3238:
33:
724:), where one doesn't want to allow the introduction of new functional symbols (nor any other new symbols, for that matter). But there is a method of replacing functional symbols with relational symbols wherever the former may occur; furthermore, this is algorithmic and thus suitable for applying most metalogical theorems to the result.
1469:
This version of the axiom schema of replacement is now suitable for use in a formal language that doesn't allow the introduction of new function symbols. Alternatively, one may interpret the original statement as a statement in such a formal language; it was merely an abbreviation for the statement
1464:
851:
Because the elimination of functional predicates is both convenient for some purposes and possible, many treatments of formal logic do not deal explicitly with function symbols but instead use only relation symbols; another way to think of this is that a functional predicate is a
1300:
1186:
1009:
1311:
1095:
560:
that doesn't allow you to introduce new symbols after proving theorems, then you will have to use relation symbols to get around this, as in the next section.) Specifically, if you can prove that for every
384:
1617:
1201:
2292:
2375:
1516:
864:; how do you know ahead of time whether it satisfies that condition? To get an equivalent formulation of the schema, first replace anything of the form
1114:
2689:
1459:{\displaystyle (\forall X,\exists !Y,P(X,Y))\rightarrow (\forall A,\exists B,\forall C,\forall D,P(C,D)\rightarrow (C\in A\rightarrow D\in B)).}
937:
405:
that allows one to introduce new predicate symbols, one will also want to be able to introduce new function symbols. Given the function symbols
848:(Of course, this is the same proposition that had to be proven as a theorem before introducing a new function symbol in the previous section.)
549:
that satisfies the same equation; there are additional function symbols associated with other ways of constructing new types out of old ones.
240:. One can similarly define function symbols of more than one variable, analogous to functions of more than one variable; a function symbol in
2847:
1032:
54:
1635:
153:, is a logical symbol that may be applied to an object term to produce another object term. Functional predicates are also sometimes called
2702:
2025:
721:
2287:
2707:
2697:
2434:
1640:
2185:
1631:
856:
predicate, specifically one that satisfies the proposition above. This may seem to be a problem if you wish to specify a proposition
2843:
124:
105:
2940:
2684:
1509:
77:
2245:
1938:
1679:
920:
3201:
2903:
2666:
2661:
2486:
1907:
1591:
58:
84:
3196:
2979:
2896:
2609:
2540:
2417:
1659:
2267:
3121:
2947:
2633:
1866:
2272:
2604:
2343:
1601:
1502:
916:
91:
2999:
2994:
2928:
2518:
1912:
1880:
1571:
281:
1645:
3218:
3167:
3064:
2562:
2523:
2000:
73:
3059:
1674:
2989:
2528:
2380:
2363:
2086:
1566:
43:
2891:
2868:
2829:
2715:
2656:
2302:
2222:
2066:
2010:
1623:
1479:
62:
47:
3181:
2908:
2886:
2853:
2746:
2592:
2577:
2550:
2501:
2385:
2320:
2145:
2111:
2106:
1980:
1811:
1788:
390:
166:
466:. Of course, the right side of this equation doesn't make sense in typed logic unless the domain type of
3262:
3111:
2964:
2756:
2474:
2210:
2116:
1975:
1960:
1841:
1816:
158:
3237:
3084:
3046:
2923:
2727:
2567:
2469:
2297:
2255:
2154:
2121:
1985:
1773:
1684:
1295:{\displaystyle \forall A,\exists B,\forall C,\forall D,P(C,D)\rightarrow (C\in A\rightarrow D\in B).}
924:
909:
877:
822:
574:
423:
3213:
3104:
3089:
3069:
3026:
2913:
2863:
2789:
2734:
2671:
2464:
2459:
2407:
2175:
2164:
1836:
1736:
1664:
1655:
1651:
1586:
1581:
98:
3242:
3011:
2974:
2959:
2952:
2935:
2721:
2587:
2513:
2496:
2449:
2262:
2171:
2005:
1990:
1950:
1902:
1887:
1875:
1831:
1806:
1576:
1525:
1484:
2739:
2195:
3177:
2984:
2794:
2784:
2676:
2557:
2392:
2368:
2149:
2133:
2038:
2015:
1892:
1861:
1826:
1721:
1556:
713:
593:
260:
393:
with domain and codomain . It is a requirement of a consistent model that = whenever = .
