3227:
22:
713:), 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.
1458:
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
1453:
840:
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
1289:
1175:
998:
1300:
1084:
549:
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
373:
1190:
1606:
853:; 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
1103:
2281:
1448:{\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)).}
926:
394:
that allows one to introduce new predicate symbols, one will also want to be able to introduce new function symbols. Given the function symbols
837:(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.)
538:
that satisfies the same equation; there are additional function symbols associated with other ways of constructing new types out of old ones.
229:. One can similarly define function symbols of more than one variable, analogous to functions of more than one variable; a function symbol in
1021:
142:, is a logical symbol that may be applied to an object term to produce another object term. Functional predicates are also sometimes called
2364:
1505:
845:
predicate, specifically one that satisfies the proposition above. This may seem to be a problem if you wish to specify a proposition
94:
2678:
66:
2836:
43:
1624:
73:
2691:
2014:
710:
2696:
2686:
2423:
2276:
1629:
1620:
80:
2832:
113:
2174:
51:
2929:
2673:
1498:
270:
2234:
1927:
909:
62:
1668:
3190:
2892:
2655:
2650:
2475:
1896:
1580:
47:
3185:
2968:
2885:
2598:
2529:
2406:
1648:
3110:
2936:
2622:
2256:
1855:
2988:
2983:
2593:
2332:
2261:
1590:
1491:
905:
455:. Of course, the right side of this equation doesn't make sense in typed logic unless the domain type of
2917:
2507:
1901:
1869:
1560:
3207:
3156:
3053:
2551:
2512:
1989:
1634:
1284:{\displaystyle \forall A,\exists B,\forall C,\forall D,P(C,D)\rightarrow (C\in A\rightarrow D\in B).}
1663:
3048:
2978:
2517:
2369:
2352:
2075:
1555:
32:
87:
2880:
2857:
2818:
2704:
2645:
2291:
2211:
2055:
1999:
1612:
1468:
36:
3170:
2897:
2875:
2842:
2735:
2581:
2566:
2539:
2490:
2374:
2309:
2134:
2100:
2095:
1969:
1800:
1777:
379:
155:
3251:
3100:
2953:
2745:
2463:
2199:
2105:
1964:
1949:
1830:
1805:
382:
with domain and codomain . It is a requirement of a consistent model that = whenever = .
147:
3073:
3035:
2912:
2716:
2556:
2458:
2286:
2244:
2143:
2110:
1974:
1762:
1673:
913:
898:
866:
811:
563:
412:
1294:
This is almost correct, but it applies to too many predicates; what we actually want is:
8:
3202:
3093:
3078:
3058:
3015:
2902:
2852:
2778:
2723:
2660:
2453:
2448:
2396:
2164:
2153:
1825:
1725:
1653:
1644:
1640:
1575:
1570:
3231:
3000:
2963:
2948:
2941:
2924:
2728:
2710:
2576:
2502:
2485:
2438:
2251:
2160:
1994:
1979:
1939:
1891:
1876:
1864:
1820:
1795:
1565:
1514:
1473:
2184:
701:
Many treatments of predicate logic don't allow functional predicates, only relational
3226:
3166:
2973:
2783:
2773:
2665:
2546:
2381:
2357:
2138:
2122:
2027:
2004:
1881:
1850:
1815:
1710:
1545:
702:
582:
249:
466:
One also gets certain function symbols automatically. In untyped logic, there is an
3180:
3175:
3068:
3025:
2847:
2808:
2803:
2788:
2614:
2571:
2468:
2266:
2216:
1790:
1752:
1478:
234:
3161:
3151:
3105:
3088:
3043:
3005:
2907:
2827:
2634:
2561:
2534:
2522:
2428:
2342:
2316:
2271:
2239:
2040:
1842:
1785:
1735:
1700:
1658:
1170:{\displaystyle \forall A,\exists B,\forall C,\forall D,C\in A\rightarrow D\in B.}
391:
166:
541:
Additionally, one can define functional predicates after proving an appropriate
3146:
3125:
3083:
3063:
2958:
2813:
2411:
2401:
2391:
2386:
2320:
2194:
2070:
1959:
1954:
1932:
1533:
846:
796:). To be able to make the same deductions, you need an additional proposition:
678:
3245:
3120:
2798:
2305:
2090:
2080:
2050:
2035:
1705:
546:
151:
3020:
2867:
2768:
2760:
2640:
2588:
2497:
2433:
2416:
2347:
2206:
2065:
1767:
1550:
559:
127:
993:{\displaystyle \forall A,\exists B,\forall C,C\in A\rightarrow F(C)\in B.}
3130:
3010:
2189:
2179:
2126:
1810:
1730:
1715:
1595:
1540:
186:
131:
2060:
1915:
1886:
1692:
1079:{\displaystyle \forall A,\exists B,\forall C,C\in A\rightarrow D\in B.}
230:
3212:
3115:
2168:
2085:
2045:
2009:
1945:
1757:
1747:
1720:
1483:
706:
523:
170:
776:) would appear in a statement, you can replace it with a new symbol
21:
3197:
2995:
2443:
2148:
1742:
725:
916:.) This schema states (in one form), for any functional predicate
2793:
1585:
800:
542:
449:
185:) is again a symbol representing an object in that language. In
901:
of the uniqueness condition for a functional predicate above.
