3286:
20:
1346:
801:
574:
1153:
1260:
represents a member of that set. Examining the differences between the image and codomain can often be useful for discovering properties of the function in question. For example, it can be concluded that
334:
1362:
747:
680:
509:
1354:
1250:
1219:
1068:
1037:
999:
618:
383:
911:
711:
465:
1342:
1327:
1378:
1665:
434:
870:
840:
2340:
2423:
1564:
940:, which is a consequence of the function and could be unknown at the level of the composition) is a subset of the domain of the function on the left side.
275:. With such a definition functions do not have a codomain, although some authors still use it informally after introducing a function in the form
2737:
3315:
2895:
1534:
1509:
1491:
1470:
1449:
1428:
1370:
1683:
2750:
2073:
755:
531:
2335:
2755:
2745:
2482:
1688:
1099:
2233:
1679:
1070:. However, the image is uncertain. Some transformations may have image equal to the whole codomain (in this case the matrices with
2891:
1406:
303:
2988:
2732:
1557:
2293:
1986:
1727:
3310:
3249:
2951:
2714:
2709:
2534:
1955:
1639:
1437:
3244:
3027:
2944:
2657:
2588:
2465:
1707:
635:
to the same number, they are not, in this view, the same function because they have different codomains. A third function
2315:
3169:
2995:
2681:
1914:
1416:
2320:
2652:
2391:
1649:
1550:
1518:
3047:
3042:
716:
645:
478:
2976:
2566:
1960:
1928:
1619:
1693:
3266:
3215:
3112:
2610:
2571:
2048:
1224:
1193:
1042:
1011:
973:
3107:
1722:
921:, might receive an argument for which no output is defined â negative numbers are not elements of the domain of
3037:
2576:
2428:
2411:
2134:
1614:
580:
345:
892:
692:
446:
2939:
2916:
2877:
2763:
2704:
2350:
2270:
2114:
2058:
1671:
3229:
2956:
2934:
2901:
2794:
2640:
2625:
2598:
2549:
2433:
2368:
2193:
2159:
2154:
2028:
1859:
1836:
512:
84:
3159:
3012:
2804:
2522:
2258:
2164:
2023:
2008:
1889:
1864:
1071:
963:
3285:
3132:
3094:
2971:
2775:
2615:
2539:
2517:
2345:
2303:
2202:
2169:
2033:
1821:
1732:
1005:
926:
807:
388:
156:
60:
947:, in that the function is surjective if and only if its codomain equals its image. In the example,
3261:
3152:
3137:
3117:
3074:
2961:
2911:
2837:
2782:
2719:
2512:
2507:
2455:
2223:
2212:
1884:
1784:
1712:
1703:
1699:
1634:
1629:
216:
203:
178:
118:
112:
48:
3290:
3059:
3022:
3007:
3000:
2983:
2769:
2635:
2561:
2544:
2497:
2310:
2219:
2053:
2038:
1998:
1950:
1935:
1923:
1879:
1854:
1624:
1573:
956:
253:
2787:
2243:
846:
816:
3225:
3032:
2842:
2832:
2724:
2605:
2440:
2416:
2197:
2181:
2086:
2063:
1940:
1909:
1874:
1769:
1604:
1530:
1505:
1487:
1466:
1458:
1445:
1424:
1402:
88:
3239:
3234:
3127:
3084:
2906:
2867:
2862:
2847:
2673:
2630:
2527:
2325:
2275:
1849:
1811:
1394:
3220:
3210:
3164:
3147:
3102:
3064:
2966:
2886:
2693:
2620:
2593:
2581:
2487:
2401:
2375:
2330:
2298:
2099:
1901:
1844:
1794:
1759:
1717:
1078:
3205:
3184:
3142:
3122:
3017:
2872:
2470:
2460:
2450:
2445:
2379:
2253:
2129:
2018:
2013:
1991:
1592:
3304:
3179:
2857:
2364:
2149:
2139:
2109:
2094:
1764:
962:
A second example of the difference between codomain and image is demonstrated by the
3079:
2926:
2827:
2819:
2699:
2647:
2556:
2492:
2475:
2406:
2265:
2124:
1826:
1609:
967:
91:
into which all of the output of the function is constrained to fall. It is the set
215:
of its codomain so it might not coincide with it. Namely, a function that is not
3189:
3069:
2248:
2238:
2185:
1869:
1789:
1774:
1654:
1599:
72:
2119:
1974:
1945:
1751:
1522:
1479:
944:
249:
3271:
3174:
2227:
2144:
2104:
2068:
2004:
1816:
1806:
1779:
1274:
1265:
does not have full rank since its image is smaller than the whole codomain.