3191:
3186:
3079:
3036:
2858:
2819:
2814:
2799:
2625:
2582:
2479:
2277:
2227:
1801:
1763:
1489:
245:
3172:
3162:
3116:
3099:
3054:
3016:
2918:
2838:
2645:
2572:
2545:
2533:
2439:
2353:
2327:
2282:
2250:
2051:
1853:
1796:
1746:
1711:
1669:
1305:
This is almost correct, but it applies to too many predicates; what we actually want is:
402:
177:
3157:
3136:
3094:
3074:
2969:
2824:
2422:
2412:
2402:
2397:
2331:
2205:
2081:
1970:
1965:
1943:
1544:
857:
689:
712:
Many treatments of predicate logic don't allow functional predicates, only relational
3256:
3131:
2809:
2316:
2101:
2091:
2061:
2046:
1716:
557:
162:
477:
One also gets certain function symbols automatically. In untyped logic, there is an
3031:
2878:
2779:
2771:
2651:
2599:
2508:
2444:
2427:
2358:
2217:
2076:
1778:
1561:
570:
138:
1181:{\displaystyle \forall A,\exists B,\forall C,\forall D,C\in A\rightarrow D\in B.}
3141:
3021:
2200:
2190:
2137:
1821:
1741:
1726:
1606:
1551:
552:
Additionally, one can define functional predicates after proving an appropriate
197:
142:
32:
807:). To be able to make the same deductions, you need an additional proposition:
17:
2071:
1926:
1897:
1703:
241:
3223:
3126:
2179:
2096:
2056:
2020:
1956:
1768:
1758:
1731:
717:
534:
181:
1004:{\displaystyle \forall A,\exists B,\forall C,C\in A\rightarrow F(C)\in B.}
3208:
3006:
2454:
2159:
1753:
736:
2804:
1596:
1090:{\displaystyle \forall A,\exists B,\forall C,C\in A\rightarrow D\in B.}
811:
553:
460:
1494:
787:) would appear in a statement, you can replace it with a new symbol
2348:
1694:
1539:
927:.) This schema states (in one form), for any functional predicate
196:) is again a symbol representing an object in that language. In
1498:
912:
of the uniqueness condition for a functional predicate above.
26:
892:
is quantified over, or at the beginning of the statement if
251:
Now consider a model of the formal language, with the types
716:. This is useful, for example, in the context of proving
474:, so this is required for the composition to be defined.
541:, then there is an inclusion predicate of domain type
1314:
1204:
1117:
1035:
940:
284:
3150:
3045:
2877:
2770:
2622:
2315:
2238:
2132:
2036:
1925:
1852:
1787:
1702:
1693:
1615:
1532:
1458:
1294:
1180:
1089:
1003:
378:
1510:
379:{\displaystyle :={\big \{}(,):\in {\big \}},}
368:
299:
236:) is a symbol representing an object of type
8:
896:is free), and guard the quantification with
860:that applies only to functional predicates
743:, then it can be replaced with a predicate
584:, then you can introduce a function symbol
61:. Unsourced material may be challenged and
2336:
1931:
1699:
1517:
1503:
1495:
413:, one can introduce a new function symbol
165:, a function symbol will be modelled by a
1313:
1203:
1116:
1100:Of course, this statement isn't correct;
1034:
939:
644:then you can introduce a function symbol
367:
366:
358:
298:
297:
283:
125:Learn how and when to remove this message
908:). Finally, make the entire statement a
188:representing an object in the language,
7:
884:immediately after the corresponding
59:adding citations to reliable sources
1104:must be quantified over just after
708:Doing without functional predicates
497:, there is an identity predicate id
1393:
1384:
1375:
1366:
1327:
1318:
1232:
1223:
1214:
1205:
1145:
1136:
1127:
1118:
1054:
1045:
1036:
959:
950:
941:
604:. So if there is such a predicate
271:modelled by an element in . Then
159:additional meanings in mathematics
25:
493:. In typed logic, given any type
3236:
397:Introducing new function symbols
359:
31:
1195:to guard this quantification:
722:Gödel's incompleteness theorems
224:representing an object of type
1450:
1447:
1435:
1423:
1420:
1417:
1405:
1363:
1360:
1357:
1354:
1342:
1315:
1286:
1274:
1262:
1259:
1256:
1244:
1163:
1072:
989:
983:
977:
915:Let us take as an example the
888:is introduced (that is, after
795:and include another statement
503:with domain and codomain type
363:
355:
349:
343:
337:
334:
331:
325:
319:
313:
307:
304:
291:
285:
1:
3197:History of mathematical logic
470:matches the codomain type of
3122:Primitive recursive function
592:will itself be a relational
588:to indicate this. Note that
204:is a functional symbol with
1022:) with some other variable
921:ZermeloâFraenkel set theory
917:axiom schema of replacement
275:can be modelled by the set
180:is a functional symbol if,
3279:
2186:SchröderâBernstein theorem
1913:Monadic predicate calculus
1572:Foundations of mathematics
580:satisfying some condition
556:. (If you're working in a
3232:
3219:Philosophy of mathematics
3168:Automated theorem proving
2339:
2293:Von NeumannâBernaysâGödel
1934:
172:Specifically, the symbol
1191:We still must introduce
141:and related branches of
2869:Self-verifying theories
2690:Tarski's axiomatization
1641:Tarski's undefinability
1636:incompleteness theorems
1480:Function symbol (logic)
1014:First, we must replace
3243:Mathematics portal
2854:Proof of impossibility
2502:propositional variable
1812:Propositional calculus
1460:
1296:
1182:
1091:
1005:
872:) with a new variable
380:
244:variables is simply a
74:"Functional predicate"
3112:Kolmogorov complexity
3065:Computably enumerable
2965:Model complete theory
2757:Principia Mathematica
1817:Propositional formula
1646:BanachâTarski paradox
1470:produced at the end.