2337:
1683:
1528:
881:
is quantified over, or at the beginning of the statement if
240:
Now consider a model of the formal language, with the types
705:. This is useful, for example, in the context of proving
463:, so this is required for the composition to be defined.
696:
530:, then there is an inclusion predicate of domain type
1303:
1193:
1106:
1024:
929:
273:
385:
1447:
1283:
1169:
1078:
992:
367:
3243:
1499:
368:{\displaystyle :={\big \{}(,):\in {\big \}},}
357:
288:
225:) is a symbol representing an object of type
885:is free), and guard the quantification with
849:that applies only to functional predicates
732:, then it can be replaced with a predicate
573:, then you can introduce a function symbol
50:. Unsourced material may be challenged and
1691:
1506:
1492:
402:, one can introduce a new function symbol
154:, a function symbol will be modelled by a
1089:Of course, this statement isn't correct;
633:then you can introduce a function symbol
114:Learn how and when to remove this message
897:). Finally, make the entire statement a
177:representing an object in the language,
3244:
1513:
1487:
873:immediately after the corresponding
48:adding citations to reliable sources
15:
1093:must be quantified over just after
697:Doing without functional predicates
486:, there is an identity predicate id
13:
1382:
1373:
1364:
1355:
1316:
1307:
1221:
1212:
1203:
1194:
1134:
1125:
1116:
1107:
1043:
1034:
1025:
948:
939:
930:
593:. So if there is such a predicate
260:modelled by an element in . Then
148:additional meanings in mathematics
14:
3263:
482:. In typed logic, given any type
3225:
386:Introducing new function symbols
348:
20:
1184:to guard this quantification:
711:Gödel's incompleteness theorems
213:representing an object of type
1439:
1436:
1424:
1412:
1409:
1406:
1394:
1352:
1349:
1346:
1343:
1331:
1304:
1275:
1263:
1251:
1248:
1245:
1233:
1152:
1061:
978:
972:
966:
904:Let us take as an example the
877:is introduced (that is, after
784:and include another statement
492:with domain and codomain type
352:
344:
338:
332:
326:
323:
320:
314:
308:
302:
296:
293:
280:
274:
1:
3186:History of mathematical logic
459:matches the codomain type of
3111:Primitive recursive function
581:will itself be a relational
577:to indicate this. Note that
193:is a functional symbol with
7:
1462:
1011:) with some other variable
910:ZermeloâFraenkel set theory
906:axiom schema of replacement
264:can be modelled by the set
169:is a functional symbol if,
10:
3268:
2175:SchröderâBernstein theorem
1902:Monadic predicate calculus
1561:Foundations of mathematics
569:satisfying some condition
545:. (If you're working in a
3221:
3208:Philosophy of mathematics
3157:Automated theorem proving
3139:
3034:
2866:
2759:
2611:
2328:
2304:
2282:Von NeumannâBernaysâGödel
2227:
2121:
2025:
1923:
1914:
1841:
1776:
1682:
1604:
1521:
161:Specifically, the symbol
1180:We still must introduce
130:and related branches of
2858:Self-verifying theories
2679:Tarski's axiomatization
1630:Tarski's undefinability
1625:incompleteness theorems
1469:Function symbol (logic)
1003:First, we must replace
3232:Mathematics portal
2843:Proof of impossibility
2491:propositional variable
1801:Propositional calculus
1449:
1285:
1171:
1080:
994:
861:) with a new variable
369:
233:variables is simply a
63:"Functional predicate"
3101:Kolmogorov complexity
3054:Computably enumerable
2954:Model complete theory
2746:Principia Mathematica
1806:Propositional formula
1635:BanachâTarski paradox
1459:produced at the end.