3256:
3054:
2502:
1801:
1280:
2852:
1644:
1421:
An
Introduction to Mathematical Reasoning: Numbers, Sets, and Functions
885:
is not useful. It is true, unless defined otherwise, that the image of
1008:
with real coefficients. Each matrix represents a map with the domain
19:
1542:
796:{\displaystyle h\colon \mathbb {R} _{0}^{+}\rightarrow \mathbb {R} .}
212:
117:
is sometimes ambiguously used to refer to either the codomain or the
569:{\displaystyle g\colon \mathbb {R} \rightarrow \mathbb {R} _{0}^{+}}
2396:
1742:
1587:
18:
1190:, but is still in the codomain since linear transformations from
1486:, Symposium in Pure Mathematics, American Mathematical Society,
1148:{\displaystyle T={\begin{pmatrix}1&0\\1&0\end{pmatrix}}}
932:
Function composition therefore is a useful notion only when the
1546:
955:
is not. The codomain does not affect whether a function is an
1158:
which represents a linear transformation that maps the point
936:
of the function on the right side of a composition (not its
329:{\displaystyle f\colon \mathbb {R} \rightarrow \mathbb {R} }
259:, in which case there is formally no such thing as a triple
252:
it is desirable to permit the domain of a function to be a
471:
does not map to any negative number. Thus the image of
1228:
1197:
1114:
1046:
1015:
977:
896:
889:
is not known; it is only known that it is a subset of
720:
696:
482:
450:
1227:
1196:
1102:
1077:) but many do not, instead mapping into some smaller
1045:
1014:
976:
970:â in particular, all the linear transformations from
895:
849:
819:
758:
719:
695:
648:
583:
534:
481:
449:
391:
348:
306:
3198:
3093:
2925:
2818:
2670:
2363:
2286:
2180:
2084:
1973:
1900:
1835:
1750:
1741:
1663:
1580:
1244:
1213:
1147:
1062:
1031:
993:
905:
864:
834:
795:
741:
705:
674:
612:
568:
503:
459:
428:
377:
328:
742:{\displaystyle \textstyle \mathbb {R} _{0}^{+}}
675:{\displaystyle h\colon \,x\mapsto {\sqrt {x}}.}
504:{\displaystyle \textstyle \mathbb {R} _{0}^{+}}
1558:
943:The codomain affects whether a function is a
8:
1374:
1245:{\displaystyle \textstyle \mathbb {R} ^{2}}
1214:{\displaystyle \textstyle \mathbb {R} ^{2}}
1063:{\displaystyle \textstyle \mathbb {R} ^{2}}
1032:{\displaystyle \textstyle \mathbb {R} ^{2}}
1001:to itself, which can be represented by the
994:{\displaystyle \textstyle \mathbb {R} ^{2}}
248:is defined as just a graph. For example in
2384:
1979:
1747:
1565:
1551:
1543:
1252:are of explicit relevance. Just like all
1358:
1235:
1231:
1230:
1226:
1204:
1200:
1199:
1195:
1109:
1101:
1053:
1049:
1048:
1044:
1022:
1018:
1017:
1013:
984:
980:
979:
975:
898:
897:
894:
848:
818:
786:
785:
776:
771:
767:
766:
757:
732:
727:
723:
722:
718:
698:
697:
694:
662:
655:
647:
613:{\displaystyle g\colon \,x\mapsto x^{2}.}
601:
590:
582:
560:
555:
551:
550:
542:
541:
533:
494:
489:
485:
484:
480:
452:
451:
448:
417:
390:
378:{\displaystyle f\colon \,x\mapsto x^{2},}
366:
355:
347:
322:
321:
314:
313:
305:
1463:Categories for the working mathematician
1350:
1311:
1299:
1323:
1292:
913:. For this reason, it is possible that
906:{\displaystyle \textstyle \mathbb {R} }
706:{\displaystyle \textstyle \mathbb {R} }
460:{\displaystyle \textstyle \mathbb {R} }
223:in its codomain for which the equation
197:ranges over the elements of the domain
1401:. ĂlĂ©ments de mathĂ©matique. Springer.
1366:
1338:
240:A codomain is not part of a function
182:. The set of all elements of the form
16:Target set of a mathematical function
7:
639:can be defined to demonstrate why:
126:A codomain is part of a function
14:
1277: â One-to-one correspondence
3284:
1089:). Take for example the matrix
429:{\displaystyle f(x)\ =\ x^{2},}
211:. The image of a function is a
1527:The foundations of mathematics
1504:, Discovery Publishing House,
1444:, Cambridge University Press,
1423:, Cambridge University Press,
782:
659:
594:
546:
401:
395:
359:
318:
1:
3245:History of mathematical logic
3316:Basic concepts in set theory
3170:Primitive recursive function
1529:, Oxford University Press,
3332:
2234:SchröderâBernstein theorem
1961:Monadic predicate calculus
1620:Foundations of mathematics
1502:Introduction To Set Theory
1482:; Jech, Thomas J. (1967),
1465:(2nd ed.), Springer,
237:does not have a solution.