1461:
1297:
1183:
1092:
1006:
923:. (This example uses
481:id that satisfies id(
381:
263:and and each symbol
220:if, given any symbol
3060:ChurchâTuring thesis
3047:Computability theory
2256:continuum hypothesis
1774:Square of opposition
1632:Gödel's completeness
1312:
1202:
1115:
1033:
938:
925:mathematical symbols
910:material consequence
878:universally quantify
569:of a certain type),
282:
157:, but that term has
147:functional predicate
55:improve this article
3214:Mathematical object
3105:P versus NP problem
3070:Computable function
2864:Reverse mathematics
2790:Logical consequence
2667:primitive recursive
2662:elementary function
2435:Free/bound variable
2288:TarskiâGrothendieck
1807:Logical connectives
1737:Logical equivalence
1587:Logical consequence
3012:Transfer principle
2975:Semantics of logic
2960:Categorical theory
2936:Non-standard model
2450:Logical connective
1577:Information theory
1526:Mathematical logic
1485:Logical connective
1456:
1292:
1178:
1087:
1001:
720:theorems (such as
652:and codomain type
620:, for some unique
545:and codomain type
479:identity predicate
401:In a treatment of
389:which is simply a
376:
3250:
3249:
3182:Abstract category
2985:Theories of truth
2795:Rule of inference
2785:Natural deduction
2766:
2765:
2311:
2310:
2016:Cartesian product
1921:
1920:
1827:Many-valued logic
1802:Boolean functions
1685:Russell's paradox
1660:diagonal argument
1557:First-order logic
931:in one variable:
727:Specifically, if
507:; it satisfies id
135:
134:
127:
109:
16:(Redirected from
3270:
3241:
3240:
3192:History of logic
3187:Category of sets
3080:Decision problem
2859:Ordinal analysis
2800:Sequent calculus
2698:Boolean algebras
2638:
2637:
2612:
2583:logical/constant
2337:
2323:
2246:ZermeloâFraenkel
1997:Set operations:
1932:
1869:
1700:
1680:LöwenheimâSkolem
1567:Formal semantics
1519:
1512:
1505:
1496:
1490:Logical constant
1465:
1463:
1462:
1457:
1301:
1299:
1298:
1293:
1187:
1185:
1184:
1179:
1096:
1094:
1093:
1088:
1010:
1008:
1007:
1002:
779:. Then whenever
755:). Intuitively,
731:has domain type
656:that satisfies:
529:. Similarly, if
385:
383:
382:
377:
372:
371:
362:
303:
302:
130:
123:
119:
116:
110:
108:
67:
35:
27:
21:
3278:
3277:
3273:
3272:
3271:
3269:
3268:
3267:
3253:
3252:
3251:
3246:
3235:
3228:
3173:Category theory
3163:Algebraic logic
3146:
3117:Lambda calculus
3055:Church encoding
3041:
3017:Truth predicate
2873:
2839:Complete theory
2762:
2631:
2627:
2623:
2618:
2610:
2330: and
2326:
2321:
2307:
2283:New Foundations
2251:axiom of choice
2234:
2196:Gödel numbering
2136: and
2128:
2032:
1917:
1867:
1848:
1797:Boolean algebra
1783:
1747:Equiconsistency
1712:Classical logic
1689:
1670:Halting problem
1658: and
1634: and
1622: and
1621:
1616:Theorems (
1611:
1528:
1523:
1476:
1310:
1309:
1200:
1199:
1113:
1112:
1031:
1030:
936:
935:
854:special kind of
710:
648:of domain type
608:and a theorem:
596:involving both
512:
502:
403:predicate logic
399:
280:
279:
178:formal language
