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2073:(the concept that corresponds to size, that is, the number of elements, of a finite set) as the whole; such cases can run counter to one's initial intuition.
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2069:. These are two examples in which both the subset and the whole set are infinite, and the subset has the same
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3408: – Subset T of a topological vector space X where the linear span of T is a dense subset of X
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3366: – Partial order that arises as the subset-inclusion relation on some collection of objects
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2084:. In this example, both sets are infinite, but the latter set has a larger cardinality (or
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3357: – In geometry, set whose intersection with every line is a single line segment
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superset respectively; that is, with the same meaning as and instead of the symbols
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The set A = {1, 2} is a proper subset of B = {1, 2, 3}, thus both expressions
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respectively; that is, with the same meaning as and instead of the symbols
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3927:
3162:{\displaystyle A\subseteq B{\text{ if and only if }}|A\cap B|=|A|.}
191:. The relationship of one set being a subset of another is called
48:
6301:
4978:
3770:
3096:
of their intersection is equal to the cardinality of A. Formally:
1903:
5680:
3638:
3065:{\displaystyle A\subseteq B{\text{ if and only if }}A\cup B=B.}
2996:{\displaystyle A\subseteq B{\text{ if and only if }}A\cap B=A.}
526:{\displaystyle \forall x\left(x\in A\Rightarrow x\in B\right),}
418:
The validity of this technique can be seen as a consequence of
313:{\displaystyle \forall x\left(x\in A\Rightarrow x\in B\right).}
3393: – System of elements that are subordinated to each other
6177:
5730:
5490:
5426:
4522:
3868:
3713:
2359:{\displaystyle A\leq B{\text{ if and only if }}B\subseteq A.}
348:
by applying a proof technique known as the element argument:
2589:{\displaystyle S=\left\{s_{1},s_{2},\ldots ,s_{k}\right\},}
1497:
For example, for these authors, it is true of every set
3401:
Pages displaying short descriptions of redirect targets
3380:
Pages displaying short descriptions of redirect targets
3359:
Pages displaying short descriptions of redirect targets
2785:
3310:
3284:
3254:
3240:
to some collection of sets ordered by inclusion. The
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2007:
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1948:
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705:
at least one element of B which is not an element of
653:
613:
539:
482:
428:
366:
328:
269:
243:
108:
74:
3378: – Connected open subset of a topological space
3177:
on sets. In fact, the subsets of a given set form a
3019:
if and only if their union is equal to B. Formally:
399:
is a particular but arbitrarily chosen element of A
3399: – Mathematical set with some added structure
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120:
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3204:, in the sense that every partially ordered set
2929:
2397:{\displaystyle \operatorname {\mathcal {P}} (S)}
3649:
3434:(7th ed.). New York: McGraw-Hill. p.
3372: – Study of parts and the wholes they form
30:"Superset" redirects here. For other uses, see
37:"⊃" redirects here. For the logic symbol, see
6102:
5442:
3684:
2801:
2788:
2065:is a proper subset of the set of points in a
3608:
3387: – Decision problem in computer science
2673:
2660:
2479:
2467:
830:
827:
465:{\displaystyle (c\in A)\Rightarrow (c\in B)}
27:Set whose elements all belong to another set
2950:their intersection is equal to A. Formally:
2120:C is a subset but not a proper subset of B.
6116:
6109:
6095:
5449:
5435:
3876:
3691:
3677:
2408:, the inclusion partial order is—up to an
2270:
2266:
3488:. San Francisco, CA: Dover Publications.
3430:Discrete Mathematics and Its Applications
3193:, and the subset relation itself is the
3181:under the subset relation, in which the
2283:{\displaystyle A\leq B\iff A\subseteq B}
2042:greater than 10} is a proper subset of {
1975:a proper subset) of E = {1, 2, 3}, thus
1902:
1801:). Similarly, using the convention that
1537:Other authors prefer to use the symbols
881:
476:. Universal generalisation then implies
47:
2518:This can be illustrated by enumerating
1971:The set D = {1, 2, 3} is a subset (but
179:to be equal; if they are unequal, then
14:
6406:
3698:
3461:Discrete Mathematics with Applications
3274:of all ordinals less than or equal to
3244:are a simple example: if each ordinal
1898:
6090:
5430:
3672:
3650:
3579:
3508:
3425:
2323:by reverse set inclusion by defining
2818:, in analogue with the notation for
3458:
2596:, and associating with each subset
2061:; likewise, the set of points in a
877:
24:
2792:
2699:th coordinate is 1 if and only if
2377:
2299:
2226:
2163:
1401:
866:has no elements, and therefore is
483:
472:for an arbitrarily chosen element
270:
25:
6425:
3631:
3483:
2460:) copies of the partial order on
2316:{\displaystyle {\mathcal {P}}(S)}
2243:{\displaystyle {\mathcal {P}}(S)}
2180:{\displaystyle {\mathcal {P}}(S)}
2080:is a proper subset of the set of
2057:is a proper subset of the set of
2050:is an odd number greater than 10}
6388:
6377:
5480:
5410:
3637:
3463:(Fourth ed.). p. 337.