3280:
3267:Philosophy of mathematics
3216:Automated theorem proving
2387:
2341:Von NeumannâBernaysâGödel
1982:
1442:Logic, Induction and Sets
865:{\displaystyle h\circ g.}
835:{\displaystyle h\circ f,}
713:but can be defined to be
43:. The yellow oval inside
1081:(the matrices with rank
521:An alternative function
2917:Self-verifying theories
2738:Tarski's axiomatization
1689:Tarski's undefinability
1684:incompleteness theorems
1375:Stewart & Tall 1977
1186:is not in the image of
134:is defined as a triple
3311:Functions and mappings
3291:Mathematics portal
2902:Proof of impossibility
2550:propositional variable
1860:Propositional calculus
1246:
1215:
1149:
1064:
1033:
995:
964:linear transformations
951:is a surjection while
907:
866:
836:
797:
743:
707:
676:
614:
570:
505:
461:
430:
379:
330:
68:
3160:Kolmogorov complexity
3113:Computably enumerable
3013:Model complete theory
2805:Principia Mathematica
1865:Propositional formula
1694:BanachâTarski paradox
1500:Sharma, A.K. (2004),
1399:Théorie des ensembles
1359:Scott & Jech 1967
1247:
1216:
1150:
1065:
1034:
996:
917:, when composed with
908:
867:
837:
798:
744:
708:
677:
615:
571:
506:
462:
431:
380:
331:
22:
3108:ChurchâTuring thesis
3095:Computability theory
2304:continuum hypothesis
1822:Square of opposition
1680:Gödel's completeness
1484:Axiomatic set theory
1225:
1194:
1100:
1043:
1012:
974:
927:square root function
893:
847:
817:
756:
717:
693:
646:
581:
532:
479:
447:
389:
346:
304:
3262:Mathematical object
3153:P versus NP problem
3118:Computable function
2912:Reverse mathematics
2838:Logical consequence
2715:primitive recursive
2710:elementary function
2483:Free/bound variable
2336:TarskiâGrothendieck
1855:Logical connectives
1785:Logical equivalence
1635:Logical consequence
781:
737:
565:
499:
55:, and the red oval
39:is the codomain of
3060:Transfer principle
3023:Semantics of logic
3008:Categorical theory
2984:Non-standard model
2498:Logical connective
1625:Information theory
1574:Mathematical logic
1459:Mac Lane, Saunders
1242:
1241:
1211:
1210:
1145:
1139:
1060:
1059:
1029:
1028:
991:
990:
903:
902:
862:
832:
793:
765:
739:
738:
721:
703:
702:
672:
610:
566:
549:
525:is defined thus:
501:
500:
483:
457:
456:
426:
375:
326:
81:set of destination
69:
3298:
3297:
3230:Abstract category
3033:Theories of truth
2843:Rule of inference
2833:Natural deduction
2814:
2813:
2359:
2358:
2064:Cartesian product
1969:
1968:
1875:Many-valued logic
1850:Boolean functions
1733:Russell's paradox
1708:diagonal argument
1605:First-order logic
1536:978-0-19-853165-4
1511:978-81-7141-877-0
1493:978-0-8218-0245-8
1472:978-0-387-98403-2
1451:978-0-521-53361-4
1430:978-0-521-59718-0
1395:Bourbaki, Nicolas
1281:Morphism#Codomain
667:
412:
406:
3323:
3289:
3288:
3240:History of logic
3235:Category of sets
3128:Decision problem
2907:Ordinal analysis
2848:Sequent calculus
2746:Boolean algebras
2686:
2685:
2660:
2631:logical/constant
2385:
2371:
2294:ZermeloâFraenkel
2045:Set operations:
1980:
1917:
1748:
1728:LöwenheimâSkolem
1615:Formal semantics
1567:
1560:
1553:
1544:
1539:
1523:Tall, David Orme
1514:
1496:
1475:
1454:
1433:
1417:Eccles, Peter J.
