3123:
3193:
353:
1376:- Both the notion of set (a collection of members), membership or element-hood, the axiom of extension, the axiom of separation, and the union axiom (Suppes calls it the sum axiom) are needed for a more thorough understanding of "set element".
275:
223:
116:
595:
267:
156:
548:
437:
1109:
1078:
1502:
1016:
is a set with a finite number of elements. The above examples are examples of finite sets. An example of an infinite set is the set of positive integers
855:
3657:
2177:
614:
2260:
1401:
2574:
3807:
2732:
1222:
1520:
3346:
3159:
2587:
1910:
3674:
2592:
2582:
2319:
2172:
1525:
1320:
1516:
2728:
1370:
1328:
1197:
2070:
2825:
2569:
1394:
3652:
3246:
2130:
1823:
3532:
1564:
3086:
2788:
2551:
2546:
2371:
1792:
1476:
1334:- "Naive" means that it is not fully axiomatized, not that it is silly or easy (Halmos's treatment is neither).
893:
3426:
3305:
3081:
2864:
2781:
2494:
2425:
2302:
1544:
17:
348:{\displaystyle C=\{\mathrm {\color {Red}red} ,\mathrm {\color {green}green} ,\mathrm {\color {blue}blue} \}}
3669:
3006:
2832:
2518:
2152:
1751:
3662:
3300:
3263:
2884:
2879:
2489:
2228:
2157:
1486:
1387:
816:
675:
514:
strongly urged that "contains" be used for membership only, and "includes" for the subset relation only.
2813:
2403:
1797:
1765:
1456:
172:
3317:
3351:
3236:
3224:
3219:
3103:
3052:
2949:
2447:
2408:
1885:
1530:
31:
1559:
3152:
2944:
2874:
2413:
2265:
2248:
1971:
1451:
71:
3771:
3689:
3564:
3516:
3330:
3253:
2776:
2753:
2714:
2600:
2541:
2187:
2107:
1951:
1895:
1508:
1136:
1000:; informally, this is the size of a set. In the above examples, the cardinality of the set
3723:
3604:
3416:
3229:
3066:
2793:
2771:
2738:
2631:
2477:
2462:
2435:
2386:
2270:
2205:
2030:
1996:
1991:
1865:
1696:
1673:
1033:
1029:
574:
403:
54:
3639:
3609:
3553:
3473:
3453:
2996:
2849:
2641:
2359:
2095:
2001:
1860:
1726:
1701:
240:
129:
527:
416:
3713:
3703:
3537:
3468:
3421:
3361:
3241:
2969:
2931:
2808:
2612:
2452:
2376:
2354:
2182:
2140:
2039:
2006:
1870:
1658:
1569:
1266:
1313:
1094:
1063:
8:
3708:
3619:
3527:
3522:
3336:
3278:
3209:
3145:
3098:
2989:
2974:
2954:
2911:
2798:
2748:
2674:
2619:
2556:
2349:
2344:
2292:
2060:
2049:
1721:
1621:
1549:
1540:
1536:
1471:
1466:
1032:, set membership must have a domain and a range. Conventionally the domain is called the
57:
498:" are also used to mean set membership, although some authors use them to mean instead "
3631:
3626:
3411:
3366:
3273:
3127:
2896:
2859:
2844:
2837:
2820:
2624:
2606:
2472:
2398:
2381:
2334:
2147:
2056:
1890:
1875:
1835:
1787:
1772:
1760:
1716:
1691:
1461:
1410:
1359:
1012:
are both 3. An infinite set is a set with an infinite number of elements, while a
2080:
3488:
3325:
3288:
3258:
3182:
3122:
3062:
2869:
2679:
2669:
2561:
2442:
2277:
2253:
2034:
2018:
1923:
1900:
1777:
1746:
1711:
1606:
1441:
1366:
1324:
1193:
518:
50:
3776:
3766:
3751:
3746:
3614:
3268:
3076:
3071:
2964:
2921:
2743:
2704:
2699:
2684:
2510:
2467:
2364:
2162:
2112:
1686:
1648:
1252:
1131:
3645:
3583:
3401:
3214:
3057:
3047:
3001:
2984:
2939:
2901:
2803:
2723:
2530:
2457:
2430:
2418:
2324:
2238:
2212:
2167:
2135:
1936:
1738:
1681:
1631:
1596:
1554:
1262:
1342:
3781:
3578:
3559:
3463:
3448:
3405:
3341:
3283:
3042:
3021:
2979:
2959:
2854:
2709:
2307:
2297:
2287:
2282:
2216:
2090:
1966:
1855:
1850:
1828:
1429:
1354:
1308:
1257:
1240:
1189:
1181:
3801:
3786:
3588:
3502:
3497:
3016:
2694:
2201:
1986:
1976:
1946:
1931:
1601:
1214:
1157:
511:
3756:
3736:
3731:
3549:
3478:
3436:
3295:
3192:
2916:
2763:
2664:
2656:
2536:
2484:
2393:
2329:
2312:
2243:
2102:
1961:
1663:
1446:
3761:
3396:
3026:
2906:
2085:
2075:
2022:
1706:
1626:
1611:
1491:
1436:
1338:
997:
991:
38:
3741:
3512:
3168:
1956:
1811:
1782:
1588:
1280:
1013:
3544:
3507:
3458:
3356:
3108:
3011:
2064:
1981:
1941:
1905:
1841:
1653:
1643:
1616:
1379:
1049:
3093:
2891:
2339:
2044:
1638:
565:
996:
The number of elements in a particular set is a property known as
2689:
1481:
754:
670:
611:
The symbol â was first used by
Giuseppe Peano, in his 1889 work
27:
Any one of the distinct objects that make up a set in set theory
3569:
3391:
1041:
503:
233:
1, 2, 3, and 4. Rather, there are only three elements of
169:
Sets can themselves be elements. For example, consider the set
159:
3441:
3201:
3137:
2233:
1579:
1424:
874:
785:
568:
of set membership is denoted by the symbol "â". Writing
930:= {red, green, blue}, the following statements are true:
866:∋, ∋, ∋, ∋
669:
The symbol itself is a stylized lowercase Greek letter
1281:"Sets - Elements | Brilliant Math & Science Wiki"
1097:
1066:
577:
530:
419:
278:
243:
175:
132:
74:
869:∌, ∌, ∌
272:
The elements of a set can be anything. For example,
122:
are the numbers 1, 2, 3 and 4. Sets of elements of
1358:
1312:
1103:
1072:
589:
542:
431:
347:
261:
217:
150:
110:
3799:
860:∈, ∈, ∈, ∈
1349:, Metaphysics Research Lab, Stanford University
1180:
3153:
1395:
1213:
615:Arithmetices principia, nova methodo exposita
410:, is denoted by the symbol "â". Writing
863:∉, ∉, ∉
342:
285:
256:
244:
212:
209:
197:
182:
145:
133:
105:
81:
1323:(Hardcover ed.), NY: Springer-Verlag,
623:
612:
397:
3160:
3146:
1587:
1402:
1388:
237:, namely the numbers 1 and 2, and the set
1256:
355:is the set whose elements are the colors
1186:Handbook of Analysis and Its Foundations
1004:is 4, while the cardinality of set
1238:
14:
3800:
1409:
1353:
1307:
985:
3141:
1383:
1251:(3). Duke University Press: 367â372.
1234:
1232:
1223:Massachusetts Institute of Technology
1219:24.243 Classical Set Theory (lecture)
1207:
1174:
1155:
918:Using the sets defined above, namely
327:
305:
289:
30:For elements in category theory, see
1337:
673:("Ï”"), the first letter of the word
1347:Stanford Encyclopedia of Philosophy
118:means that the elements of the set
24:
1321:Undergraduate Texts in Mathematics
1301:
1245:Notre Dame Journal of Formal Logic
1229:
1023:
337:
334:
331:
328:
318:
315:
312:
309:
306:
296:
293:
290:
25:
3819:
1241:"What Russell learned from Peano"
218:{\displaystyle B=\{1,2,\{3,4\}\}}
3191:
3121:
1365:, NY: Dover Publications, Inc.,
406:"is an element of", also called
450:". Equivalent expressions are "
3167:
1273:
1149:
13:
1:
3082:History of mathematical logic
1142:
111:{\displaystyle A=\{1,2,3,4\}}
3808:Basic concepts in set theory
3007:Primitive recursive function
1239:Kennedy, H. C. (July 1973).