2899:is also common, especially when
2811:{\displaystyle {\tbinom {A}{k}}}
2113:
2101:
1581:(also called strict) subset and
1195:
3602:
5456:
3573:
3541:
3502:
3477:
3452:
3419:
3329:
3323:
3317:
3311:
3261:
3255:
3223:
3211:
3173:The subset relation defines a
3152:
3144:
3136:
3122:
2880:
2873:
2438:
2430:
2391:
2385:
2310:
2304:
2290:. We may also partially order
2267:
2237:
2231:
2174:
2168:
1911:form a subset of the polygons.
804:{\displaystyle B\supsetneq A.}
503:
459:
447:
444:
441:
429:
290:
219:is included (or contained) in
13:
1:
5371:History of mathematical logic
3412:
2930:Other properties of inclusion
2020:{\displaystyle D\subsetneq E}
1961:{\displaystyle A\subsetneq B}
1843:{\displaystyle A\subseteq B,}
1406:Some authors use the symbols
1390:{\displaystyle B\subsetneq A}
1364:{\displaystyle A\subsetneq B}
1331:{\displaystyle A\subsetneq C}
1305:{\displaystyle B\subsetneq C}
1279:{\displaystyle A\subsetneq B}
1245:{\displaystyle A\subsetneq A}
959:{\displaystyle A\subseteq C.}
859:{\displaystyle \varnothing ,}
753:{\displaystyle A\subsetneq B}
669:{\displaystyle B\supseteq A.}
565:
555:{\displaystyle A\subseteq B,}
382:{\displaystyle A\subseteq B,}
6414:Basic concepts in set theory
5296:Primitive recursive function
3229:{\displaystyle (X,\preceq )}
2822:, which count the number of
2688:{\displaystyle \{0,1\}^{k},}
2615:{\displaystyle T\subseteq S}
2125:
1994:{\displaystyle D\subseteq E}
1935:{\displaystyle A\subseteq B}
1621:{\displaystyle \supsetneq .}
1158:{\displaystyle B\subseteq A}
1132:{\displaystyle A\subseteq B}
1099:{\displaystyle A\subseteq C}
1073:{\displaystyle B\subseteq C}
1047:{\displaystyle A\subseteq B}
1014:{\displaystyle A\subseteq A}
930:{\displaystyle B\subseteq C}
904:{\displaystyle A\subseteq B}
626:{\displaystyle A\subseteq B}
341:{\displaystyle A\subseteq B}
322:One can prove the statement
256:{\displaystyle A\subseteq B}
121:{\displaystyle B\supseteq A}
87:{\displaystyle A\subseteq B}
7:
3348:
3338:{\displaystyle \subseteq .}
3248:is identified with the set
3200:Inclusion is the canonical
1880:{\displaystyle A\subset B,}
1598:{\displaystyle \subsetneq }
1523:{\displaystyle A\subset A.}
1490:{\displaystyle \supseteq .}
10:
6430:
5947:von Neumann–Bernays–Gödel
4360:Schröder–Bernstein theorem
4087:Monadic predicate calculus
3746:Foundations of mathematics
3550:Subsets and Proper Subsets
3516:(3rd ed.), New York:
3426:Rosen, Kenneth H. (2012).
3195:Boolean inclusion relation
3118: if and only if
3041: if and only if
2972: if and only if
2341: if and only if
2206:{\displaystyle \subseteq }
2130:The set of all subsets of
2108:A is a proper subset of B.
1891:definitely does not equal
1785:definitely does not equal
1641:{\displaystyle \subseteq }
1467:{\displaystyle \subseteq }
43:horseshoe (disambiguation)
36:
29:
6374:
6125:
6011:
5974:
5886:
5776:
5748:One-to-one correspondence
5664:
5605:
5489:
5478:
5464:
5406:
5393:Philosophy of mathematics
5342:Automated theorem proving
5324:
5219:
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4513:
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4467:Von Neumann–Bernays–Gödel
4412:
4306:
4210:
4108:
4099:
4026:
3961:
3867:
3789:
3706:
3514:Real and complex analysis
207:may also be expressed as
32:Superset (disambiguation)
3459:Epp, Susanna S. (2011).
1814:{\displaystyle \subset }
1737:{\displaystyle x\leq y,}
1661:{\displaystyle \subset }
1570:{\displaystyle \supset }
1550:{\displaystyle \subset }
1439:{\displaystyle \supset }
1419:{\displaystyle \subset }
420:universal generalization
360:be given. To prove that
5043:Self-verifying theories
4864:Tarski's axiomatization
3815:Tarski's undefinability
3810:incompleteness theorems
3391:Subsumptive containment
3297:{\displaystyle a\leq b}
2622:(i.e., each element of
2511:{\displaystyle 0<1.}
2485:{\displaystyle \{0,1\}}
1774:{\displaystyle x<y,}
533:which is equivalent to
211:includes (or contains)
6395:Mathematics portal
5706:Constructible universe
5526:Constructibility (V=L)
5417:Mathematics portal
5028:Proof of impossibility
4676:propositional variable
3986:Propositional calculus
3339:
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3268:
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2512:
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2446:
2398:
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2181:
2144:
2092:Another example in an
2088:) than the former set.