1412:
1381:
1336:
1330:
1321:
1315:
1309:
1303:
1297:
1264:
1259:
1255:
1251:
1249:
1248:
1243:
1240:
1239:
1234:
1220:
1218:
1217:
1212:
1209:
1208:
1203:
1189:
1185:
1181:
1169:
1154:
1152:
1151:
1146:
1144:
1143:
1092:
1088:
1084:
1076:
1069:
1067:
1066:
1061:
1058:
1057:
1052:
1038:
1036:
1035:
1030:
1027:
1026:
1021:
1004:
1000:
998:
997:
992:
989:
988:
983:
954:
950:
924:
920:
916:
912:
910:
909:
904:
901:
888:
884:
871:
869:
868:
863:
841:
839:
838:
833:
802:
800:
799:
794:
789:
780:
775:
770:
748:
746:
745:
740:
736:
731:
726:
712:
710:
709:
704:
701:
688:
681:
679:
678:
673:
668:
663:
638:
634:
630:
626:
619:
617:
616:
611:
606:
605:
575:
573:
572:
567:
564:
559:
554:
545:
524:
517:
510:
508:
507:
502:
498:
493:
488:
474:
470:
466:
464:
463:
458:
455:
442:
439:the codomain of
435:
433:
432:
427:
422:
421:
410:
404:
385:or equivalently
384:
382:
381:
376:
371:
370:
335:
333:
332:
327:
325:
317:
288:
274:
258:
247:
243:
236:
222:
210:
201:, is called the
200:
196:
192:
175:
167:
163:
153:
149:
133:
129:
108:
95:in the notation
94:
66:
58:
54:
46:
42:
38:
35:. The blue oval
34:
30:
26:
3331:
3330:
3326:
3325:
3324:
3322:
3321:
3320:
3301:
3300:
3299:
3294:
3283:
3276:
3221:Category theory
3211:Algebraic logic
3194:
3165:Lambda calculus
3103:Church encoding
3089:
3065:Truth predicate
2921:
2887:Complete theory
2810:
2679:
2675:
2671:
2666:
2658:
2378: and
2374:
2369:
2355:
2331:New Foundations
2299:axiom of choice
2282:
2244:Gödel numbering
2184: and
2176:
2080:
1965:
1915:
1896:
1845:Boolean algebra
1831:
1795:Equiconsistency
1760:Classical logic
1737:
1718:Halting problem
1706: and
1682: and
1670: and
1669:
1664:Theorems (
1659:
1576:
1571:
1537:
1517:
1512:
1499:
1494:
1478:
1473:
1457:
1452:
1438:Forster, Thomas
1436:
1431:
1415:
1409:
1393:
1390:
1385:
1384:
1357:; Mac Lane, in
1337:
1333:
1328:pp. 10–11
1322:
1318:
1310:
1306:
1298:
1294:
1289:
1271:
1262:
1257:
1253:
1229:
1223:
1222:
1198:
1192:
1191:
1187:
1183:
1171:
1159:
1138:
1137:
1132:
1126:
1125:
1120:
1110:
1098:
1097:
1090:
1086:
1082:
1074:
1047:
1041:
1040:
1016:
1010:
1009:
1002:
978:
972:
971:
952:
948:
925:, which is the
922:
918:
914:
891:
890:
886:
876:
875:On inspection,
845:
844:
815:
814:
754:
753:
715:
714:
691:
690:
686:
644:
643:
636:
632:
628:
624:
597:
579:
578:
530:
529:
522:
515:
477:
476:
472:
468:
445:
444:
440:
413:
387:
386:
362:
344:
343:
302:
301:
297:For a function
295:
276:
260:
256:
245:
241:
224:
220:
208:
198:
194:
183:
173:
165:
161:
151:
135:
131:
127:
123:of a function.