7:
1125:
913:
817:Numeric character reference
720:DOES NOT CONTAIN AS MEMBER
619:. Here he wrote on page X:
10:
3824:
3658:von NeumannâBernaysâGödel
2071:SchröderâBernstein theorem
1798:Monadic predicate calculus
1457:Foundations of mathematics
1040:. The range is the set of
989:
674:
637:legitur a est quoddam b; âŠ
604:is not an element of
29:
3722:
3685:
3597:
3487:
3459:One-to-one correspondence
3375:
3316:
3200:
3189:
3175:
3117:
3104:Philosophy of mathematics
3053:Automated theorem proving
3035:
2930:
2762:
2655:
2507:
2224:
2200:
2178:Von NeumannâBernaysâGödel
2123:
2017:
1921:
1819:
1810:
1737:
1672:
1578:
1500:
1417:
906:
903:
900:
897:
887:
884:
881:
878:
868:
865:
862:
859:
856:Named character reference
719:
716:
713:
710:
702:
699:
696:
693:
690:
590:{\displaystyle x\notin A}
517:For the relation â , the
60:that belong to that set.
32:Element (category theory)
1258:10.1305/ndjfl/1093891001
1091:. The converse relation
398:Notation and terminology
2754:Self-verifying theories
2575:Tarski's axiomatization
1526:Tarski's undefinability
1521:incompleteness theorems
1137:Singleton (mathematics)
262:{\displaystyle \{3,4\}}
151:{\displaystyle \{1,2\}}
63:
3417:Constructible universe
3237:Constructibility (V=L)
3128:Mathematics portal
2739:Proof of impossibility
2387:propositional variable
1697:Propositional calculus
1105:
1074:
686:Character information
667:
641:
624:
613:
591:
544:
543:{\displaystyle A\ni x}
446:is an element of
433:
432:{\displaystyle x\in A}
349:
263:
219:
152:
112:
3640:Principia Mathematica
3474:Transfinite induction
3333:(i.e. set difference)
2997:Kolmogorov complexity
2950:Computably enumerable
2850:Model complete theory
2642:Principia Mathematica
1702:Propositional formula
1531:BanachâTarski paradox
1162:mathworld.wolfram.com
1106:
1075:
1060:). Thus the relation
926:= {1, 2, {3, 4}} and
645:
621:
592:
557:contains or includes
545:
434:
350:
264:
220:
153:
113:
3714:Burali-Forti paradox
3469:Set-builder notation
3422:Continuum hypothesis
3362:Symmetric difference
2945:ChurchâTuring thesis
2932:Computability theory
2141:continuum hypothesis
1659:Square of opposition
1517:Gödel's completeness
1361:Axiomatic Set Theory
1217:(February 4, 1992).
1104:{\displaystyle \ni }
1095:
1073:{\displaystyle \in }
1064:
679:, which means "is".
629:significat est. Ita
575:
528:
482:". The expressions "
454:is a member of
417:
276:
241:
173:
130:
72:
3675:TarskiâGrothendieck
3099:Mathematical object
2990:P versus NP problem
2955:Computable function
2749:Reverse mathematics
2675:Logical consequence
2552:primitive recursive
2547:elementary function
2320:Free/bound variable
2173:TarskiâGrothendieck
1692:Logical connectives
1622:Logical equivalence
1472:Logical consequence
1156:Weisstein, Eric W.
986:Cardinality of sets
894:Wolfram Mathematica
717:CONTAINS AS MEMBER
687:
647:The symbol â means
3264:Limitation of size
2897:Transfer principle
2860:Semantics of logic
2845:Categorical theory
2821:Non-standard model
2335:Logical connective
1462:Information theory
1411:Mathematical logic