2021:
1995:
1962:
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905:
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837:
805:
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670:
627:
586:is also an element of
556:
527:
466:
422:: the technique shows
416:
383:
342:
314:
257:
129:
122:
88:
41:. For other uses, see
6384:Philosophy portal
5929:Principia Mathematica
5763:Transfinite induction
5622:(i.e. set difference)
5286:Kolmogorov complexity
5239:Computably enumerable
5139:Model complete theory
4931:Principia Mathematica
3991:Propositional formula
3820:Banach–Tarski paradox
3586:mathworld.wolfram.com
3340:
3299:
3269:
3231:
3164:
3092:, if and only if the
3067:
2998:
2914:
2894:
2857:
2837:
2820:binomial coefficients
2813:
2772:
2752:
2721:
2719:{\displaystyle s_{i}}
2690:
2644:
2642:{\displaystyle 2^{S}}
2617:
2591:
2513:
2487:
2447:
2445:{\displaystyle k=|S|}
2399:
2361:
2318:
2285:
2245:
2208:
2182:
2145:
2022:
1996:
1963:
1937:
1906:
1882:
1854:may or may not equal
1845:
1821:is proper subset, if
1816:
1776:
1748:may or may not equal
1739:
1710:
1708:{\displaystyle <.}
1687:
1685:{\displaystyle \leq }
1663:
1643:
1623:
1600:
1572:
1552:
1525:
1492:
1469:
1441:
1421:
1392:
1366:
1333:
1307:
1281:
1247:
1221:
1186:
1160:
1134:
1101:
1075:
1049:
1016:
990:
961:
932:
906:
885:
861:
838:
806:
755:
671:
628:
557:
528:
467:
384:
350:
343:
315:
258:
171:. It is possible for
155:are also elements of
123:
89:
51:
6003:Burali-Forti paradox
5758:Set-builder notation
5711:Continuum hypothesis
5651:Symmetric difference
5234:Church–Turing thesis
5221:Computability theory
4430:continuum hypothesis
3948:Square of opposition
3806:Gödel's completeness
3646:at Wikimedia Commons
3486:Set Theory and Logic
3308:
3282:
3252:
3208:
3104:
3027:
2958:
2903:
2892:{\displaystyle ^{k}}
2870:
2846:
2826:
2781:
2761:
2741:
2703:
2657:
2626:
2600:
2522:
2496:
2464:
2420:
2372:
2327:
2294:
2254:
2221:
2197:
2158:
2154:, and is denoted by
2134:
2027:is not true (false).
2005:
1979:
1946:
1920:
1862:
1825:
1805:
1799:irreflexive relation
1756:
1719:
1696:
1676:
1652:
1632:
1609:
1589:
1561:
1541:
1505:
1478:
1458:
1430:
1410:
1375:
1349:
1316:
1290:
1264:
1230:
1210:
1169:
1143:
1117:
1084:
1058:
1032:
999:
979:
941:
915:
889:
870:a subset of any set
847:
836:{\displaystyle \{\}}
824:
786:
738:
651:
611:
537:
480:
426:
364:
326:
267:
241:
106:
72:
5964:Tarski–Grothendieck
5388:Mathematical object
5279:P versus NP problem
5244:Computable function
5038:Reverse mathematics
4964:Logical consequence
4841:primitive recursive
4836:elementary function
4609:Free/bound variable
4462:Tarski–Grothendieck
3981:Logical connectives
3911:Logical equivalence
3761:Logical consequence
3616:. Springer-Verlag.
3580:Weisstein, Eric W.
1899:Examples of subsets
1184:{\displaystyle A=B}
578:are sets and every
94:) and, conversely,
5553:Limitation of size
5186:Transfer principle
5149:Semantics of logic
5134:Categorical theory
5110:Non-standard model
4624:Logical connective
3751:Information theory
3700:Mathematical logic
3652:Weisstein, Eric W.
3385:Subset sum problem
3335:
3294:
3264:
3226:
3159:
3062:
2993:
2909:
2889:
2852:
2832:
2808:
2806:
2767:
2747:
2716:
2685:
2639:
2612:
2586:
2508:
2482:
2442:
2394:
2368:For the power set
2356:
2313:
2280:
2240:
2203:
2177:
2140:
2017:
1991:
1958:
1932:
1913:
1877:
1840:
1811:
1771:
1734:
1705:
1682:
1658:
1638:
1618:
1595:
1567:
1547:
1532:reflexive relation
1520:
1487:
1464:
1436:
1416:
1387:
1361:
1328:
1302:
1276:
1242:
1216:
1181:
1155:
1129:
1096:
1070:
1044:
1011:
985:
967:
956:
927:
901:
856:
833:
801:
760:, or equivalently,
750:
666:
633:, or equivalently,
623:
552:
523:
462:
379:
338:
310:
263:is represented as
253:
132:In mathematics, a
130:
118:
84:
39:horseshoe (symbol)
6401:
6400:
6369:
6368:
6084:
6083:
5993:Russell's paradox
5942:Zermelo–Fraenkel
5843:Dedekind-infinite
5716:Diagonal argument
5615:Cartesian product
5472:Set (mathematics)
5424:
5423:
5356:Abstract category
5159:Theories of truth
4969:Rule of inference
4959:Natural deduction
4940:
4939:
4485:
4484:
4190:Cartesian product
4095:
4094:
4001:Many-valued logic
3976:Boolean functions
3859:Russell's paradox
3834:diagonal argument
3731:First-order logic
3642:Media related to
3527:978-0-07-054234-1
3495:978-0-486-63829-4
3484:Stoll, Robert R.