96:
92:
64:
56:
52:
44:
40:
36:
32:
28:
24:
17:
12:
11:
5:
3329:
3327:
3319:
3318:
3313:
3303:
3302:
3296:
3295:
3281:
3278:
3277:
3275:
3274:
3269:
3264:
3259:
3254:
3253:
3252:
3242:
3237:
3232:
3223:
3218:
3213:
3208:
3206:Abstract logic
3202:
3200:
3196:
3195:
3193:
3192:
3187:
3185:Turing machine
3182:
3177:
3172:
3167:
3162:
3157:
3156:
3155:
3150:
3145:
3140:
3135:
3125:
3123:Computable set
3120:
3115:
3110:
3105:
3099:
3097:
3091:
3090:
3088:
3087:
3082:
3077:
3072:
3067:
3062:
3057:
3052:
3051:
3050:
3045:
3040:
3030:
3025:
3020:
3018:Satisfiability
3015:
3010:
3005:
3004:
3003:
2993:
2992:
2991:
2981:
2980:
2979:
2974:
2969:
2964:
2959:
2949:
2948:
2947:
2942:
2935:Interpretation
2931:
2929:
2923:
2922:
2920:
2919:
2914:
2909:
2904:
2899:
2889:
2884:
2883:
2882:
2881:
2880:
2870:
2865:
2855:
2850:
2845:
2840:
2835:
2830:
2824:
2822:
2816:
2815:
2812:
2811:
2809:
2808:
2800:
2799:
2798:
2797:
2792:
2791:
2790:
2785:
2780:
2760:
2759:
2758:
2756:minimal axioms
2753:
2742:
2741:
2740:
2729:
2728:
2727:
2722:
2717:
2712:
2707:
2702:
2689:
2687:
2668:
2667:
2665:
2664:
2663:
2662:
2650:
2645:
2644:
2643:
2638:
2633:
2628:
2618:
2613:
2608:
2603:
2602:
2601:
2596:
2586:
2585:
2584:
2579:
2574:
2569:
2559:
2554:
2553:
2552:
2547:
2542:
2532:
2531:
2530:
2525:
2520:
2515:
2510:
2505:
2495:
2490:
2485:
2480:
2479:
2478:
2473:
2468:
2463:
2453:
2448:
2446:Formation rule
2443:
2438:
2437:
2436:
2431:
2421:
2420:
2419:
2409:
2404:
2399:
2394:
2388:
2382:
2365:Formal systems
2361:
2360:
2357:
2356:
2354:
2353:
2348:
2343:
2338:
2333:
2328:
2323:
2318:
2313:
2308:
2307:
2306:
2301:
2290:
2288:
2284:
2283:
2281:
2280:
2279:
2278:
2268:
2263:
2262:
2261:
2254:Large cardinal
2251:
2246:
2241:
2236:
2231:
2217:
2216:
2215:
2210:
2205:
2190:
2188:
2178:
2177:
2175:
2174:
2173:
2172:
2167:
2162:
2152:
2147:
2142:
2137:
2132:
2127:
2122:
2117:
2112:
2107:
2102:
2097:
2091:
2089:
2082:
2081:
2079:
2078:
2077:
2076:
2071:
2066:
2061:
2056:
2051:
2043:
2042:
2041:
2036:
2026:
2021:
2019:Extensionality
2016:
2014:Ordinal number
2011:
2001:
1996:
1995:
1994:
1983:
1977:
1971:
1970:
1967:
1966:
1964:
1963:
1958:
1953:
1948:
1943:
1938:
1933:
1932:
1931:
1921:
1920:
1919:
1906:
1904:
1898:
1897:
1895:
1894:
1893:
1892:
1887:
1882:
1872:
1867:
1862:
1857:
1852:
1847:
1841:
1839:
1833:
1832:
1830:
1829:
1824:
1819:
1814:
1809:
1804:
1799:
1798:
1797:
1787:
1782:
1777:
1772:
1767:
1762:
1756:
1754:
1745:
1739:
1738:
1736:
1735:
1730:
1725:
1720:
1715:
1710:
1698:Cantor's
1696:
1691:
1686:
1676:
1674:
1661:
1660:
1658:
1657:
1652:
1647:
1642:
1637:
1632:
1627:
1622:
1617:
1612:
1607:
1602:
1597:
1596:
1595:
1584:
1582:
1578:
1577:
1572:
1570:
1569:
1562:
1555:
1547:
1541:
1540:
1535:
1515:
1510:
1497:
1492:
1480:Scott, Dana S.