1450:
1286:
1172:
1081:
995:
912:. (This example uses
470:id that satisfies id(
370:
252:and and each symbol
209:if, given any symbol
3049:ChurchâTuring thesis
3036:Computability theory
2245:continuum hypothesis
1763:Square of opposition
1621:Gödel's completeness
1301:
1191:
1104:
1022:
927:
914:mathematical symbols
899:material consequence
867:universally quantify
558:of a certain type),
271:
146:, but that term has
136:functional predicate
44:improve this article
3203:Mathematical object
3094:P versus NP problem
3059:Computable function
2853:Reverse mathematics
2779:Logical consequence
2656:primitive recursive
2651:elementary function
2424:Free/bound variable
2277:TarskiâGrothendieck
1796:Logical connectives
1726:Logical equivalence
1576:Logical consequence
3001:Transfer principle
2964:Semantics of logic
2949:Categorical theory
2925:Non-standard model
2439:Logical connective
1566:Information theory
1515:Mathematical logic
1474:Logical connective
1445:
1281:
1167:
1076:
990:
709:theorems (such as
641:and codomain type
609:, for some unique
534:and codomain type
468:identity predicate
390:In a treatment of
378:which is simply a
365:
3239:
3238:
3171:Abstract category
2974:Theories of truth
2784:Rule of inference
2774:Natural deduction
2755:
2754:
2300:
2299:
2005:Cartesian product
1910:
1909:
1816:Many-valued logic
1791:Boolean functions
1674:Russell's paradox
1649:diagonal argument
1546:First-order logic
920:in one variable:
716:Specifically, if
496:; it satisfies id
124:
123:
116:
98:
3259:
3230:
3229:
3181:History of logic
3176:Category of sets
3069:Decision problem
2848:Ordinal analysis
2789:Sequent calculus
2687:Boolean algebras
2627:
2626:
2601:
2572:logical/constant
2326:
2325:
2312:
2235:ZermeloâFraenkel
1986:Set operations:
1921:
1920:
1858:
1689:
1688:
1669:LöwenheimâSkolem
1556:Formal semantics
1508:
1501:
1494:
1485:
1484:
1479:Logical constant
1454:
1452:
1451:
1446:
1290:
1288:
1287:
1282:
1176:
1174:
1173:
1168:
1085:
1083:
1082:
1077:
999:
997:
996:
991:
768:. Then whenever
744:). Intuitively,
720:has domain type
645:that satisfies:
518:. Similarly, if
374:
372:
371:
366:
361:
360:
351:
292:
291:
119:
112:
108:
105:
99:
97:
56:
24:
16:
3267:
3266:
3262:
3261:
3260:
3258:
3257:
3256:
3242:
3241:
3240:
3235:
3224:
3217:
3162:Category theory
3152:Algebraic logic
3135:
3106:Lambda calculus
3044:Church encoding
3030:
3006:Truth predicate
2862:
2828:Complete theory
2751:
2620:
2616:
2612:
2607:
2599:
2319: and
2315:
2310:
2296:
2272:New Foundations
2240:axiom of choice
2223:
2185:Gödel numbering
2125: and
2117:
2021:
1906:
1856:
1837:
1786:Boolean algebra
1772:
1736:Equiconsistency
1701:Classical logic
1678:
1659:Halting problem
1647: and
1623: and
1611: and
1610:
1605:Theorems (
1600:
1517:
1512:
1465:
1302:
1299:
1298:
1192:
1189:
1188:
1105:
1102:
1101:
1023:
1020:
1019:
928:
925:
924:
843:special kind of
699:
637:of domain type
597:and a theorem:
585:involving both
501:
491:
392:predicate logic
388:
356:
355:
347:
287:
286:
272:
269:
268:
167:formal language
140:function symbol