3470:978-0-495-39132-6
3445:978-0-07-338309-5
3119:
3042:
2973:
2912:{\displaystyle k}
2862:-element set. In
2855:{\displaystyle n}
2835:{\displaystyle k}
2799:
2770:{\displaystyle A}
2750:{\displaystyle k}
2414:Cartesian product
2410:order isomorphism
2342:
2143:{\displaystyle S}
1668:analogous to the
1628:This usage makes
1219:{\displaystyle A}
988:{\displaystyle A}
562:as stated above.
409:is an element of
237:When quantified,
230:is a subset with
98:is a superset of
16:(Redirected from
6421:
6393:
6392:
6382:
6381:
6380:
6226:
6175:
6141:
6128:
6127:
6111:
6104:
6097:
6088:
6087:
6066:Bertrand Russell
6056:John von Neumann
6041:Abraham Fraenkel
6036:Richard Dedekind
5998:Suslin's problem
5909:Cantor's theorem
5626:De Morgan's laws
5484:
5451:
5444:
5437:
5428:
5427:
5415:
5414:
5366:History of logic
5361:Category of sets
5254:Decision problem
5033:Ordinal analysis
4974:Sequent calculus
4872:Boolean algebras
4812:
4811:
4786:
4757:logical/constant
4511:
4510:
4497:
4420:Zermelo–Fraenkel
4171:Set operations:
4106:
4105:
4043:
3874:
3873:
3854:Löwenheim–Skolem
3741:Formal semantics
3693:
3686:
3679:
3670:
3669:
3665:
3664:
3641:
3627:
3596:
3595:
3593:
3592:
3577:
3571:
3570:
3569:
3568:
3562:
3556:, archived from
3555:
3545:
3539:
3538:
3506:
3500:
3499:
3481:
3475:
3474:
3456:
3450:
3449:
3433:
3423:
3402:
3381:
3360:
3344:
3342:
3341:
3336:
3303:
3301:
3300:
3295:
3273:
3271:
3270:
3267:{\displaystyle }
3265:
3235:
3233:
3232:
3227:
3168:
3166:
3165:
3160:
3155:
3147:
3139:
3125:
3120:
3117:
3071:
3069:
3068:
3063:
3043:
3040:
3002:
3000:
2999:
2994:
2974:
2971:
2918:
2916:
2915:
2910:
2898:
2896:
2895:
2890:
2888:
2887:
2861:
2859:
2858:
2853:
2841:
2839:
2838:
2833:
2817:
2815:
2814:
2809:
2807:
2805:
2804:
2791:
2776:
2774:
2773:
2768:
2756:
2754:
2753:
2748:
2725:
2723:
2722:
2717:
2715:
2714:
2694:
2692:
2691:
2686:
2681:
2680:
2648:
2646:
2645:
2640:
2638:
2637:
2621:
2619:
2618:
2613:
2595:
2593:
2592:
2587:
2582:
2578:
2577:
2576:
2558:
2557:
2545:
2544:
2517:
2515:
2514:
2509:
2491:
2489:
2488:
2483:
2451:
2449:
2448:
2443:
2441:
2433:
2403:
2401:
2400:
2395:
2381:
2380:
2365:
2363:
2362:
2357:
2343:
2340:
2322:
2320:
2319:
2314:
2303:
2302:
2289:
2287:
2286:
2281:
2249:
2247:
2246:
2241:
2230:
2229:
2212:
2210:
2209:
2204:
2186:
2184:
2183:
2178:
2167:
2166:
2149:
2147:
2146:
2141:
2117:
2105:
2078:rational numbers
2059:rational numbers
2026:
2024:
2023:
2018:
2000:
1998:
1997:
1992:
1967:
1965:
1964:
1959:
1941:
1939:
1938:
1933:
1909:regular polygons
1886:
1884:
1883:
1878:
1849:
1847:
1846:
1841:
1820:
1818:
1817:
1812:
1780:
1778:
1777:
1772:
1743:
1741:
1740:
1735:
1715:For example, if
1714:
1712:
1711:
1706:
1691:
1689:
1688:
1683:
1667:
1665:
1664:
1659:
1647:
1645:
1644:
1639:
1627:
1625:
1624:
1619:
1604:
1602:
1601:
1596:
1576:
1574:
1573:
1568:
1556:
1554:
1553:
1548:
1529:
1527:
1526:
1521:
1496:
1494:
1493:
1488:
1473:
1471:
1470:
1465:
1445:
1443:
1442:
1437:
1425:
1423:
1422:
1417:
1396:
1394:
1393:
1388:
1370:
1368:
1367:
1362:
1337:
1335:
1334:
1329:
1311:
1309:
1308:
1303:
1285:
1283:
1282:
1277:
1251:
1249:
1248:
1243:
1225:
1223:
1222:
1217:
1206:: Given any set
1190:
1188:
1187:
1182:
1164:
1162:
1161:
1156:
1138:
1136:
1135:
1130:
1105:
1103:
1102:
1097:
1079:
1077:
1076:
1071:
1053:
1051:
1050:
1045:
1020:
1018:
1017:
1012:
994:
992:
991:
986:
975:: Given any set
965:
963:
962:
957:
936:
934:
933:
928:
910:
908:
907:
902:
878:Basic properties
865:
863:
862:
857:
842:
840:
839:
834:
810:
808:
807:
802:
759:
757:
756:
751:
675:
673:
672:
667:
632:
630:
629:
624:
561:
559:
558:
553:
532:
530:
529:
524:
519:
515:
471:
469:
468:
463:
388:
386:
385:
380:
347:
345:
344:
339:
319:
317:
316:
311:
306:
302:
262:
260:
259:
254:
127:
125:
124:
119:
93:
91:
90:
85:
21:
6429:
6428:
6424:
6423:
6422:
6420:
6419:
6418:
6404:
6403:
6402:
6397:
6387:
6386:
6378:
6376:
6370:
6365:
6361:
6353:
6349:
6341:
6338:
6335:
6327:
6324:
6321:
6313:
6309:
6304:
6296:
6292:
6287:
6279:
6278:
6275:
6271:
6263:
6262:
6259:
6255:
6247:
6243:
6235:
6231:
6222:
6213:
6209:
6204:
6196:
6192:
6184:
6180:
6171:
6162:
6158:
6150:
6146:
6137:
6121:
6119:logical symbols
6115:
6085:
6080:
6007:
5986:
5970:
5935:New Foundations
5882:
5772:
5691:Cardinal number
5674:
5660:
5601:
5485:
5476:
5460:
5455:
5425:
5420:
5409:
5402:
5347:Category theory
5337:Algebraic logic
5320:
5291:Lambda calculus
5229:Church encoding
5215:
5191:Truth predicate
5047:
5013:Complete theory
4936:
4805:
4801:
4797:
4792:
4784:
4504: and
4500:
4495:
4481:
4457:New Foundations
4425:axiom of choice
4408:
4370:Gödel numbering
4310: and
4302:
4206:
4091:
4041:
4022:
3971:Boolean algebra
3957:
3921:Equiconsistency
3886:Classical logic
3863:
3844:Halting problem
3832: and
3808: and
3796: and
3795:
3790:Theorems (
3785:
3702:
3697:
3634:
3624:
3605:
3600:
3599:
3590:
3588:
3578:
3574:
3566:
3564:
3560:
3553:
3547:
3546:
3542:
3528:
3507:
3503:
3496:
3482:
3478:
3471:
3457:
3453:
3446:
3424:
3420:
3415:
3400:
3379:
3364:Inclusion order
3358:
3351:
3309:
3306:
3305:
3304:if and only if
3283:
3280:
3279:
3253:
3250:
3249:
3242:ordinal numbers
3209:
3206:
3205:
3179:Boolean algebra
3151:
3143:
3135:
3121:
3116:
3105:
3102:
3101:
3039:
3028:
3025:
3024:
2970:
2959:
2956:
2955:
2932:
2924:cardinal number
2904:
2901:
2900:
2883:
2879:
2871:
2868:
2867:
2866:, the notation
2847:
2844:
2843:
2842:-subsets of an
2827:
2824:
2823:
2800:
2787:
2786:
2784:
2782:
2779:
2778:
2762:
2759:
2758:
2742:
2739:
2738:
2737:The set of all
2710:
2706:
2704:
2701:
2700:
2676:
2672:
2658:
2655:
2654:
2633:
2629:
2627:
2624:
2623:
2601:
2598:
2597:
2572:
2568:
2553:
2549:
2540:
2536:
2535:
2531:
2523:
2520:
2519:
2497:
2494:
2493:
2465:
2462:
2461:
2437:
2429:
2421:
2418:
2417:
2376:
2375:
2373:
2370:
2369:
2339:
2328:
2325:
2324:
2298:
2297:
2295:
2292:
2291:
2255:
2252:
2251:
2225:
2224:
2222:
2219:
2218:
2198:
2195:
2194:
2162:
2161:
2159:
2156:
2155:
2135:
2132:
2131:
2128:
2121:
2118:
2109:
2106:
2055:natural numbers
2006:
2003:
2002:
1980:
1977:
1976:
1947:
1944:
1943:
1921:
1918:
1917:
1901:
1863:
1860:
1859:
1826:
1823:
1822:
1806:
1803:
1802:
1757:
1754:
1753:
1720:
1717:
1716:
1697:
1694:
1693:
1677:
1674:
1673:
1653:
1650:
1649:
1633:
1630:
1629:
1610:
1607:
1606:
1590:
1587:
1586:
1562:
1559:
1558:
1542:
1539:
1538:
1506:
1503:
1502:
1479:
1476:
1475:
1459:
1456:
1455:
1431:
1428:
1427:
1411:
1408:
1407:
1404:
1402:⊂ and ⊃ symbols
1376:
1373:
1372:
1350:
1347:
1346:
1317:
1314:
1313:
1291:
1288:
1287:
1265:
1262:
1261:
1231:
1228:
1227:
1211:
1208:
1207:
1198:
1170:
1167:
1166:
1144:
1141:
1140:
1118:
1115:
1114:
1085:
1082:
1081:
1059:
1056:
1055:
1033:
1030:
1029:
1000:
997:
996:
980:
977:
976:
942:
939:
938:
916:
913:
912:
890:
887:
886:
880:
848:
845:
844:
825:
822:
821:
787:
784:
783:
739:
736:
735:
685:is a subset of
652:
649:
648:
612:
609:
608:
568:
538:
535:
534:
493:
489:
481:
478:
477:
427:
424:
423:
365:
362:
361:
327:
324:
323:
280:
276:
268:
265:
264:
242:
239:
238:
203:is a subset of
107:
104:
103:
73:
70:
69:
56:
46:
35:
28:
23:
22:
15:
12:
11:
5:
6427:
6417:
6416:
6399:
6398:
6375:
6372:
6371:
6367:
6366:
6357:
6356:
6354:
6345:
6344:
6342:
6331:
6330:
6328:
6317:
6316:
6314:
6300:
6299:
6297:
6283:
6282:
6280:
6276:quantification
6272:
6267:
6266:
6264:
6260:quantification
6256:
6251:
6250:
6248:
6239:
6238:
6236:
6217:
6216:
6214:
6200:
6199:
6197:
6188:
6187:
6185:
6166:
6165:
6163:
6154:
6153:
6151:
6132:
6131:
6126:
6123:
6122:
6114:
6113:
6106:
6099:
6091:
6082:
6081:
6079:
6078:
6073:
6071:Thoralf Skolem
6068:
6063:
6058:
6053:
6048:
6043:
6038:
6033:
6028:
6023:
6017:
6015:
6009:
6008:
6006:
6005:
6000:
5995:
5989:
5987:
5985:
5984:
5981:
5975:
5972:
5971:
5969:
5968:
5967:
5966:
5961:
5956:
5955:
5954:
5939:
5938:
5937:
5925:
5924:
5923:
5912:
5911:
5906:
5901:
5896:
5890:
5888:
5884:
5883:
5881:
5880:
5875:
5870:
5865:
5856:
5851:
5846:
5836:
5831:
5830:
5829:
5824:
5819:
5809:
5799:
5794:
5789:
5783:
5781:
5774:
5773:
5771:
5770:
5765:
5760:
5755:
5753:Ordinal number
5750:
5745:
5740:
5735:
5734:
5733:
5728:
5718:
5713:
5708:
5703:
5698:
5688:
5683:
5677:
5675:
5673:
5672:
5669:
5665:
5662:
5661:
5659:
5658:
5653:
5648:
5643:
5638:
5633:
5631:Disjoint union
5628:
5623:
5617:
5611:
5609:
5603:
5602:
5600:
5599:
5598:
5597:
5592:
5581:
5580:
5578:Martin's axiom
5575:
5570:
5565:
5560:
5555:
5550:
5545:
5543:Extensionality
5540:
5539:
5538:
5528:
5523:
5522:
5521:
5516:
5511:
5501:
5495:
5493:
5487:
5486:
5479:
5477:
5475:
5474:
5468:
5466:
5462:
5461:
5454:
5453:
5446:
5439:
5431:
5422:
5421:
5407:
5404:
5403:
5401:
5400:
5395:
5390:
5385:
5380:
5379:
5378:
5368:
5363:
5358:
5349:
5344:
5339:
5334:
5332:Abstract logic
5328:
5326:
5322:
5321:
5319:
5318:
5313:
5311:Turing machine
5308:
5303:
5298:
5293:
5288:
5283:
5282:
5281:
5276:
5271:
5266:
5261:
5251:
5249:Computable set
5246:
5241:
5236:
5231:
5225:
5223:
5217:
5216:
5214:
5213:
5208:
5203:
5198:
5193:
5188:
5183:
5178:
5177:
5176:
5171:
5166:
5156:
5151:
5146:
5144:Satisfiability
5141:
5136:
5131:
5130:
5129:
5119:
5118:
5117:
5107:
5106:
5105:
5100:
5095:
5090:
5085:
5075:
5074:
5073:
5068:
5061:Interpretation
5057:
5055:
5049:
5048:
5046:
5045:
5040:
5035:
5030:
5025:
5015:
5010:
5009:
5008:
5007:
5006:
4996:
4991:
4981:
4976:
4971:
4966:
4961:
4956:
4950:
4948:
4942:
4941:
4938:
4937:
4935:
4934:
4926:
4925:
4924:
4923:
4918:
4917:
4916:
4911:
4906:
4886:
4885:
4884:
4882:minimal axioms
4879:
4868:
4867:
4866:
4855:
4854:
4853:
4848:
4843:
4838:
4833:
4828:
4815:
4813:
4794:
4793:
4791:
4790:
4789:
4788:
4776:
4771:
4770:
4769:
4764:
4759:
4754:
4744:
4739:
4734:
4729:
4728:
4727:
4722:
4712:
4711:
4710:
4705:
4700:
4695:
4685:
4680:
4679:
4678:
4673:
4668:
4658:
4657:
4656:
4651:
4646:
4641:
4636:
4631:
4621:
4616:
4611:
4606:
4605:
4604:
4599:
4594:
4589:
4579:
4574:
4572:Formation rule
4569:
4564:
4563:
4562:
4557:
4547:
4546:
4545:
4535:
4530:
4525:
4520:
4514:
4508:
4491:Formal systems
4487:
4486:
4483:
4482:
4480:
4479:
4474:
4469:
4464:
4459:
4454:
4449:
4444:
4439:
4434:
4433:
4432:
4427:
4416:
4414:
4410:
4409:
4407:
4406:
4405:
4404:
4394:
4389:
4388:
4387:
4380:Large cardinal
