
725:
718:
711:
703:
680:
675:
668:
659:
652:
642:
635:
628:
621:
609:
515:
512:
403:
400:
380:
379:
368:
361:
350:
345:the domain of
319:
299:
296:
293:
290:
287:
238:
234:
211:
201:such that the
175:
151:
148:
145:
142:
139:
113:interpretation
108:
105:
32:interpretation
15:
9:
6:
4:
3:
2:
3321:
3310:
3307:
3305:
3302:
3301:
3299:
3284:
3281:
3278:
3274:
3271:
3269:
3266:
3264:
3254:
3253:
3250:
3240:
3239:Logic symbols
3237:
3235:
3232:
3230:
3227:
3225:
3222:
3220:
3217:
3216:
3214:
3210:
3204:
3201:
3199:
3196:
3194:
3191:
3190:
3188:
3186:
3182:
3179:
3175:
3169:
3166:
3164:
3161:
3159:
3156:
3154:
3151:
3149:
3146:
3144:
3141:
3139:
3136:
3134:
3131:
3129:
3126:
3124:
3121:
3119:
3118:Logical truth
3116:
3114:
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3109:
3106:
3102:
3099:
3098:
3097:
3094:
3092:
3089:
3087:
3084:
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3079:
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3074:
3070:
3067:
3065:
3062:
3061:
3060:
3059:Contradiction
3057:
3055:
3052:
3050:
3047:
3045:
3042:
3040:
3037:
3036:
3034:
3030:
3020:
3017:
3015:
3012:
3010:
3007:
3005:
3004:Argumentation
3002:
3001:
2999:
2995:
2989:
2988:Philosophical
2986:
2984:
2983:Non-classical
2981:
2979:
2976:
2972:
2969:
2967:
2964:
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2962:
2959:
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2755:
2753:
2750:
2749:
2747:
2743:
2737:
2734:
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2729:
2727:
2726:Recursive set
2724:
2722:
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2712:
2709:
2707:
2704:
2700:
2697:
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2565:
2562:
2560:
2557:
2555:
2552:
2548:
2545:
2544:
2543:
2540:
2536:
2535:of arithmetic
2533:
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2464:
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2459:
2456:
2454:
2451:
2449:
2446:
2443:
2442:from ZFC
2439:
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2431:
2425:
2422:
2421:
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2417:
2415:
2412:
2410:
2407:
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2400:
2397:
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2377:
2375:
2372:
2371:
2369:
2367:
2363:
2353:
2352:
2348:
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2342:
2341:non-Euclidean
2339:
2335:
2332:
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2327:
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2320:
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2241:
2237:
2236:
2234:
2229:
2223:
2218:Example
2215:
2207:
2202:
2201:
2200:
2197:
2195:
2192:
2188:
2185:
2183:
2180:
2178:
2175:
2173:
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2169:
2168:
2165:
2163:
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2158:
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2150:
2146:
2143:
2141:
2138:
2137:
2136:
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2129:
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2121:
2119:
2116:
2114:
2111:
2110:
2109:
2106:
2104:
2101:
2097:
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2089:
2087:
2084:
2083:
2082:
2079:
2075:
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2057:
2055:
2052:
2050:
2047:
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2042:
2040:
2037:
2035:
2032:
2030:
2027:
2023:
2020:
2018:
2015:
2013:
2010:
2008:
2005:
2004:
2003:
2000:
1998:
1995:
1993:
1990:
1988:
1985:
1981:
1978:
1976:
1975:by definition
1973:
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1956:
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1878:
1875:
1873:
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1868:
1865:
1863:
1862:KripkeâPlatek
1860:
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1855:
1851:
1848:
1846:
1843:
1842:
1841:
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1835:
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1823:
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1813:
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1801:
1798:
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1791:
1788:
1786:
1783:
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1778:
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1767:
1764:
1760:
1757:
1755:
1752:
1750:
1747:
1746:
1745:
1741:
1738:
1737:
1735:
1733:
1729:
1725:
1717:
1714:
1712:
1709:
1707:
1706:constructible
1704:
1703:
1702:
1699:
1697:
1694:
1692:
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1679:
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1568:
1566:
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1524:
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1518:
1508:
1505:
1503:
1500:
1498:
1495:
1493:
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1488:
1485:
1483:
1480:
1476:
1473:
1472:
1471:
1468:
1464:
1459:
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1457:
1454:
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1449:
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1437:
1434:
1432:
1429:
1427:
1424:
1423:
1422:
1419:
1417:
1414:
1412:
1409:
1407:
1404:
1402:
1399:
1397:
1394:
1392:
1389:
1388:
1386:
1384:
1383:Propositional
1380:
1374:
1371:
1369:
1366:
1364:
1361:
1359:
1356:
1354:
1351:
1349:
1346:
1342:
1339:
1338:
1337:
1334:
1332:
1329:
1327:
1324:
1322:
1319:
1317:
1314:
1312:
1311:Logical truth
1309:
1307:
1304:
1303:
1301:
1299:
1295:
1292:
1290:
1286:
1280:
1277:
1275:
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1260:
1257:
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1247:
1243:
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1236:
1233:
1231:
1227:
1224:
1223:
1221:
1219:
1213:
1208:
1202:
1199:
1197:
1194:
1192:
1189:
1187:
1184:
1182:
1179:
1177:
1174:
1172:
1169:
1167:
1164:
1162:
1159:
1157:
1154:
1152:
1149:
1147:
1144:
1140:
1137:
1136:
1135:
1132:
1131:
1129:
1125:
1121:
1114:
1109:
1107:
1102:
1100:
1095:
1094:
1091:
1083:
1077:
1073:
1068:
1067:
1060:
1056:
1050:
1046:
1043:, Cambridge:
1042:
1038:
1034:
1030:
1025:
1021:
1017:
1016:
1010:
1009:
1005:
994:
981:
975:
971:
964:
957:
953:
949:
943:
939:
938:
933:
927:
923:
908:
901:
894:
885:
878:
871:
861:
854:
847:
840:
833:
826:
820:
813:
806:
799:
790:
783:
776:
769:
762:
755:
745:
738:
731:
724:
717:
710:
704:
701:
698: =
697:
693:
689:
685:
681:
674:
667:
658:
651:
641:
634:
627:
620:
614:
610:
607:
603:
599:
595:
591:
590:
589:
587:
583:
579:
575:
571:
568:
564:
560:
557:
553:
549:
545:
541:
537:
533:
529:
525:
521:
511:
509:
505:
501:
497:
493:
489:
485:
481:
477:
473:
469:
465:
461:
456:
454:
450:
445:
441:
437:
433:
429:
425:
421:
417:
413:
409:
399:
397:
393:
389:
385:
377:
373:
369:
366:
362:
359:
355:
351:
348:
344:
343:
342:
340:
336:
331:
317:
294:
291:
288:
277:
273:
269:
265:
261:
258:
255: â
254:
236:
232:
223:
209:
200:
196:
192:
189:
173:
165:
146:
143:
140:
129:
125:
122:
118:
114:
104:
102:
98:
93:
91:
87:
83:
79:
75:
70:
68:
64:
60:
56:
52:
48:
44:
40:
37:
33:
29:
22:
3304:Model theory
3158:Substitution
2978:Mathematical
2903:Major fields
2828:
2626:Ultraproduct
2490:
2473:Model theory
2438:Independence
2374:Formal proof
2366:Proof theory
2349:
2322:
2279:real numbers
2251:second-order
2162:Substitution
2039:Metalanguage
1980:conservative
1953:Axiom schema
1897:Constructive
1867:MorseâKelley
1833:Set theories
1812:Aleph number
1805:inaccessible
1711:Grothendieck
1595:intersection
1482:Higher-order
1470:Second-order
1416:Truth tables
1373:Venn diagram
1156:Formal proof
1065:
1040:
1019:
1013:
969:
963:
936:
926:
906:
899:
892:
883:
876:
869:
859:
852:
845:
838:
831:
824:
811:
804:
797:
788:
781:
774:
767:
760:
753:
743:
736:
729:
722:
715:
708:
699:
695:
691:
687:
683:
672:
665:
656:
649:
639:
632:
625:
618:
612:
605:
601:
597:
593:
585:
581:
577:
569:
558:
551:
547:
543:
539:
535:
531:
527:
523:
519:
517:
507:
503:
499:
495:
491:
487:
483:
479:
475:
471:
467:
463:
459:
457:
443:
439:
435:
431:
427:
423:
419:
415:
411:
407:
405:
395:
391:
387:
384:model theory
381:
375:
364:
357:
346:
338:
334:
332:
275:
271:
263:
256:
252:
198:
194:
163:
127:
123:
120:
116:
112:
110:
94:
89:
85:
81:
73:
71:
66:
62:
54:
50:
42:
38:
31:
28:model theory
25:
3273:WikiProject
3143:Proposition
3138:Probability
3091:Description
3032:Foundations
2736:Type theory
2684:undecidable
2616:Truth value
2503:equivalence
2182:non-logical
1795:Enumeration
1785:Isomorphism
1732:cardinality
1716:Von Neumann
1681:Ultrafilter
1646:Uncountable
1580:equivalence
1497:Quantifiers
1487:Fixed-point
1456:First-order
1336:Consistency
1321:Proposition
1298:Traditional
1269:Lindström's
1259:Compactness
1201:Type theory
1146:Cardinality
690:) given by
534:that maps (
392:0-definable
3298:Categories
3203:Set theory
3101:Linguistic
3096:Entailment
3086:Definition
3054:Consequent
3049:Antecedent
2547:elementary
2240:arithmetic
2108:Quantifier
2086:functional
1958:Expression
1676:Transitive
1620:identities
1605:complement
1538:hereditary
1521:Set theory
918:References
188:surjective
107:Definition
3234:Fallacies
3229:Paradoxes
3219:Logicians
3153:Statement
3148:Reference
3113:Induction
3076:Deduction
3039:Abduction
3009:Metalogic
2956:Classical
2920:Inference
2818:Supertask
2721:Recursion
2679:decidable
2513:saturated
2491:of models
2414:deductive
2409:axiomatic
2329:Hilbert's
2316:Euclidean
2297:canonical
2220:axiomatic
2152:Signature
2081:Predicate
1970:Extension
1892:Ackermann
1817:Operation
1696:Universal
1686:Recursive
1661:Singleton
1656:Inhabited
1641:Countable
1631:Types of
1615:power set
1585:partition
1502:Predicate
1448:Predicate
1363:Syllogism
1353:Soundness
1326:Inference
1316:Tautology
1218:paradoxes
1022:: 63â82,
956:853624133
867:given by
751:given by
647:given by
388:definable
386:the term
260:definable
222:-preimage
101:structure
47:signature
36:structure
3268:Category
3168:Validity
3069:Antinomy
2997:Theories
2961:Informal
2803:Logicism
2796:timeline
2772:Concrete
2631:Validity
2601:T-schema
2594:Kripke's
2589:Tarski's
2584:semantic
2574:Strength
2523:submodel
2518:spectrum
2486:function
2334:Tarski's
2323:Elements
2310:geometry
2266:Robinson
2187:variable
2172:function
2145:spectrum
2135:Sentence
2091:variable
2034:Language
1987:Relation
1948:Automata
1938:Alphabet
1922:language
1776:-jection
1754:codomain
1740:Function
1701:Universe
1671:Infinite
1575:Relation
1358:Validity
1348:Argument
1246:theorem,
1072:Springer
1039:(1997),
574:integers
354:diagonal
3283:changes
3275: (
3133:Premise
3064:Paradox
2894:History
2889:Outline
2745:Related
2542:Diagram
2440: (
2419:Hilbert
2404:Systems
2399:Theorem
2277:of the
2222:systems
2002:Formula
1997:Grammar
1913: (
1857:General
1570:Forcing
1555:Element
1475:Monadic
1250:paradox
1191:Theorem
1127:General
686:,
671:×
655:×
600:,
565:in the
538:,
514:Example
506:and on
498:and in
53:inside
3185:topics
2971:Reason
2949:Logics
2940:Syntax
2508:finite
2271:Skolem
2224:
2199:Theory
2167:Symbol
2157:String
2140:atomic
2017:ground
2012:closed
2007:atomic
1963:ground
1926:syntax
1822:binary
1749:domain
1666:Finite
1431:finite
1289:Logics
1248:
1196:Theory
1078:
1051:
976:
954:
944:
905:×
898:×
882:×
875:×
810:×
803:×
787:×
780:×
766:×
759:×
582:define
451:among
162:where
78:stable
59:reduct
3212:other
3177:Lists
3163:Truth
2930:Proof
2878:Logic
2498:Model
2246:Peano
2103:Proof
1943:Arity
1872:Naive
1759:image
1691:Fuzzy
1651:Empty
1600:union
1545:Class
1186:Model
1176:Lemma
1134:Axiom
556:field
542:) to
530:onto
522:from
372:graph
266:by a
197:onto
186:is a
34:of a
3277:talk
3123:Name
3108:Form
2621:Type
2424:list
2228:list
2205:list
2194:Term
2128:rank
2022:open
1916:list
1728:Maps
1633:sets
1492:Free
1462:list
1212:list
1139:list
1076:ISBN
1049:ISBN
974:ISBN
952:OCLC
942:ISBN
567:ring
466:are
462:and
442:and
422:and
410:and
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