1101:
1070:
888:\not\ni or \notni
714:NOT AN ELEMENT OF
685:
587:
540:
429:
370:In logical terms,
345:
340:
321:
299:
259:
225:. The elements of
215:
148:
108:
53:is any one of the
3795:
3794:
3704:Russell's paradox
3653:ZermeloâFraenkel
3554:Dedekind-infinite
3427:Diagonal argument
3326:Cartesian product
3183:Set (mathematics)
3135:
3134:
3067:Abstract category
2870:Theories of truth
2680:Rule of inference
2670:Natural deduction
2651:
2650:
2196:
2195:
1901:Cartesian product
1806:
1805:
1712:Many-valued logic
1687:Boolean functions
1570:Russell's paradox
1545:diagonal argument
1442:First-order logic
1018:{1, 2, 3, 4, ...}
911:
910:
521:â may be written
519:converse relation
16:(Redirected from
3815:
3777:Bertrand Russell
3767:John von Neumann
3752:Abraham Fraenkel
3747:Richard Dedekind
3709:Suslin's problem
3620:Cantor's theorem
3337:De Morgan's laws
3195:
3162:
3155:
3148:
3139:
3138:
3126:
3125:
3077:History of logic
3072:Category of sets
2965:Decision problem
2744:Ordinal analysis
2685:Sequent calculus
2583:Boolean algebras
2523:
2522:
2497:
2468:logical/constant
2222:
2221:
2208:
2131:ZermeloâFraenkel
1882:Set operations:
1817:
1816:
1754:
1585:
1584:
1565:LöwenheimâSkolem
1452:Formal semantics
1404:
1397:
1390:
1381:
1380:
1375:
1364:
1350:
1333:
1318:
1315:Naive Set Theory
1295:
1294:
1292:
1291:
1277:
1271:
1270:
1260:
1236:
1227:
1226:
1211:
1205:
1203:
1178:
1172:
1171:
1169:
1168:
1153:
1132:Identity element
1121:
1110:
1108:
1107:
1102:
1090:
1079:
1077:
1076:
1071:
1019:
981:
973:
965:
957:
947:
939:
922:= {1, 2, 3, 4},
850:
846:
842:
838:
834:
830:
826:
822:
688:
684:
678:
660:
639:
636:
634:
628:
618:
596:
594:
593:
588:
549:
547:
546:
541:
462:belongs to
438:
436:
435:
430:
393:
366:
362:
358:
354:
352:
351:
346:
341:
322:
300:
268:
266:
265:
260:
236:
228:
224:
222:
221:
216:
165:
157:
155:
154:
149:
125:
121:
117:
115:
114:
109:
21:
3823:
3822:
3818:
3817:
3816:
3814:
3813:
3812:
3798:
3797:
3796:
3791:
3718:
3697:
3681:
3646:New Foundations
3593:
3483:
3402:Cardinal number
3385:
3371:
3312:
3196:
3187:
3171:
3166:
3136:
3131:
3120:
3113:
3058:Category theory
3048:Algebraic logic
3031:
3002:Lambda calculus
2940:Church encoding
2926:
2902:Truth predicate
2758:
2724:Complete theory
2647:
2516:
2512:
2508:
2503:
2495:
2215: and
2211:
2206:
2192:
2168:New Foundations
2136:axiom of choice
2119:
2081:Gödel numbering
2021: and
2013:
1917:
1802:
1752:
1733:
1682:Boolean algebra
1668:
1632:Equiconsistency
1597:Classical logic
1574:
1555:Halting problem
1543: and
1519: and
1507: and
1506:
1501:Theorems (
1496:
1413:
1408:
1373:
1355:Suppes, Patrick
1331:
1309:Halmos, Paul R.
1304:
1302:Further reading
1299:
1298:
1289:
1287:
1279:
1278:
1274:
1237:
1230:
1212:
1208:
1200:
1179:
1175:
1166:
1164:
1154:
1150:
1145:
1128:
1112:
1111:is a subset of
1096:
1093:
1092:
1081:
1080:is a subset of
1065:
1062:
1061:
1026:
1024:Formal relation
1017:
994:
988:
976:
968:
960:
952:
942:
934:
916:
682:
652:
632:
630:
626:
576:
573:
572:
529:
526:
525:
418:
415:
414:
400:
371:
364:
360:
356:
326:
304:
288:
277:
274:
273:
242:
239:
238:
234:
226:
174:
171:
170:
163:
131:
128:
127:
123:
119:
73:
70:
69:
66:
35:
28:
23:
22:
15:
12:
11:
5:
3821:
3811:
3810:
3793:
3792:
3790:
3789:
3784:
3782:Thoralf Skolem
3779:
3774:
3769:
3764:
3759:
3754:
3749:
3744:
3739:
3734:
3728:
3726:
3720:
3719:
3717:
3716:
3711:
3706:
3700:
3698:
3696:
3695:
3692:
3686:
3683:
3682:
3680:
3679:
3678:
3677:
3672:
3667:
3666:
3665:
3650:
3649:
3648:
3636:
3635:
3634:
3623:
3622:
3617:
3612:
3607:
3601:
3599:
3595:
3594:
3592:
3591:
3586:
3581:
3576:
3567:
3562:
3557:
3547:
3542:
3541:
3540:
3535:
3530:
3520:
3510:
3505:
3500:
3494:
3492:
3485:
3484:
3482:
3481:
3476:
3471:
3466:
3464:Ordinal number
3461:
3456:
3451:
3446:
3445:
3444:
3439:
3429:
3424:
3419:
3414:
3409:
3399:
3394:
3388:
3386:
3384:
3383:
3380:
3376:
3373:
3372:
3370:
3369:
3364:
3359:
3354:
3349:
3344:
3342:Disjoint union
3339:
3334:
3328:
3322:
3320:
3314:
3313:
3311:
3310:
3309:
3308:
3303:
3292:
3291:
3289:Martin's axiom
3286:
3281:
3276:
3271:
3266:
3261:
3256:
3254:Extensionality
3251:
3250:
3249:
3239:
3234:
3233:
3232:
3227:
3222:
3212:
3206:
3204:
3198:
3197:
3190:
3188:
3186:
3185:
3179:
3177:
3173:
3172:
3165:
3164:
3157:
3150:
3142:
3133:
3132:
3118:
3115:
3114:
3112:
3111:
3106:
3101:
3096:
3091:
3090:
3089:
3079:
3074:
3069:
3060:
3055:
3050:
3045:
3043:Abstract logic
3039:
3037:
3033:
3032:
3030:
3029:
3024:
3022:Turing machine
3019:
3014:
3009:
3004:
2999:
2994:
2993:
2992:
2987:
2982:
2977:
2972:
2962:
2960:Computable set
2957:
2952:
2947:
2942:
2936:
2934:
2928:
2927:
2925:
2924:
2919:
2914:
2909:
2904:
2899:
2894:
2889:
2888:
2887:
2882:
2877:
2867:
2862:
2857:
2855:Satisfiability
2852:
2847:
2842:
2841:
2840:
2830:
2829:
2828:
2818:
2817:
2816:
2811:
2806:
2801:
2796:
2786:
2785:
2784:
2779:
2772:Interpretation
2768:
2766:
2760:
2759:
2757:
2756:
2751:
2746:
2741:
2736:
2726:
2721:
2720:
2719:
2718:
2717:
2707:
2702:
2692:
2687:
2682:
2677:
2672:
2667:
2661:
2659:
2653:
2652:
2649:
2648:
2646:
2645:
2637:
2636:
2635:
2634:
2629:
2628:
2627:
2622:
2617:
2597:
2596:
2595:
2593:minimal axioms
2590:
2579:
2578:
2577:
2566:
2565:
2564:
2559:
2554:
2549:
2544:
2539:
2526:
2524:
2505:
2504:
2502:
2501:
2500:
2499:
2487:
2482:
2481:
2480:
2475:
2470:
2465:
2455:
2450:
2445:
2440:
2439:
2438:
2433:
2423:
2422:
2421:
2416:
2411:
2406:
2396:
2391:
2390:
2389:
2384:
2379:
2369:
2368:
2367:
2362:
2357:
2352:
2347:
2342:
2332:
2327:
2322:
2317:
2316:
2315:
2310:
2305:
2300:
2290:
2285:
2283:Formation rule
2280:
2275:
2274:
2273:
2268:
2258:
2257:
2256:
2246:
2241:
2236:
2231:
2225:
2219:
2202:Formal systems
2198:
2197:
2194:
2193:
2191:
2190:
2185:
2180:
2175:
2170:
2165:
2160:
2155:
2150:
2145:
2144:
2143:
2138:
2127:
2125:
2121:
2120:
2118:
2117:
2116:
2115:
2105:
2100:
2099:
2098:
2091:Large cardinal
2088:
2083:
2078:
2073:
2068:
2054:
2053:
2052:
2047:
2042:
2027:
2025:
2015:
2014:
2012:
2011:
2010:
2009:
2004:
1999:
1989:
1984:
1979:
1974:
1969:
1964:
1959:
1954:
1949:
1944:
1939:
1934:
1928:
1926:
1919:
1918:
1916:
1915:
1914:
1913:
1908:
1903:
1898:
1893:
1888:
1880:
1879:
1878:
1873:
1863:
1858:
1856:Extensionality
1853:
1851:Ordinal number
1848:
1838:
1833:
1832:
1831:
1820:
1814:
1808:
1807:
1804:
1803:
1801:
1800:
1795:
1790:
1785:
1780:
1775:
1770:
1769:
1768:
1758:
1757:
1756:
1743:
1741:
1735:
1734:
1732:
1731:
1730:
1729:
1724:
1719:
1709:
1704:
1699:
1694:
1689:
1684:
1678:
1676:
1670:
1669:
1667:
1666:
1661:
1656:
1651:
1646:
1641:
1636:
1635:
1634:
1624:
1619:
1614:
1609:
1604:
1599:
1593:
1591:
1582:
1576:
1575:
1573:
1572:
1567:
1562:
1557:
1552:
1547:
1535:Cantor's
1533:
1528:
1523:
1513:
1511:
1498:
1497:
1495:
1494:
1489:
1484:
1479:
1474:
1469:
1464:
1459:
1454:
1449:
1444:
1439:
1434:
1433:
1432:
1421:
1419:
1415:
1414:
1407:
1406:
1399:
1392:
1384:
1378:
1377:
1371:
1351:
1335:
1329:
1303:
1300:
1297:
1296:
1272:
1228:
1206:
1198:
1190:Academic Press
1182:Eric Schechter
1173:
1147:
1146:
1144:
1141:
1140:
1139:
1134:
1127:
1124:
1100:
1069:
1056:and denoted P(
1025:
1022:
990:Main article:
987:
984:
983:
982:
974:
966:
958:
949:
948:
940:
915:
912:
909:
908:
905:
902:
899:
896:
890:
889:
886:
883:
880:
877:
871:
870:
867:
864:
861:
858:
852:
851:
847:
843:
839:
835:
831:
827:
823:
819:
813:
812:
809:
806:
803:
800:
797:
794:
791:
788:
782:
781:
778:
775:
772:
769:
766:
763:
760:
757:
751:
750:
747:
744:
741:
738:
735:
732:
729:
726:
722:
721:
718:
715:
712:
709:
705:
704:
701:
698:
695:
692:
598:
597:
586:
583:
580:
551:
550:
539:
536:
533:
440:
439:
428:
425:
422:
408:set membership
399:
396:
344:
339:
336:
333:
330:
325:
320:
317:
314:
311:
308:
303:
298:
295:
292:
287:
284:
281:
258:
255:
252:
249:
246:
214:
211:
208:
205:
202:
199:
196:
193:
190:
187:
184:
181:
178:
147:
144:
141:
138:
135:
126:, for example
107:
104:
101:
98:
95:
92:
89:
86:
83:
80:
77:
65:
62:
26:
9:
6:
4:
3:
2:
3820:
3809:
3806:
3805:
3803:
3788:
3787:Ernst Zermelo
3785:
3783:
3780:
3778:
3775:
3773:
3772:Willard Quine
3770:
3768:
3765:
3763:
3760:
3758:
3755:
3753:
3750:
3748:
3745:
3743:
3740:
3738:
3735:
3733:
3730:
3729:
3727:
3725:
3724:Set theorists
3721:
3715:
3712:
3710:
3707:
3705:
3702:
3701:
3699:
3693:
3691:
3688:
3687:
3684:
3676:
3673:
3671:
3670:KripkeâPlatek
3668:
3664:
3661:
3660:
3659:
3656:
3655:
3654:
3651:
3647:
3644:
3643:
3642:
3641:
3637:
3633:
3630:
3629:
3628:
3625:
3624:
3621:
3618:
3616:
3613:
3611:
3608:
3606:
3603:
3602:
3600:
3596:
3590:
3587:
3585:
3582:
3580:
3577:
3575:
3573:
3568:
3566:
3563:
3561:
3558:
3555:
3551:
3548:
3546:
3543:
3539:
3536:
3534:
3531:
3529:
3526:
3525:
3524:
3521:
3518:
3514:
3511:
3509:
3506:
3504:
3501:
3499:
3496:
3495:
3493:
3490:
3486:
3480:
3477:
3475:
3472:
3470:
3467:
3465:
3462:
3460:
3457:
3455:
3452:
3450:
3447:
3443:
3440:
3438:
3435:
3434:
3433:
3430:
3428:
3425:
3423:
3420:
3418:
3415:
3413:
3410:
3407:
3403:
3400:
3398:
3395:
3393:
3390:
3389:
3387:
3381:
3378:
3377:
3374:
3368:
3365:
3363:
3360:
3358:
3355:
3353:
3350:
3348:
3345:
3343:
3340:
3338:
3335:
3332:
3329:
3327:
3324:
3323:
3321:
3319:
3315:
3307:
3306:specification