151:function symbol
131:
120:
114:
111:
68:
66:
52:
36:
23:
22:
18:Mapping (logic)
15:
12:
11:
5:
3276:
3274:
3266:
3265:
3255:
3254:
3248:
3247:
3233:
3230:
3229:
3227:
3226:
3221:
3216:
3211:
3206:
3205:
3204:
3194:
3189:
3184:
3175:
3170:
3165:
3160:
3158:Abstract logic
3154:
3152:
3148:
3147:
3145:
3144:
3139:
3137:Turing machine
3134:
3129:
3124:
3119:
3114:
3109:
3108:
3107:
3102:
3097:
3092:
3087:
3077:
3075:Computable set
3072:
3067:
3062:
3057:
3051:
3049:
3043:
3042:
3040:
3039:
3034:
3029:
3024:
3019:
3014:
3009:
3004:
3003:
3002:
2997:
2992:
2982:
2977:
2972:
2970:Satisfiability
2967:
2962:
2957:
2956:
2955:
2945:
2944:
2943:
2933:
2932:
2931:
2926:
2921:
2916:
2911:
2901:
2900:
2899:
2894:
2887:Interpretation
2883:
2881:
2875:
2874:
2872:
2871:
2866:
2861:
2856:
2851:
2841:
2836:
2835:
2834:
2833:
2832:
2822:
2817:
2807:
2802:
2797:
2792:
2787:
2782:
2776:
2774:
2768:
2767:
2764:
2763:
2761:
2760:
2752:
2751:
2750:
2749:
2744:
2743:
2742:
2737:
2732:
2712:
2711:
2710:
2708:minimal axioms
2705:
2694:
2693:
2692:
2681:
2680:
2679:
2674:
2669:
2664:
2659:
2654:
2641:
2639:
2620:
2619:
2617:
2616:
2615:
2614:
2602:
2597:
2596:
2595:
2590:
2585:
2580:
2570:
2565:
2560:
2555:
2554:
2553:
2548:
2538:
2537:
2536:
2531:
2526:
2521:
2511:
2506:
2505:
2504:
2499:
2494:
2484:
2483:
2482:
2477:
2472:
2467:
2462:
2457:
2447:
2442:
2437:
2432:
2431:
2430:
2425:
2420:
2415:
2405:
2400:
2398:Formation rule
2395:
2390:
2389:
2388:
2383:
2373:
2372:
2371:
2361:
2356:
2351:
2346:
2340:
2334:
2317:Formal systems
2313:
2312:
2309:
2308:
2306:
2305:
2300:
2295:
2290:
2285:
2280:
2275:
2270:
2265:
2260:
2259:
2258:
2253:
2242:
2240:
2236:
2235:
2233:
2232:
2231:
2230:
2220:
2215:
2214:
2213:
2206:Large cardinal
2203:
2198:
2193:
2188:
2183:
2169:
2168:
2167:
2162:
2157:
2142:
2140:
2130:
2129:
2127:
2126:
2125:
2124:
2119:
2114:
2104:
2099:
2094:
2089:
2084:
2079:
2074:
2069:
2064:
2059:
2054:
2049:
2043:
2041:
2034:
2033:
2031:
2030:
2029:
2028:
2023:
2018:
2013:
2008:
2003:
1995:
1994:
1993:
1988:
1978:
1973:
1971:Extensionality
1968:
1966:Ordinal number
1963:
1953:
1948:
1947:
1946:
1935:
1929:
1923:
1922:
1919:
1918:
1916:
1915:
1910:
1905:
1900:
1895:
1890:
1885:
1884:
1883:
1873:
1872:
1871:
1858:
1856:
1850:
1849:
1847:
1846:
1845:
1844:
1839:
1834:
1824:
1819:
1814:
1809:
1804:
1799:
1793:
1791:
1785:
1784:
1782:
1781:
1776:
1771:
1766:
1761:
1756:
1751:
1750:
1749:
1739:
1734:
1729:
1724:
1719:
1714:
1708:
1706:
1697:
1691:
1690:
1688:
1687:
1682:
1677:
1672:
1667:
1662:
1650:Cantor's
1648:
1643:
1638:
1628:
1626:
1613:
1612:
1610:
1609:
1604:
1599:
1594:
1589:
1584:
1579:
1574:
1569:
1564:
1559:
1554:
1549:
1548:
1547:
1536:
1534:
1530:
1529:
1524:
1522:
1521:
1514:
1507:
1499:
1493:
1492:
1487:
1482:
1475:
1472:
1467:
1466:
1455:
1452:
1449:
1446:
1443:
1440:
1437:
1434:
1431:
1428:
1425:
1422:
1419:
1416:
1413:
1410:
1407:
1404:
1401:
1398:
1395:
1392:
1389:
1386:
1383:
1380:
1377:
1374:
1371:
1368:
1365:
1362:
1359:
1356:
1353:
1350:
1347:
1344:
1341:
1338:
1335:
1332:
1329:
1326:
1323:
1320:
1317:
1303:
1302:
1291:
1288:
1285:
1282:
1279:
1276:
1273:
1270:
1267:
1264:
1261:
1258:
1255:
1252:
1249:
1246:
1243:
1240:
1237:
1234:
1231:
1228:
1225:
1222:
1219:
1216:
1213:
1210:
1207:
1189:
1188:
1177:
1174:
1171:
1168:
1165:
1162:
1159:
1156:
1153:
1150:
1147:
1144:
1141:
1138:
1135:
1132:
1129:
1126:
1123:
1120:
1098:
1097:
1086:
1083:
1080:
1077:
1074:
1071:
1068:
1065:
1062:
1059:
1056:
1053:
1050:
1047:
1044:
1041:
1038:
1012:
1011:
1000:
997:
994:
991:
988:
985:
982:
979:
976:
973:
970:
967:
964:
961:
958:
955:
952:
949:
946:
943:
846:
845:
709:
706:
705:
704:
690:if and only if
642:
641:
508:
498:
435:, satisfying (
398:
395:
387:
386:
375:
370:
365:
361:
357:
354:
351:
348:
345:
342:
339:
336:
333:
330:
327:
324:
321:
318:
315:
312:
309:
306:
301:
296:
293:
290:
287:
133:
132:
39:
37:
30:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3275:
3264:
3261:
3260:
3258:
3245:
3244:
3239:
3231:
3225:
3222:
3220:
3217:
3215:
3212:
3210:
3207:
3203:
3200:
3199:
3198:
3195:
3193:
3190:
3188:
3185:
3183:
3179:
3176:
3174:
3171:
3169:
3166:
3164:
3161:
3159:
3156:
3155:
3153:
3149:
3143:
3140:
3138:
3135:
3133:
3132:Recursive set
3130:
3128:
3125:
3123:
3120:
3118:
3115:
3113:
3110:
3106:
3103:
3101:
3098:
3096:
3093:
3091:
3088:
3086:
3083:
3082:
3081:
3078:
3076:
3073:
3071:
3068:
3066:
3063:
3061:
3058:
3056:
3053:
3052:
3050:
3048:
3044:
3038:
3035:
3033:
3030:
3028:
3025:
3023:
3020:
3018:
3015:
3013:
3010:
3008:
3005:
3001:
2998:
2996:
2993:
2991:
2988:
2987:
2986:
2983:
2981:
2978:
2976:
2973:
2971:
2968:
2966:
2963:
2961:
2958:
2954:
2951:
2950:
2949:
2946:
2942:
2941:of arithmetic
2939:
2938:
2937:
2934:
2930:
2927:
2925:
2922:
2920:
2917:
2915:
2912:
2910:
2907:
2906:
2905:
2902:
2898:
2895:
2893:
2890:
2889:
2888:
2885:
2884:
2882:
2880:
2876:
2870:
2867:
2865:
2862:
2860:
2857:
2855:
2852:
2849:
2848:from ZFC
2845:
2842:
2840:
2837:
2831:
2828:
2827:
2826:
2823:
2821:
2818:
2816:
2813:
2812:
2811:
2808:
2806:
2803:
2801:
2798:
2796:
2793:
2791:
2788:
2786:
2783:
2781:
2778:
2777:
2775:
2773:
2769:
2759:
2758:
2754:
2753:
2748:
2747:non-Euclidean
2745:
2741:
2738:
2736:
2733:
2731:
2730:
2726:
2725:
2723:
2720:
2719:
2717:
2713:
2709:
2706:
2704:
2701:
2700:
2699:
2695:
2691:
2688:
2687:
2686:
2682:
2678:
2675:
2673:
2670:
2668:
2665:
2663:
2660:
2658:
2655:
2653:
2650:
2649:
2647:
2643:
2642:
2640:
2635:
2629:
2624:Example
2621:
2613:
2608:
2607:
2606:
2603:
2601:
2598:
2594:
2591:
2589:
2586:
2584:
2581:
2579:
2576:
2575:
2574:
2571:
2569:
2566:
2564:
2561:
2559:
2556:
2552:
2549:
2547:
2544:
2543:
2542:
2539:
2535:
2532:
2530:
2527:
2525:
2522:
2520:
2517:
2516:
2515:
2512:
2510:
2507:
2503:
2500:
2498:
2495:
2493:
2490:
2489:
2488:
2485:
2481:
2478:
2476:
2473:
2471:
2468:
2466:
2463:
2461:
2458:
2456:
2453:
2452:
2451:
2448:
2446:
2443:
2441:
2438:
2436:
2433:
2429:
2426:
2424:
2421:
2419:
2416:
2414:
2411:
2410:
2409:
2406:
2404:
2401:
2399:
2396:
2394:
2391:
2387:
2384:
2382:
2381:by definition
2379:
2378:
2377:
2374:
2370:
2367:
2366:
2365:
2362:
2360:
2357:
2355:
2352:
2350:
2347:
2345:
2342:
2341:
2338:
2335:
2333:
2329:
2324:
2318:
2314:
2304:
2301:
2299:
2296:
2294:
2291:
2289:
2286:
2284:
2281:
2279:
2276:
2274:
2271:
2269:
2268:KripkeâPlatek
2266:
2264:
2261:
2257:
2254:
2252:
2249:
2248:
2247:
2244:
2243:
2241:
2237:
2229:
2226:
2225:
2224:
2221:
2219:
2216:
2212:
2209:
2208:
2207:
2204:
2202:
2199:
2197:
2194:
2192:
2189:
2187:
2184:
2181:
2177:
2173:
2170:
2166:
2163:
2161:
2158:
2156:
2153:
2152:
2151:
2147:
2144:
2143:
2141:
2139:
2135:
2131:
2123:
2120:
2118:
2115:
2113:
2112:constructible
2110:
2109:
2108:
2105:
2103:
2100:
2098:
2095:
2093:
2090:
2088:
2085:
2083:
2080:
2078:
2075:
2073:
2070:
2068:
2065:
2063:
2060:
2058:
2055:
2053:
2050:
2048:
2045:
2044:
2042:
2040:
2035:
2027:
2024:
2022:
2019:
2017:
2014:
2012:
2009:
2007:
2004:
2002:
1999:
1998:
1996:
1992:
1989:
1987:
1984:
1983:
1982:
1979:
1977:
1974:
1972:
1969:
1967:
1964:
1962:
1958:
1954:
1952:
1949:
1945:
1942:
1941:
1940:
1937:
1936:
1933:
1930:
1928:
1924:
1914:
1911:
1909:
1906:
1904:
1901:
1899:
1896:
1894:
1891:
1889:
1886:
1882:
1879:
1878:
1877:
1874:
1870:
1865:
1864:
1863:
1860:
1859:
1857:
1855:
1851:
1843:
1840:
1838:
1835:
1833:
1830:
1829:
1828:
1825:
1823:
1820:
1818:
1815:
1813:
1810:
1808:
1805:
1803:
1800:
1798:
1795:
1794:
1792:
1790:
1789:Propositional
1786:
1780:
1777:
1775:
1772:
1770:
1767:
1765:
1762:
1760:
1757:
1755:
1752:
1748:
1745:
1744:
1743:
1740:
1738:
1735:
1733:
1730:
1728:
1725:
1723:
1720:
1718:
1717:Logical truth
1715:
1713:
1710:
1709:
1707:
1705:
1701:
1698:
1696:
1692:
1686:
1683:
1681:
1678:
1676:
1673:
1671:
1668:
1666:
1663:
1661:
1657:
1653:
1649:
1647:
1644:
1642:
1639:
1637:
1633:
1630:
1629:
1627:
1625:
1619:
1614:
1608:
1605:
1603:
1600:
1598:
1595:
1593:
1590:
1588:
1585:
1583:
1580:
1578:
1575:
1573:
1570:
1568:
1565:
1563:
1560:
1558:
1555:
1553:
1550:
1546:
1543:
1542:
1541:
1538:
1537:
1535:
1531:
1527:
1520:
1515:
1513:
1508:
1506:
1501:
1500:
1497:
1491:
1488:
1486:
1483:
1481:
1478:
1477:
1473:
1471:
1453:
1444:
1441:
1438:
1432:
1429:
1426:
1414:
1411:
1408:
1402:
1399:
1396:
1390:
1387:
1381:
1378:
1372:
1369:
1351:
1348:
1345:
1339:
1336:
1333:
1330:
1324:
1321:
1308:
1307:
1306:
1289:
1283:
1280:
1277:
1271:
1268:
1265:
1253:
1250:
1247:
1241:
1238:
1235:
1229:
1226:
1220:
1217:
1211:
1208:
1198:
1197:
1196:
1194:
1175:
1172:
1169:
1166:
1160:
1157:
1154:
1151:
1148:
1142:
1139:
1133:
1130:
1124:
1121:
1111:
1110:
1109:
1107:
1103:
1084:
1081:
1078:
1075:
1069:
1066:
1063:
1060:
1057:
1051:
1048:
1042:
1039:
1029:
1028:
1027:
1025:
1021:
1017:
998:
995:
992:
986:
980:
974:
971:
968:
965:
962:
956:
953:
947:
944:
934:
933:
932:
930:
926:
922:
918:
913:
911:
907:
903:
899:
895:
891:
887:
883:
879:
875:
871:
867:
863:
859:
855:
849:
843:
839:
835:
831:
827:
824:
820:
816:
813:
810:
809:
808:
806:
802:
798:
794:
790:
786:
782:
778:
774:
770:
766:
762:
758:
754:
750:
746:
742:
738:
734:
730:
725:
723:
719:
715:
707:
702:
698:
694:
691:
687:
683:
679:
675:
671:
667:
663:
659:
658:
657:
655:
651:
647:
639:
635:
631:
627:
623:
619:
615:
611:
610:
609:
607:
603:
599:
595:
591:
587:
583:
579:
576:
572:
568:
564:
559:
558:formal system
555:
550:
548:
544:
540:
536:
532:
528:
524:
520:
516:
511:
506:
501:
496:
492:
488:
484:
480:
475:
473:
469:
465:
462:
458:
454:
450:
446:
442:
438:
434:
430:
426:
425:
420:
416:
412:
408:
404:
396:
394:
392:
373:
352:
346:
340:
328:
322:
316:
310:
294:
288:
278:
277:
276:
274:
270:
266:
262:
258:
254:
249:
247:
243:
239:
235:
231:
227:
223:
219:
215:
211:
207:
203:
199:
195:
191:
187:
183:
179:
175:
170:
168:
164:
160:
156:
152:
148:
144:
140:
129:
126:
118:
115:December 2009
107:
104:
100:
97:
93:
90:
86:
83:
79:
76: â
75:
71:
70:Find sources:
64:
60:
56:
50:
49:
45:
40:This article
38:
34:
29:
28:
19:
3263:Model theory
3234:
3032:Ultraproduct
2879:Model theory
2844:Independence
2780:Formal proof
2772:Proof theory
2755:
2728:
2685:real numbers
2657:second-order
2568:Substitution
2491:
2445:Metalanguage
2386:conservative
2359:Axiom schema
2303:Constructive
2273:MorseâKelley
2239:Set theories
2218:Aleph number
2211:inaccessible
2117:Grothendieck
2001:intersection
1888:Higher-order
1876:Second-order
1822:Truth tables
1779:Venn diagram
1562:Formal proof
1468:
1304:
1192:
1190:
1105:
1101:
1099:
1023:
1019:
1015:
1013:
928:
914:
905:
901:
897:
893:
889:
885:
881:
873:
869:
865:
861:
853:
850:
847:
841:
837:
833:
829:
825:
818:
814:
804:
800:
796:
792:
788:
784:
780:
776:
772:
768:
764:
760:
756:
752:
748:
744:
740:
732:
728:
726:
711:
700:
696:
692:
685:
681:
677:
673:
669:
665:
661:
653:
649:
645:
643:
637:
633:
629:
625:
621:
617:
613:
605:
601:
597:
589:
585:
581:
577:
571:there exists
566:
562:
551:
546:
542:
538:
530:
526:
522:
518:
514:
509:
504:
499:
494:
490:
486:
482:
478:
476:
471:
467:
463:
456:
452:
448:
444:
440:
436:
432:
428:
422:
418:
414:
410:
406:
400:
388:
272:
268:
264:
259:modelled by
256:
252:
250:
237:
233:
229:
225:
221:
217:
213:
209:
205:
201:
193:
189:
185:
173:
171:
154:
150:
146:
139:formal logic
136:
121:
112:
102:
95:
88:
81:
69:
53:Please help
41:
3142:Type theory
3090:undecidable
3022:Truth value
2909:equivalence
2588:non-logical
2201:Enumeration
2191:Isomorphism
2138:cardinality
2122:Von Neumann
2087:Ultrafilter
2052:Uncountable
1986:equivalence
1903:Quantifiers
1893:Fixed-point
1862:First-order
1742:Consistency
1727:Proposition
1704:Traditional
1675:Lindström's
1665:Compactness
1607:Type theory
1552:Cardinality
821:, for some
718:metalogical
424:composition
198:typed logic
143:mathematics
2953:elementary
2646:arithmetic
2514:Quantifier
2492:functional
2364:Expression
2082:Transitive
2026:identities
2011:complement
1944:hereditary
1927:Set theory
880:over each
714:predicates
668:, for all
565:(or every
85:newspapers
3224:Supertask
3127:Recursion
3085:decidable
2919:saturated
2897:of models