120:
109:
103:
100:
57:
55:
41:
25:
12:
11:
5:
3265:
3255:
3254:
3237:
3236:
3222:
3219:
3218:
3216:
3215:
3210:
3205:
3200:
3195:
3194:
3193:
3183:
3178:
3173:
3164:
3159:
3154:
3149:
3147:Abstract logic
3143:
3141:
3137:
3136:
3134:
3133:
3128:
3126:Turing machine
3123:
3118:
3113:
3108:
3103:
3098:
3097:
3096:
3091:
3086:
3081:
3076:
3066:
3064:Computable set
3061:
3056:
3051:
3046:
3040:
3038:
3032:
3031:
3029:
3028:
3023:
3018:
3013:
3008:
3003:
2998:
2993:
2992:
2991:
2986:
2981:
2971:
2966:
2961:
2959:Satisfiability
2956:
2951:
2946:
2945:
2944:
2934:
2933:
2932:
2922:
2921:
2920:
2915:
2910:
2905:
2900:
2890:
2889:
2888:
2883:
2876:Interpretation
2872:
2870:
2864:
2863:
2861:
2860:
2855:
2850:
2845:
2840:
2830:
2825:
2824:
2823:
2822:
2821:
2811:
2806:
2796:
2791:
2786:
2781:
2776:
2771:
2765:
2763:
2757:
2756:
2753:
2752:
2750:
2749:
2741:
2740:
2739:
2738:
2733:
2732:
2731:
2726:
2721:
2701:
2700:
2699:
2697:minimal axioms
2694:
2683:
2682:
2681:
2670:
2669:
2668:
2663:
2658:
2653:
2648:
2643:
2630:
2628:
2609:
2608:
2606:
2605:
2604:
2603:
2591:
2586:
2585:
2584:
2579:
2574:
2569:
2559:
2554:
2549:
2544:
2543:
2542:
2537:
2527:
2526:
2525:
2520:
2515:
2510:
2500:
2495:
2494:
2493:
2488:
2483:
2473:
2472:
2471:
2466:
2461:
2456:
2451:
2446:
2436:
2431:
2426:
2421:
2420:
2419:
2414:
2409:
2404:
2394:
2389:
2387:Formation rule
2384:
2379:
2378:
2377:
2372:
2362:
2361:
2360:
2350:
2345:
2340:
2335:
2329:
2323:
2306:Formal systems
2302:
2301:
2298:
2297:
2295:
2294:
2289:
2284:
2279:
2274:
2269:
2264:
2259:
2254:
2249:
2248:
2247:
2242:
2231:
2229:
2225:
2224:
2222:
2221:
2220:
2219:
2209:
2204:
2203:
2202:
2195:Large cardinal
2192:
2187:
2182:
2177:
2172:
2158:
2157:
2156:
2151:
2146:
2131:
2129:
2119:
2118:
2116:
2115:
2114:
2113:
2108:
2103:
2093:
2088:
2083:
2078:
2073:
2068:
2063:
2058:
2053:
2048:
2043:
2038:
2032:
2030:
2023:
2022:
2020:
2019:
2018:
2017:
2012:
2007:
2002:
1997:
1992:
1984:
1983:
1982:
1977:
1967:
1962:
1960:Extensionality
1957:
1955:Ordinal number
1952:
1942:
1937:
1936:
1935:
1924:
1918:
1912:
1911:
1908:
1907:
1905:
1904:
1899:
1894:
1889:
1884:
1879:
1874:
1873:
1872:
1862:
1861:
1860:
1847:
1845:
1839:
1838:
1836:
1835:
1834:
1833:
1828:
1823:
1813:
1808:
1803:
1798:
1793:
1788:
1782:
1780:
1774:
1773:
1771:
1770:
1765:
1760:
1755:
1750:
1745:
1740:
1739:
1738:
1728:
1723:
1718:
1713:
1708:
1703:
1697:
1695:
1686:
1680:
1679:
1677:
1676:
1671:
1666:
1661:
1656:
1651:
1639:Cantor's
1637:
1632:
1627:
1617:
1615:
1602:
1601:
1599:
1598:
1593:
1588:
1583:
1578:
1573:
1568:
1563:
1558:
1553:
1548:
1543:
1538:
1537:
1536:
1525:
1523:
1519:
1518:
1511:
1510:
1503:
1496:
1488:
1482:
1481:
1476:
1471:
1464:
1461:
1456:
1455:
1444:
1441:
1438:
1435:
1432:
1429:
1426:
1423:
1420:
1417:
1414:
1411:
1408:
1405:
1402:
1399:
1396:
1393:
1390:
1387:
1384:
1381:
1378:
1375:
1372:
1369:
1366:
1363:
1360:
1357:
1354:
1351:
1348:
1345:
1342:
1339:
1336:
1333:
1330:
1327:
1324:
1321:
1318:
1315:
1312:
1309:
1306:
1292:
1291:
1280:
1277:
1274:
1271:
1268:
1265:
1262:
1259:
1256:
1253:
1250:
1247:
1244:
1241:
1238:
1235:
1232:
1229:
1226:
1223:
1220:
1217:
1214:
1211:
1208:
1205:
1202:
1199:
1196:
1178:
1177:
1166:
1163:
1160:
1157:
1154:
1151:
1148:
1145:
1142:
1139:
1136:
1133:
1130:
1127:
1124:
1121:
1118:
1115:
1112:
1109:
1087:
1086:
1075:
1072:
1069:
1066:
1063:
1060:
1057:
1054:
1051:
1048:
1045:
1042:
1039:
1036:
1033:
1030:
1027:
1001:
1000:
989:
986:
983:
980:
977:
974:
971:
968:
965:
962:
959:
956:
953:
950:
947:
944:
941:
938:
935:
932:
835:
834:
698:
695:
694:
693:
679:if and only if
631:
630:
497:
487:
424:, satisfying (
387:
384:
376:
375:
364:
359:
354:
350:
346:
343:
340:
337:
334:
331:
328:
325:
322:
319:
316:
313:
310:
307:
304:
301:
298:
295:
290:
285:
282:
279:
276:
122:
121:
28:
26:
19:
9:
6:
4:
3:
2:
3264:
3253:
3250:
3249:
3247:
3234:
3233:
3228:
3220:
3214:
3211:
3209:
3206:
3204:
3201:
3199:
3196:
3192:
3189:
3188:
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3179:
3177:
3174:
3172:
3168:
3165:
3163:
3160:
3158:
3155:
3153:
3150:
3148:
3145:
3144:
3142:
3138:
3132:
3129:
3127:
3124:
3122:
3121:Recursive set
3119:
3117:
3114:
3112:
3109:
3107:
3104:
3102:
3099:
3095:
3092:
3090:
3087:
3085:
3082:
3080:
3077:
3075:
3072:
3071:
3070:
3067:
3065:
3062:
3060:
3057:
3055:
3052:
3050:
3047:
3045:
3042:
3041:
3039:
3037:
3033:
3027:
3024:
3022:
3019:
3017:
3014:
3012:
3009:
3007:
3004:
3002:
2999:
2997:
2994:
2990:
2987:
2985:
2982:
2980:
2977:
2976:
2975:
2972:
2970:
2967:
2965:
2962:
2960:
2957:
2955:
2952:
2950:
2947:
2943:
2940:
2939:
2938:
2935:
2931:
2930:of arithmetic
2928:
2927:
2926:
2923:
2919:
2916:
2914:
2911:
2909:
2906:
2904:
2901:
2899:
2896:
2895:
2894:
2891:
2887:
2884:
2882:
2879:
2878:
2877:
2874:
2873:
2871:
2869:
2865:
2859:
2856:
2854:
2851:
2849:
2846:
2844:
2841:
2838:
2837:from ZFC
2834:
2831:
2829:
2826:
2820:
2817:
2816:
2815:
2812:
2810:
2807:
2805:
2802:
2801:
2800:
2797:
2795:
2792:
2790:
2787:
2785:
2782:
2780:
2777:
2775:
2772:
2770:
2767:
2766:
2764:
2762:
2758:
2748:
2747:
2743:
2742:
2737:
2736:non-Euclidean
2734:
2730:
2727:
2725:
2722:
2720:
2719:
2715:
2714:
2712:
2709:
2708:
2706:
2702:
2698:
2695:
2693:
2690:
2689:
2688:
2684:
2680:
2677:
2676:
2675:
2671:
2667:
2664:
2662:
2659:
2657:
2654:
2652:
2649:
2647:
2644:
2642:
2639:
2638:
2636:
2632:
2631:
2629:
2624:
2618:
2613:Example
2610:
2602:
2597:
2596:
2595:
2592:
2590:
2587:
2583:
2580:
2578:
2575:
2573:
2570:
2568:
2565:
2564:
2563:
2560:
2558:
2555:
2553:
2550:
2548:
2545:
2541:
2538:
2536:
2533:
2532:
2531:
2528:
2524:
2521:
2519:
2516:
2514:
2511:
2509:
2506:
2505:
2504:
2501:
2499:
2496:
2492:
2489:
2487:
2484:
2482:
2479:
2478:
2477:
2474:
2470:
2467:
2465:
2462:
2460:
2457:
2455:
2452:
2450:
2447:
2445:
2442:
2441:
2440:
2437:
2435:
2432:
2430:
2427:
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2422:
2418:
2415:
2413:
2410:
2408:
2405:
2403:
2400:
2399:
2398:
2395:
2393:
2390:
2388:
2385:
2383:
2380:
2376:
2373:
2371:
2370:by definition
2368:
2367:
2366:
2363:
2359:
2356:
2355:
2354:
2351:
2349:
2346:
2344:
2341:
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2336:
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2331:
2330:
2327:
2324:
2322:
2318:
2313:
2307:
2303:
2293:
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2285:
2283:
2280:
2278:
2275:
2273:
2270:
2268:
2265:
2263:
2260:
2258:
2257:KripkeâPlatek
2255:
2253:
2250:
2246:
2243:
2241:
2238:
2237:
2236:
2233:
2232:
2230:
2226:
2218:
2215:
2214:
2213:
2210:
2208:
2205:
2201:
2198:
2197:
2196:
2193:
2191:
2188:
2186:
2183:
2181:
2178:
2176:
2173:
2170:
2166:
2162:
2159:
2155:
2152:
2150:
2147:
2145:
2142:
2141:
2140:
2136:
2133:
2132:
2130:
2128:
2124:
2120:
2112:
2109:
2107:
2104:
2102:
2101:constructible
2099:
2098:
2097:
2094:
2092:
2089:
2087:
2084:
2082:
2079:
2077:
2074:
2072:
2069:
2067:
2064:
2062:
2059:
2057:
2054:
2052:
2049:
2047:
2044:
2042:
2039:
2037:
2034:
2033:
2031:
2029:
2024:
2016:
2013:
2011:
2008:
2006:
2003:
2001:
1998:
1996:
1993:
1991:
1988:
1987:
1985:
1981:
1978:
1976:
1973:
1972:
1971:
1968:
1966:
1963:
1961:
1958:
1956:
1953:
1951:
1947:
1943:
1941:
1938:
1934:
1931:
1930:
1929:
1926:
1925:
1922:
1919:
1917:
1913:
1903:
1900:
1898:
1895:
1893:
1890:
1888:
1885:
1883:
1880:
1878:
1875:
1871:
1868:
1867:
1866:
1863:
1859:
1854:
1853:
1852:
1849:
1848:
1846:
1844:
1840:
1832:
1829:
1827:
1824:
1822:
1819:
1818:
1817:
1814:
1812:
1809:
1807:
1804:
1802:
1799:
1797:
1794:
1792:
1789:
1787:
1784:
1783:
1781:
1779:
1778:Propositional
1775:
1769:
1766:
1764:
1761:
1759:
1756:
1754:
1751:
1749:
1746:
1744:
1741:
1737:
1734:
1733:
1732:
1729:
1727:
1724:
1722:
1719:
1717:
1714:
1712:
1709:
1707:
1706:Logical truth
1704:
1702:
1699:
1698:
1696:
1694:
1690:
1687:
1685:
1681:
1675:
1672:
1670:
1667:
1665:
1662:
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1657:
1655:
1652:
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1646:
1642:
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1636:
1633:
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1628:
1626:
1622:
1619:
1618:
1616:
1614:
1608:
1603:
1597:
1594:
1592:
1589:
1587:
1584:
1582:
1579:
1577:
1574:
1572:
1569:
1567:
1564:
1562:
1559:
1557:
1554:
1552:
1549:
1547:
1544:
1542:
1539:
1535:
1532:
1531:
1530:
1527:
1526:
1524:
1520:
1516:
1509:
1504:
1502:
1497:
1495:
1490:
1489:
1486:
1480:
1477:
1475:
1472:
1470:
1467:
1466:
1460:
1442:
1433:
1430:
1427:
1421:
1418:
1415:
1403:
1400:
1397:
1391:
1388:
1385:
1379:
1376:
1370:
1367:
1361:
1358:
1340:
1337:
1334:
1328:
1325:
1322:
1319:
1313:
1310:
1297:
1296:
1295:
1278:
1272:
1269:
1266:
1260:
1257:
1254:
1242:
1239:
1236:
1230:
1227:
1224:
1218:
1215:
1209:
1206:
1200:
1197:
1187:
1186:
1185:
1183:
1164:
1161:
1158:
1155:
1149:
1146:
1143:
1140:
1137:
1131:
1128:
1122:
1119:
1113:
1110:
1100:
1099:
1098:
1096:
1092:
1073:
1070:
1067:
1064:
1058:
1055:
1052:
1049:
1046:
1040:
1037:
1031:
1028:
1018:
1017:
1016:
1014:
1010:
1006:
987:
984:
981:
975:
969:
963:
960:
957:
954:
951:
945:
942:
936:
933:
923:
922:
921:
919:
915:
911:
907:
902:
900:
896:
892:
888:
884:
880:
876:
872:
868:
864:
860:
856:
852:
848:
844:
838:
832:
828:
824:
820:
816:
813:
809:
805:
802:
799:
798:
797:
795:
791:
787:
783:
779:
775:
771:
767:
763:
759:
755:
751:
747:
743:
739:
735:
731:
727:
723:
719:
714:
712:
708:
704:
691:
687:
683:
680:
676:
672:
668:
664:
660:
656:
652:
648:
647:
646:
644:
640:
636:
628:
624:
620:
616:
612:
608:
604:
600:
599:
598:
596:
592:
588:
584:
580:
576:
572:
568:
565:
561:
557:
553:
548:
547:formal system
544:
539:
537:
533:
529:
525:
521:
517:
513:
509:
505:
500:
495:
490:
485:
481:
477:
473:
469:
464:
462:
458:
454:
451:
447:
443:
439:
435:
431:
427:
423:
419:
415:
414:
409:
405:
401:
397:
393:
383:
381:
362:
341:
335:
329:
317:
311:
305:
299:
283:
277:
267:
266:
265:
263:
259:
255:
251:
247:
243:
238:
236:
232:
228:
224:
220:
216:
212:
208:
204:
200:
196:
192:
188:
184:
180:
176:
172:
168:
164:
159:
157:
153:
149:
145:
141:
137:
133:
129:
118:
115:
107:
104:December 2009
96:
93:
89:
86:
82:
79:
75:
72:
68:
65: â
64:
60:
59:Find sources:
53:
49:
45:
39:
38:
34:
29:This article
27:
23:
18:
17:
3252:Model theory
3223:
3021:Ultraproduct
2868:Model theory
2833:Independence
2769:Formal proof
2761:Proof theory
2744:
2717:
2674:real numbers
2646:second-order
2557:Substitution
2480:
2434:Metalanguage
2375:conservative
2348:Axiom schema
2292:Constructive
2262:MorseâKelley
2228:Set theories
2207:Aleph number
2200:inaccessible
2106:Grothendieck
1990:intersection
1877:Higher-order
1865:Second-order
1811:Truth tables
1768:Venn diagram
1551:Formal proof
1457:
1293:
1181:
1179:
1094:
1090:
1088:
1012:
1008:
1004:
1002:
917:
903:
894:
890:
886:
882:
878:
874:
870:
862:
858:
854:
850:
842:
839:
836:
830:
826:
822:
818:
814:
807:
803:
793:
789:
785:
781:
777:
773:
769:
765:
761:
757:
753:
749:
745:
741:
737:
733:
729:
721:
717:
715:
700:
689:
685:
681:
674:
670:
666:
662:
658:
654:
650:
642:
638:
634:
632:
626:
622:
618:
614:
610:
606:
602:
594:
590:
586:
578:
574:
570:
566:
560:there exists
555:
551:
540:
535:
531:
527:
519:
515:
511:
507:
503:
498:
493:
488:
483:
479:
475:
471:
467:
465:
460:
456:
452:
445:
441:
437:
433:
429:
425:
421:
417:
411:
407:
403:
399:
395:
389:
377:
261:
257:
253:
248:modelled by
245:
241:
239:
226:
222:
218:
214:
210:
206:
202:
198:
194:
190:
182:
178:
174:
162:
160:
143:
139:
135:
128:formal logic
125:
110:
101:
91:
84:
77:
70:
58:
42:Please help
30:
3131:Type theory
3079:undecidable
3011:Truth value
2898:equivalence
2577:non-logical
2190:Enumeration
2180:Isomorphism
2127:cardinality
2111:Von Neumann
2076:Ultrafilter
2041:Uncountable
1975:equivalence
1892:Quantifiers
1882:Fixed-point
1851:First-order
1731:Consistency
1716:Proposition
1693:Traditional
1664:Lindström's
1654:Compactness
1596:Type theory
1541:Cardinality
810:, for some
707:metalogical
413:composition
187:typed logic
132:mathematics
2942:elementary
2635:arithmetic
2503:Quantifier
2481:functional
2353:Expression
2071:Transitive
2015:identities
2000:complement
1933:hereditary
1916:Set theory
869:over each
703:predicates
657:, for all
554:(or every
74:newspapers
3213:Supertask
3116:Recursion
3074:decidable
2908:saturated
2886:of models