1476:
1471:
1455:
1450:
1434:
1429:
1413:
1407:
1389:
1386:
1383:
1382:
1331:
1316:
1304:
1291:
1290:
1288:
1285:
1284:
1283:
1278:
1270:
1267:
1238:
1233:
1207:
1202:
1156:
1155:
1142:
1136:
1133:
1131:
1128:
1127:
1124:
1121:
1119:
1116:
1115:
1113:
1108:
1105:
1056:
1051:
1025:
1020:
987:
982:
900:
873:
872:
861:
858:
855:
852:
842:
831:
828:
825:
822:
804:
803:
792:
788:
784:
779:
774:
769:
764:
761:
735:
730:
725:
700:
685:The domain of
683:
682:
671:
666:
661:
658:
654:
651:
621:
620:
609:
604:
600:
596:
593:
589:
586:
576:
563:
558:
553:
548:
544:
540:
537:
497:
492:
487:
454:
437:
436:
425:
420:
416:
409:
403:
400:
397:
394:
374:
369:
365:
361:
358:
354:
351:
337:
336:
324:
320:
316:
312:
309:
294:
291:
154:is called the
15:
13:
10:
9:
6:
4:
3:
2:
3328:
3317:
3314:
3312:
3309:
3308:
3306:
3293:
3292:
3287:
3279:
3273:
3270:
3268:
3265:
3263:
3260:
3258:
3255:
3251:
3248:
3247:
3246:
3243:
3241:
3238:
3236:
3233:
3231:
3227:
3224:
3222:
3219:
3217:
3214:
3212:
3209:
3207:
3204:
3203:
3201:
3197:
3191:
3188:
3186:
3183:
3181:
3180:Recursive set
3178:
3176:
3173:
3171:
3168:
3166:
3163:
3161:
3158:
3154:
3151:
3149:
3146:
3144:
3141:
3139:
3136:
3134:
3131:
3130:
3129:
3126:
3124:
3121:
3119:
3116:
3114:
3111:
3109:
3106:
3104:
3101:
3100:
3098:
3096:
3092:
3086:
3083:
3081:
3078:
3076:
3073:
3071:
3068:
3066:
3063:
3061:
3058:
3056:
3053:
3049:
3046:
3044:
3041:
3039:
3036:
3035:
3034:
3031:
3029:
3026:
3024:
3021:
3019:
3016:
3014:
3011:
3009:
3006:
3002:
2999:
2998:
2997:
2994:
2990:
2989:of arithmetic
2987:
2986:
2985:
2982:
2978:
2975:
2973:
2970:
2968:
2965:
2963:
2960:
2958:
2955:
2954:
2953:
2950:
2946:
2943:
2941:
2938:
2937:
2936:
2933:
2932:
2930:
2928:
2924:
2918:
2915:
2913:
2910:
2908:
2905:
2903:
2900:
2897:
2896:from ZFC
2893:
2890:
2888:
2885:
2879:
2876:
2875:
2874:
2871:
2869:
2866:
2864:
2861:
2860:
2859:
2856:
2854:
2851:
2849:
2846:
2844:
2841:
2839:
2836:
2834:
2831:
2829:
2826:
2825:
2823:
2821:
2817:
2807:
2806:
2802:
2801:
2796:
2795:non-Euclidean
2793:
2789:
2786:
2784:
2781:
2779:
2778:
2774:
2773:
2771:
2768:
2767:
2765:
2761:
2757:
2754:
2752:
2749:
2748:
2747:
2743:
2739:
2736:
2735:
2734:
2730:
2726:
2723:
2721:
2718:
2716:
2713:
2711:
2708:
2706:
2703:
2701:
2698:
2697:
2695:
2691:
2690:
2688:
2683:
2677:
2672:Example
2669:
2661:
2656:
2655:
2654:
2651:
2649:
2646:
2642:
2639:
2637:
2634:
2632:
2629:
2627:
2624:
2623:
2622:
2619:
2617:
2614:
2612:
2609:
2607:
2604:
2600:
2597:
2595:
2592:
2591:
2590:
2587:
2583:
2580:
2578:
2575:
2573:
2570:
2568:
2565:
2564:
2563:
2560:
2558:
2555:
2551:
2548:
2546:
2543:
2541:
2538:
2537:
2536:
2533:
2529:
2526:
2524:
2521:
2519:
2516:
2514:
2511:
2509:
2506:
2504:
2501:
2500:
2499:
2496:
2494:
2491:
2489:
2486:
2484:
2481:
2477:
2474:
2472:
2469:
2467:
2464:
2462:
2459:
2458:
2457:
2454:
2452:
2449:
2447:
2444:
2442:
2439:
2435:
2432:
2430:
2429:by definition
2427:
2426:
2425:
2422:
2418:
2415:
2414:
2413:
2410:
2408:
2405:
2403:
2400:
2398:
2395:
2393:
2390:
2389:
2386:
2383:
2381:
2377:
2372:
2366:
2362:
2352:
2349:
2347:
2344:
2342:
2339:
2337:
2334:
2332:
2329:
2327:
2324:
2322:
2319:
2317:
2316:KripkeâPlatek
2314:
2312:
2309:
2305:
2302:
2300:
2297:
2296:
2295:
2292:
2291:
2289:
2285:
2277:
2274:
2273:
2272:
2269:
2267:
2264:
2260:
2257:
2256:
2255:
2252:
2250:
2247:
2245:
2242:
2240:
2237:
2235:
2232:
2229:
2225:
2221:
2218:
2214:
2211:
2209:
2206:
2204:
2201:
2200:
2199:
2195:
2192:
2191:
2189:
2187:
2183:
2179:
2171:
2168:
2166:
2163:
2161:
2160:constructible
2158:
2157:
2156:
2153:
2151:
2148:
2146:
2143:
2141:
2138:
2136:
2133:
2131:
2128:
2126:
2123:
2121:
2118:
2116:
2113:
2111:
2108:
2106:
2103:
2101:
2098:
2096:
2093:
2092:
2090:
2088:
2083:
2075:
2072:
2070:
2067:
2065:
2062:
2060:
2057:
2055:
2052:
2050:
2047:
2046:
2044:
2040:
2037:
2035:
2032:
2031:
2030:
2027:
2025:
2022:
2020:
2017:
2015:
2012:
2010:
2006:
2002:
2000:
1997:
1993:
1990:
1989:
1988:
1985:
1984:
1981:
1978:
1976:
1972:
1962:
1959:
1957:
1954:
1952:
1949:
1947:
1944:
1942:
1939:
1937:
1934:
1930:
1927:
1926:
1925:
1922:
1918:
1913:
1912:
1911:
1908:
1907:
1905:
1903:
1899:
1891:
1888:
1886:
1883:
1881:
1878:
1877:
1876:
1873:
1871:
1868:
1866:
1863:
1861:
1858:
1856:
1853:
1851:
1848:
1846:
1843:
1842:
1840:
1838:
1837:Propositional
1834:
1828:
1825:
1823:
1820:
1818:
1815:
1813:
1810:
1808:
1805:
1803:
1800:
1796:
1793:
1792:
1791:
1788:
1786:
1783:
1781:
1778:
1776:
1773:
1771:
1768:
1766:
1765:Logical truth
1763:
1761:
1758:
1757:
1755:
1753:
1749:
1746:
1744:
1740:
1734:
1731:
1729:
1726:
1724:
1721:
1719:
1716:
1714:
1711:
1709:
1705:
1701:
1697:
1695:
1692:
1690:
1687:
1685:
1681:
1678:
1677:
1675:
1673:
1667:
1662:
1656:
1653:
1651:
1648:
1646:
1643:
1641:
1638:
1636:
1633:
1631:
1628:
1626:
1623:
1621:
1618:
1616:
1613:
1611:
1608:
1606:
1603:
1601:
1598:
1594:
1591:
1590:
1589:
1586:
1585:
1583:
1579:
1575:
1568:
1563:
1561:
1556:
1554:
1549:
1548:
1545:
1538:
1532:
1528:
1524:
1520:
1516:
1513:
1507:
1503:
1498:
1495:
1489:
1485:
1481:
1477:
1474:
1468:
1464:
1460:
1456:
1453:
1447:
1443:
1439:
1435:
1432:
1426:
1422:
1418:
1414:
1410:
1408:9783540340348
1404:
1400:
1396:
1392:
1391:
1387:
1380:
1376:
1372:
1368:
1364:
1360:
1356:
1352:
1351:Mac Lane 1998
1348:
1344:
1340:
1335:
1332:
1329:
1325:
1320:
1317:
1313:
1312:Bourbaki 1970
1308:
1305:
1301:
1300:Bourbaki 1970
1296:
1293:
1286:
1282:
1279:
1276:
1273:
1272:
1268:
1266:
1236:
1205:
1182:. The point
1179:
1175:
1167:
1163:
1140:
1134:
1129:
1122:
1117:
1111:
1106:
1103:
1096:
1095:
1094:
1080:
1073:
1054:
1039:and codomain
1023:
1007:
985:
969:
968:vector spaces
965:
960:
958:
946:
941:
939:
935:
930:
928:
883:
879:
859:
856:
853:
850:
843:
829:
826:
823:
820:
813:
812:
811:
809:
790:
777:
772:
762:
759:
752:
751:
750:
733:
728:
669:
664:
656:
652:
649:
642:
641:
640:
607:
602:
598:
591:
587:
584:
577:
561:
556:
538:
535:
528:
527:
526:
519:
514:
495:
490:
423:
418:
414:
407:
398:
392:
372:
367:
363:
356:
352:
349:
342:
341:
340:
310:
307:
300:
299:
298:
292:
290:
287:
283:
279:
272:
268:
264:
255:
251:
238:
235:
231:
227:
219:has elements
218:
214:
206:
205:
190:
186:
181:
180:
171:
159:
158:
147:
143:
139:
124:
122:
121:
116:
115:
114:
107:
103:
99:
90:
86:
82:
78:
74:
62:
50:
21:
3282:
3080:Ultraproduct
2927:Model theory
2892:Independence
2828:Formal proof
2820:Proof theory
2803:
2776:
2733:real numbers
2705:second-order
2616:Substitution
2493:Metalanguage
2434:conservative
2407:Axiom schema
2351:Constructive
2321:MorseâKelley
2287:Set theories
2266:Aleph number
2259:inaccessible
2207:
2165:Grothendieck
2049:intersection
1936:Higher-order
1924:Second-order
1870:Truth tables
1827:Venn diagram
1610:Formal proof
1526:
1519:Stewart, Ian
1501:
1483:
1462:
1441:
1420:
1398:
1334:
1324:Forster 2003
1319:
1314:, p. 77
1307:
1302:, p. 76
1295:
1177:
1173:
1165:
1161:
1157:
966:between two
961:
942:
937:
933:
931:
881:
877:
874:
810:are denoted
808:compositions
805:
684:
631:map a given
622:
520:
511:; i.e., the
438:
338:
296:
285:
281:
277:
270:
266:
262:
254:proper class
239:
233:
229:
225:
202:
188:
184:
177:
169:
155:
145:
141:
137:
125:
119:
111:
110:
105:
101:
97:
80:
76:
70:
3190:Type theory
3138:undecidable
3070:Truth value
2957:equivalence
2636:non-logical
2249:Enumeration
2239:Isomorphism
2186:cardinality
2170:Von Neumann
2135:Ultrafilter
2100:Uncountable
2034:equivalence
1951:Quantifiers
1941:Fixed-point
1910:First-order
1790:Consistency
1775:Proposition
1752:Traditional
1723:Lindström's
1713:Compactness
1655:Type theory
1600:Cardinality
1367:Sharma 2004
1339:Eccles 1997
475:is the set
339:defined by
109:. The term
73:mathematics
23:A function
3305:Categories
3001:elementary
2694:arithmetic
2562:Quantifier
2540:functional
2412:Expression
2130:Transitive
2074:identities
2059:complement
1992:hereditary
1975:Set theory
1388:References
1256:matrices,
945:surjection
689:cannot be
516:[0, â)
250:set theory
217:surjective
3272:Supertask
3175:Recursion
3133:decidable
2967:saturated
2945:of models
2868:deductive
2863:axiomatic
2783:Hilbert's
2770:Euclidean
2751:canonical
2674:axiomatic
2606:Signature
2535:Predicate
2424:Extension
2346:Ackermann
2271:Operation
2150:Universal
2140:Recursive
2115:Singleton
2110:Inhabited
2095:Countable
2085:Types of
2069:power set
2039:partition
1956:Predicate
1902:Predicate
1817:Syllogism
1807:Soundness
1780:Inference
1770:Tautology
1672:paradoxes
1341:, p. 91 (
1275:Bijection
1093:given by
957:injection
854:∘
824:∘
783:→
763::
660:↦
653::
595:↦
588::
547:→
539::
360:↦
353::
319:→
311::
3257:Logicism
3250:timeline
3226:Concrete
3085:Validity
3055:T-schema
3048:Kripke's
3043:Tarski's
3038:semantic
3028:Strength
2977:submodel
2972:spectrum
2940:function
2788:Tarski's
2777:Elements
2764:geometry
2720:Robinson
2641:variable
2626:function
2599:spectrum
2589:Sentence
2545:variable
2488:Language
2441:Relation
2402:Automata
2392:Alphabet
2376:language
2230:-jection
2208:codomain
2194:Function
2155:Universe
2125:Infinite
2029:Relation
1812:Validity
1802:Argument
1700:theorem,
1525:(1977),
1461:(1998),
1440:(2003),
1419:(1997),
1397:(1970).
1269:See also
1079:subspace
1006:matrices
934:codomain
513:interval
293:Examples
193:, where
170:codomain
85:function
77:codomain
3199:Related
2996:Diagram
2894: (
2873:Hilbert
2858:Systems
2853:Theorem
2731:of the
2676:systems
2456:Formula
2451:Grammar
2367: (
2311:General
2024:Forcing
2009:Element
1929:Monadic
1704:paradox
1645:Theorem
1581:General
1347:quote 2
1343:quote 1
59:is the
47:is the
2962:finite
2725:Skolem
2678:
2653:Theory
2621:Symbol
2611:String
2594:atomic
2471:ground
2466:closed
2461:atomic
2417:ground
2380:syntax
2276:binary
2203:domain
2120:Finite
1885:finite
1743:Logics
1702:
1650:Theory
1533:
1508:
1490:
1469:
1448:
1427:
1405:
1363:p. 232
1184:(2, 3)
623:While
467:, but
411:
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213:subset
172:, and
157:domain
150:where
61:domain
2952:Model
2700:Peano
2557:Proof
2397:Arity
2326:Naive
2213:image
2145:Fuzzy
2105:Empty
2054:union
1999:Class
1640:Model
1630:Lemma
1588:Axiom
1379:p. 89
1371:p. 91
1287:Notes
938:image
204:image
179:graph
120:image
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87:is a
83:of a
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