2820:deductive
2815:axiomatic
2735:Hilbert's
2722:Euclidean
2703:canonical
2626:axiomatic
2558:Signature
2487:Predicate
2376:Extension
2298:Ackermann
2223:Operation
2102:Universal
2092:Recursive
2067:Singleton
2062:Inhabited
2047:Countable
2037:Types of
2021:power set
1991:partition
1908:Predicate
1854:Predicate
1769:Syllogism
1759:Soundness
1732:Inference
1722:Tautology
1624:paradoxes
1442:∈
1436:→
1430:∈
1421:→
1394:∀
1385:∀
1376:∃
1367:∀
1361:→
1328:∃
1319:∀
1281:∈
1275:→
1269:∈
1260:→
1233:∀
1224:∀
1215:∃
1206:∀
1170:∈
1164:→
1158:∈
1146:∀
1137:∀
1128:∃
1119:∀
1079:∈
1073:→
1067:∈
1055:∀
1046:∃
1037:∀
993:∈
978:→
972:∈
960:∀
951:∃
942:∀
747:of type (
594:predicate
353:∈
182:given any
42:does not
3257:Category
3209:Logicism
3202:timeline
3178:Concrete
3037:Validity
3007:T-schema
3000:Kripke's
2995:Tarski's
2990:semantic
2980:Strength
2929:submodel
2924:spectrum
2892:function
2740:Tarski's
2729:Elements
2716:geometry
2672:Robinson
2593:variable
2578:function
2551:spectrum
2541:Sentence
2497:variable
2440:Language
2393:Relation
2354:Automata
2344:Alphabet
2328:language
2182:-jection
2160:codomain
2146:Function
2107:Universe
2077:Infinite
1981:Relation
1764:Validity
1754:Argument
1652:theorem,
1474:See also
828:of type
817:of type
791:of type
767:) means
737:codomain
672:of type
664:of type
660:For all
624:of type
616:of type
612:For all
525:of type
521:for all
489:for all
391:function
267:of type
248:symbol.
246:constant
214:codomain
167:function
155:mappings
3151:Related
2948:Diagram
2846: (
2825:Hilbert
2810:Systems
2805:Theorem
2683:of the
2628:systems
2408:Formula
2403:Grammar
2319: (
2263:General
1976:Forcing
1961:Element
1881:Monadic
1656:paradox
1597:Theorem
1533:General
876:. Then
812:For all
554:theorem
535:subtype
461:for all
184:symbol
161:. In a
99:scholar
63:removed
48:sources
2914:finite
2677:Skolem
2630:
2605:Theory
2573:Symbol
2563:String
2546:atomic
2423:ground
2418:closed
2413:atomic
2369:ground
2332:syntax
2228:binary
2155:domain
2072:Finite
1837:finite
1695:Logics
1654:
1602:Theory
858:schema
823:unique
575:unique
421:, the
206:domain
101:
94:
87:
80:
72:
2904:Model
2652:Peano
2509:Proof
2349:Arity
2278:Naive
2165:image
2097:Fuzzy
2057:Empty
2006:union
1951:Class
1592:Model
1582:Lemma
1540:Axiom
739:type
533:is a
216:type
208:type
176:in a
163:model
149:, or
106:JSTOR
92:books
3027:Type
2830:list
2634:list
2611:list
2600:Term
2534:rank
2428:open
2322:list
2134:Maps
2039:sets
1898:Free
1868:list
1618:list
1545:list
775:) =
735:and
600:and
517:) =
485:) =
459:)),
447:) =
431:and
409:and
261:sets
255:and
242:zero
212:and
145:, a
78:news
46:any
44:cite
2714:of
2696:of
2644:of
2176:Sur
2150:Map
1957:Ur-
1939:Set
1108::
1026::
919:in
537:of
427:of
137:In
57:by
3259::
3100:NP
2724::
2718::
2648::
2325:),
2180:Bi
2172:In
844:).
832:,
703:).
695:=
688:)
676:,
640:),
628:,
573:a
443:)(
439:â
417:â
295::=
228:,
200:,
169:.
3180:/
3095:P
2850:)
2636:)
2632:(
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