3304:
3302:
3299:
3298:
3297:
3294:
3293:
3290:
3287:
3285:
3282:
3280:
3277:
3275:
3272:
3270:
3267:
3265:
3262:
3260:
3257:
3255:
3252:
3248:
3245:
3244:
3243:
3240:
3238:
3235:
3231:
3228:
3226:
3223:
3221:
3218:
3217:
3216:
3213:
3211:
3208:
3207:
3205:
3203:
3199:
3194:
3184:
3181:
3180:
3178:
3174:
3170:
3163:
3158:
3156:
3151:
3149:
3144:
3143:
3140:
3130:
3129:
3124:
3116:
3110:
3107:
3105:
3102:
3100:
3097:
3095:
3092:
3088:
3085:
3084:
3083:
3080:
3078:
3075:
3073:
3070:
3068:
3064:
3061:
3059:
3056:
3054:
3051:
3049:
3046:
3044:
3041:
3040:
3038:
3034:
3028:
3025:
3023:
3020:
3018:
3017:Recursive set
3015:
3013:
3010:
3008:
3005:
3003:
3000:
2998:
2995:
2991:
2988:
2986:
2983:
2981:
2978:
2976:
2973:
2971:
2968:
2967:
2966:
2963:
2961:
2958:
2956:
2953:
2951:
2948:
2946:
2943:
2941:
2938:
2937:
2935:
2933:
2929:
2923:
2920:
2918:
2915:
2913:
2910:
2908:
2905:
2903:
2900:
2898:
2895:
2893:
2890:
2886:
2883:
2881:
2878:
2876:
2873:
2872:
2871:
2868:
2866:
2863:
2861:
2858:
2856:
2853:
2851:
2848:
2846:
2843:
2839:
2836:
2835:
2834:
2831:
2827:
2826:of arithmetic
2824:
2823:
2822:
2819:
2815:
2812:
2810:
2807:
2805:
2802:
2800:
2797:
2795:
2792:
2791:
2790:
2787:
2783:
2780:
2778:
2775:
2774:
2773:
2770:
2769:
2767:
2765:
2761:
2755:
2752:
2750:
2747:
2745:
2742:
2740:
2737:
2734:
2733:from ZFC
2730:
2727:
2725:
2722:
2716:
2713:
2712:
2711:
2708:
2706:
2703:
2701:
2698:
2697:
2696:
2693:
2691:
2688:
2686:
2683:
2681:
2678:
2676:
2673:
2671:
2668:
2666:
2663:
2662:
2660:
2658:
2654:
2644:
2643:
2639:
2638:
2633:
2632:non-Euclidean
2630:
2626:
2623:
2621:
2618:
2616:
2615:
2611:
2610:
2608:
2605:
2604:
2602:
2598:
2594:
2591:
2589:
2586:
2585:
2584:
2580:
2576:
2573:
2572:
2571:
2567:
2563:
2560:
2558:
2555:
2553:
2550:
2548:
2545:
2543:
2540:
2538:
2535:
2534:
2532:
2528:
2527:
2525:
2520:
2514:
2509:Example
2506:
2498:
2493:
2492:
2491:
2488:
2486:
2483:
2479:
2476:
2474:
2471:
2469:
2466:
2464:
2461:
2460:
2459:
2456:
2454:
2451:
2449:
2446:
2444:
2441:
2437:
2434:
2432:
2429:
2428:
2427:
2424:
2420:
2417:
2415:
2412:
2410:
2407:
2405:
2402:
2401:
2400:
2397:
2395:
2392:
2388:
2385:
2383:
2380:
2378:
2375:
2374:
2373:
2370:
2366:
2363:
2361:
2358:
2356:
2353:
2351:
2348:
2346:
2343:
2341:
2338:
2337:
2336:
2333:
2331:
2328:
2326:
2323:
2321:
2318:
2314:
2311:
2309:
2306:
2304:
2301:
2299:
2296:
2295:
2294:
2291:
2289:
2286:
2284:
2281:
2279:
2276:
2272:
2269:
2267:
2266:by definition
2264:
2263:
2262:
2259:
2255:
2252:
2251:
2250:
2247:
2245:
2242:
2240:
2237:
2235:
2232:
2230:
2227:
2226:
2223:
2220:
2218:
2214:
2209:
2203:
2199:
2189:
2186:
2184:
2181:
2179:
2176:
2174:
2171:
2169:
2166:
2164:
2161:
2159:
2156:
2154:
2153:KripkeâPlatek
2151:
2149:
2146:
2142:
2139:
2137:
2134:
2133:
2132:
2129:
2128:
2126:
2122:
2114:
2111:
2110:
2109:
2106:
2104:
2101:
2097:
2094:
2093:
2092:
2089:
2087:
2084:
2082:
2079:
2077:
2074:
2072:
2069:
2066:
2062:
2058:
2055:
2051:
2048:
2046:
2043:
2041:
2038:
2037:
2036:
2032:
2029:
2028:
2026:
2024:
2020:
2016:
2008:
2005:
2003:
2000:
1998:
1997:constructible
1995:
1994:
1993:
1990:
1988:
1985:
1983:
1980:
1978:
1975:
1973:
1970:
1968:
1965:
1963:
1960:
1958:
1955:
1953:
1950:
1948:
1945:
1943:
1940:
1938:
1935:
1933:
1930:
1929:
1927:
1925:
1920:
1912:
1909:
1907:
1904:
1902:
1899:
1897:
1894:
1892:
1889:
1887:
1884:
1883:
1881:
1877:
1874:
1872:
1869:
1868:
1867:
1864:
1862:
1859:
1857:
1854:
1852:
1849:
1847:
1843:
1839:
1837:
1834:
1830:
1827:
1826:
1825:
1822:
1821:
1818:
1815:
1813:
1809:
1799:
1796:
1794:
1791:
1789:
1786:
1784:
1781:
1779:
1776:
1774:
1771:
1767:
1764:
1763:
1762:
1759:
1755:
1750:
1749:
1748:
1745:
1744:
1742:
1740:
1736:
1728:
1725:
1723:
1720:
1718:
1715:
1714:
1713:
1710:
1708:
1705:
1703:
1700:
1698:
1695:
1693:
1690:
1688:
1685:
1683:
1680:
1679:
1677:
1675:
1674:Propositional
1671:
1665:
1662:
1660:
1657:
1655:
1652:
1650:
1647:
1645:
1642:
1640:
1637:
1633:
1630:
1629:
1628:
1625:
1623:
1620:
1618:
1615:
1613:
1610:
1608:
1605:
1603:
1602:Logical truth
1600:
1598:
1595:
1594:
1592:
1590:
1586:
1583:
1581:
1577:
1571:
1568:
1566:
1563:
1561:
1558:
1556:
1553:
1551:
1548:
1546:
1542:
1538:
1534:
1532:
1529:
1527:
1524:
1522:
1518:
1515:
1514:
1512:
1510:
1504:
1499:
1493:
1490:
1488:
1485:
1483:
1480:
1478:
1475:
1473:
1470:
1468:
1465:
1463:
1460:
1458:
1455:
1453:
1450:
1448:
1445:
1443:
1440:
1438:
1435:
1431:
1428:
1427:
1426:
1423:
1422:
1420:
1416:
1412:
1405:
1400:
1398:
1393:
1391:
1386:
1385:
1382:
1374:
1372:0-486-61630-4
1368:
1363:
1362:
1356:
1352:
1348:
1344:
1340:
1336:
1332:
1330:0-387-90092-6
1326:
1322:
1317:
1316:
1310:
1306:
1305:
1286:
1285:brilliant.org
1282:
1276:
1268:
1264:
1259:
1254:
1250:
1246:
1242:
1235:
1233:
1224:
1220:
1216:
1215:George Boolos
1210:
1201:
1199:0-12-622760-8
1195:
1191:
1187:
1183:
1177:
1163:
1159:
1152:
1148:
1138:
1135:
1133:
1130:
1129:
1123:
1120:
1116:
1098:
1088:
1084:
1067:
1059:
1055:
1051:
1047:
1043:
1039:
1035:
1031:
1021:
1015:
1011:
1007:
1003:
999:
993:
980:
975:
972:
967:
964:
959:
956:
951:
950:
946:
941:
938:
933:
932:
931:
929:
925:
921:
895:
892:
891:
876:
873:
872:
857:
854:
853:
848:
844:
840:
836:
832:
828:
824:
820:
818:
815:
814:
810:
807:
804:
801:
798:
795:
792:
789:
787:
784:
783:
779:
776:
773:
770:
767:
764:
761:
758:
756:
753:
752:
748:
745:
742:
739:
736:
733:
730:
727:
724:
723:
708:Unicode name
707:
706:
689:
683:
680:
677:
672:
666:
664:
661:is read as a
659:
655:
650:
644:
640:
638:
620:
617:
616:
609:
607:
603:
584:
581:
578:
571:
570:
569:
567:
562:
560:
556:
537:
534:
531:
524:
523:
522:
520:
515:
513:
512:George Boolos
509:
505:
501:
497:
493:
489:
485:
481:
478:lies in
477:
473:
469:
465:
461:
457:
453:
449:
445:
426:
423:
420:
413:
412:
411:
409:
405:
395:
391:
387:
383:
379:
375:
368:
323:
301:
282:
279:
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1290:2020-08-10
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1167:2020-08-10
1143:References
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1932:Countable
1922:Types of
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