2809:deductive
2804:axiomatic
2724:Hilbert's
2711:Euclidean
2692:canonical
2615:axiomatic
2547:Signature
2476:Predicate
2365:Extension
2287:Ackermann
2212:Operation
2091:Universal
2081:Recursive
2056:Singleton
2051:Inhabited
2036:Countable
2026:Types of
2010:power set
1980:partition
1897:Predicate
1843:Predicate
1758:Syllogism
1748:Soundness
1721:Inference
1711:Tautology
1613:paradoxes
1431:∈
1425:→
1419:∈
1410:→
1383:∀
1374:∀
1365:∃
1356:∀
1350:→
1317:∃
1308:∀
1270:∈
1264:→
1258:∈
1249:→
1222:∀
1213:∀
1204:∃
1195:∀
1159:∈
1153:→
1147:∈
1135:∀
1126:∀
1117:∃
1108:∀
1068:∈
1062:→
1056:∈
1044:∀
1035:∃
1026:∀
982:∈
967:→
961:∈
949:∀
940:∃
931:∀
736:of type (
583:predicate
342:∈
171:given any
31:does not
3246:Category
3198:Logicism
3191:timeline
3167:Concrete
3026:Validity
2996:T-schema
2989:Kripke's
2984:Tarski's
2979:semantic
2969:Strength
2918:submodel
2913:spectrum
2881:function
2729:Tarski's
2718:Elements
2705:geometry
2661:Robinson
2582:variable
2567:function
2540:spectrum
2530:Sentence
2486:variable
2429:Language
2382:Relation
2343:Automata
2333:Alphabet
2317:language
2171:-jection
2149:codomain
2135:Function
2096:Universe
2066:Infinite
1970:Relation
1753:Validity
1743:Argument
1641:theorem,
1463:See also
817:of type
806:of type
780:of type
756:) means
726:codomain
661:of type
653:of type
649:For all
613:of type
605:of type
601:For all
514:of type
510:for all
478:for all
380:function
256:of type
237:symbol.
235:constant
203:codomain
156:function
144:mappings
3140:Related
2937:Diagram
2835: (
2814:Hilbert
2799:Systems
2794:Theorem
2672:of the
2617:systems
2397:Formula
2392:Grammar
2308: (
2252:General
1965:Forcing
1950:Element
1870:Monadic
1645:paradox
1586:Theorem
1522:General
865:. Then
801:For all
543:theorem
524:subtype
450:for all
173:symbol
150:. In a
88:scholar
52:removed
37:sources
2903:finite
2666:Skolem
2619:
2594:Theory
2562:Symbol
2552:String
2535:atomic
2412:ground
2407:closed
2402:atomic
2358:ground
2321:syntax
2217:binary
2144:domain
2061:Finite
1826:finite
1684:Logics
1643:
1591:Theory
847:schema
812:unique
564:unique
410:, the
195:domain
90:
83:
76:
69:
61:
2893:Model
2641:Peano
2498:Proof
2338:Arity
2267:Naive
2154:image
2086:Fuzzy
2046:Empty
1995:union
1940:Class
1581:Model
1571:Lemma
1529:Axiom
728:type
522:is a
205:type
197:type
165:in a
152:model
138:, or
95:JSTOR
81:books
3016:Type
2819:list
2623:list
2600:list
2589:Term
2523:rank
2417:open
2311:list
2123:Maps
2028:sets
1887:Free
1857:list
1607:list
1534:list
764:) =
724:and
589:and
506:) =
474:) =
448:)),
436:) =
420:and
398:and
250:sets
244:and
231:zero
201:and
134:, a
67:news
35:any
33:cite
2703:of
2685:of
2633:of
2165:Sur
2139:Map
1946:Ur-
1928:Set
1097::
1015::
908:in
526:of
416:of
126:In
46:by
3248::
3089:NP
2713::
2707::
2637::
2314:),
2169:Bi
2161:In
833:).
821:,
692:).
684:=
677:)
665:,
629:),
617:,
562:a
432:)(
428:â
406:â
284::=
217:,
189:,
158:.
3169:/
3084:P
2839:)
2625:)
2621:(
2518:â
2513:!
2508:â
2469:=
2464:â
2459:â
2454:â§
2449:âš
2444:ÂŹ
2167:/
2163:/
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