4377:
4372:
4367:
4362:
4357:
4343:
4342:
4341:
4336:
4331:
4316:
4314:
4304:
4303:
4301:
4300:
4299:
4298:
4293:
4288:
4278:
4273:
4268:
4263:
4258:
4253:
4248:
4243:
4238:
4233:
4228:
4223:
4217:
4215:
4208:
4207:
4205:
4204:
4203:
4202:
4197:
4192:
4187:
4182:
4177:
4169:
4168:
4167:
4162:
4152:
4147:
4145:Extensionality
4142:
4140:Ordinal number
4137:
4127:
4122:
4121:
4120:
4109:
4103:
4097:
4096:
4093:
4092:
4090:
4089:
4084:
4079:
4074:
4069:
4064:
4059:
4058:
4057:
4047:
4046:
4045:
4032:
4030:
4024:
4023:
4021:
4020:
4019:
4018:
4013:
4008:
3998:
3993:
3988:
3983:
3978:
3973:
3967:
3965:
3959:
3958:
3956:
3955:
3950:
3945:
3940:
3935:
3930:
3925:
3924:
3923:
3913:
3908:
3903:
3898:
3893:
3888:
3882:
3880:
3871:
3865:
3864:
3862:
3861:
3856:
3851:
3846:
3841:
3836:
3824:Cantor's
3822:
3817:
3812:
3802:
3800:
3787:
3786:
3784:
3783:
3778:
3773:
3768:
3763:
3758:
3753:
3748:
3743:
3738:
3733:
3728:
3723:
3722:
3721:
3710:
3708:
3704:
3703:
3696:
3695:
3688:
3681:
3673:
3667:
3666:
3647:
3633:
3632:External links
3630:
3629:
3628:
3622:
3604:
3601:
3598:
3597:
3572:
3540:
3526:
3501:
3494:
3476:
3469:
3451:
3444:
3417:
3416:
3414:
3411:
3410:
3409:
3403:
3394:
3388:
3382:
3373:
3367:
3361:
3350:
3347:
3346:
3345:
3334:
3331:
3328:
3325:
3322:
3319:
3316:
3313:
3293:
3290:
3287:
3263:
3260:
3257:
3225:
3222:
3219:
3216:
3213:
3198:
3170:
3169:
3158:
3154:
3150:
3146:
3142:
3138:
3134:
3131:
3128:
3124:
3115:
3112:
3109:
3098:
3097:
3073:
3072:
3061:
3058:
3055:
3052:
3049:
3046:
3038:
3035:
3032:
3021:
3020:
3004:
3003:
2992:
2989:
2986:
2983:
2980:
2977:
2969:
2966:
2963:
2952:
2951:
2948:if and only if
2931:
2928:
2908:
2886:
2882:
2878:
2875:
2851:
2831:
2803:
2798:
2795:
2790:
2777:is denoted by
2766:
2746:
2713:
2709:
2684:
2679:
2675:
2671:
2668:
2665:
2662:
2636:
2632:
2611:
2608:
2605:
2585:
2581:
2575:
2571:
2567:
2564:
2561:
2556:
2552:
2548:
2543:
2539:
2534:
2530:
2527:
2507:
2504:
2501:
2481:
2478:
2475:
2472:
2469:
2440:
2436:
2432:
2428:
2425:
2393:
2390:
2387:
2384:
2379:
2355:
2352:
2349:
2346:
2338:
2335:
2332:
2312:
2309:
2306:
2301:
2279:
2276:
2273:
2269:
2265:
2262:
2259:
2239:
2236:
2233:
2228:
2202:
2190:The inclusion
2176:
2173:
2170:
2165:
2150:is called its
2139:
2127:
2124:
2123:
2122:
2119:
2112:
2110:
2107:
2100:
2090:
2089:
2087:
2074:
2051:
2028:
2016:
2013:
2010:
1990:
1987:
1984:
1974:
1969:
1957:
1954:
1951:
1931:
1928:
1925:
1900:
1897:
1876:
1873:
1870:
1867:
1839:
1836:
1833:
1830:
1810:
1770:
1767:
1764:
1761:
1733:
1730:
1727:
1724:
1704:
1701:
1681:
1657:
1637:
1617:
1614:
1594:
1584:
1580:
1566:
1546:
1519:
1516:
1513:
1510:
1486:
1483:
1463:
1453:
1449:
1435:
1415:
1403:
1400:
1399:
1398:
1386:
1383:
1380:
1360:
1357:
1354:
1338:
1327:
1324:
1321:
1301:
1298:
1295:
1275:
1272:
1269:
1253:
1241:
1238:
1235:
1215:
1197:
1194:
1193:
1192:
1180:
1177:
1174:
1154:
1151:
1148:
1128:
1125:
1122:
1106:
1095:
1092:
1089:
1069:
1066:
1063:
1043:
1040:
1037:
1021:
1010:
1007:
1004:
984:
955:
952:
949:
946:
926:
923:
920:
900:
897:
894:
879:
876:
855:
852:
832:
829:
814:
813:
812:
811:
800:
797:
794:
791:
761:
749:
746:
743:
679:
678:
677:
676:
665:
662:
659:
656:
634:
622:
619:
616:
567:
564:
551:
548:
545:
542:
522:
518:
514:
511:
508:
505:
502:
499:
496:
492:
488:
485:
461:
458:
455:
452:
449:
446:
443:
440:
437:
434:
431:
415:
414:
400:
378:
375:
372:
369:
337:
334:
331:
309:
305:
301:
298:
295:
292:
289:
286:
283:
279:
275:
272:
252:
249:
246:
195:(or sometimes
117:
114:
111:
83:
80:
77:
26:
9:
6:
4:
3:
2:
6426:
6415:
6412:
6411:
6409:
6396:
6391:
6385:
6373:
6364:
6360:
6355:
6352:
6348:
6343:
6340:
6334:
6329:
6326:
6320:
6315:
6312:
6311:contradiction
6307:
6303:
6298:
6295:
6290:
6286:
6281:
6277:
6270:
6265:
6261:
6254:
6249:
6246:
6242:
6237:
6234:
6230:
6225:
6220:
6215:
6212:
6207:
6203:
6198:
6195:
6191:
6186:
6183:
6179:
6174:
6169:
6164:
6161:
6157:
6152:
6149:
6145:
6140:
6135:
6130:
6129:
6124:
6120:
6112:
6107:
6105:
6100:
6098:
6093:
6092:
6089:
6077:
6076:Ernst Zermelo
6074:
6072:
6069:
6067:
6064:
6062:
6061:Willard Quine
6059:
6057:
6054:
6052:
6049:
6047:
6044:
6042:
6039:
6037:
6034:
6032:
6029:
6027:
6024:
6022:
6019:
6018:
6016:
6014:
6013:Set theorists
6010:
6004:
6001:
5999:
5996:
5994:
5991:
5990:
5988:
5982:
5980:
5977:
5976:
5973:
5965:
5962:
5960:
5959:Kripke–Platek
5957:
5953:
5950:
5949:
5948:
5945:
5944:
5943:
5940:
5936:
5933:
5932:
5931:
5930:
5926:
5922:
5919:
5918:
5917:
5914:
5913:
5910:
5907:
5905:
5902:
5900:
5897:
5895:
5892:
5891:
5889:
5885:
5879:
5876:
5874:
5871:
5869:
5866:
5864:
5862:
5857:
5855:
5852:
5850:
5847:
5844:
5840:
5837:
5835:
5832:
5828:
5825:
5823:
5820:
5818:
5815:
5814:
5813:
5810:
5807:
5803:
5800:
5798:
5795:
5793:
5790:
5788:
5785:
5784:
5782:
5779:
5775:
5769:
5766:
5764:
5761:
5759:
5756:
5754:
5751:
5749:
5746:
5744:
5741:
5739:
5736:
5732:
5729:
5727:
5724:
5723:
5722:
5719:
5717:
5714:
5712:
5709:
5707:
5704:
5702:
5699:
5696:
5692:
5689:
5687:
5684:
5682:
5679:
5678:
5676:
5670:
5667:
5666:
5663:
5657:
5654:
5652:
5649:
5647:
5644:
5642:
5639:
5637:
5634:
5632:
5629:
5627:
5624:
5621:
5618:
5616:
5613:
5612:
5610:
5608:
5604:
5596:
5595:specification
5593:
5591:
5588:
5587:
5586:
5583:
5582:
5579:
5576:
5574:
5571:
5569:
5566:
5564:
5561:
5559:
5556:
5554:
5551:
5549:
5546:
5544:
5541:
5537:
5534:
5533:
5532:
5529:
5527:
5524:
5520:
5517:
5515:
5512:
5510:
5507:
5506:
5505:
5502:
5500:
5497:
5496:
5494:
5492:
5488:
5483:
5473:
5470:
5469:
5467:
5463:
5459:
5452:
5447:
5445:
5440:
5438:
5433:
5432:
5429:
5419:
5418:
5413:
5405:
5399:
5396:
5394:
5391:
5389:
5386:
5384:
5381:
5377:
5374:
5373:
5372:
5369:
5367:
5364:
5362:
5359:
5357:
5353:
5350:
5348:
5345:
5343:
5340:
5338:
5335:
5333:
5330:
5329:
5327:
5323:
5317:
5314:
5312:
5309:
5307:
5306:Recursive set
5304:
5302:
5299:
5297:
5294:
5292:
5289:
5287:
5284:
5280:
5277:
5275:
5272:
5270:
5267:
5265:
5262:
5260:
5257:
5256:
5255:
5252:
5250:
5247:
5245:
5242:
5240:
5237:
5235:
5232:
5230:
5227:
5226:
5224:
5222:
5218:
5212:
5209:
5207:
5204:
5202:
5199:
5197:
5194:
5192:
5189:
5187:
5184:
5182:
5179:
5175:
5172:
5170:
5167:
5165:
5162:
5161:
5160:
5157:
5155:
5152:
5150:
5147:
5145:
5142:
5140:
5137:
5135:
5132:
5128:
5125:
5124:
5123:
5120:
5116:
5115:of arithmetic
5113:
5112:
5111:
5108:
5104:
5101:
5099:
5096:
5094:
5091:
5089:
5086:
5084:
5081:
5080:
5079:
5076:
5072:
5069:
5067:
5064:
5063:
5062:
5059:
5058:
5056:
5054:
5050:
5044:
5041:
5039:
5036:
5034:
5031:
5029:
5026:
5023:
5022:from ZFC
5019:
5016:
5014:
5011:
5005:
5002:
5001:
5000:
4997:
4995:
4992:
4990:
4987:
4986:
4985:
4982:
4980:
4977:
4975:
4972:
4970:
4967:
4965:
4962:
4960:
4957:
4955:
4952:
4951:
4949:
4947:
4943:
4933:
4932:
4928:
4927:
4922:
4921:non-Euclidean
4919:
4915:
4912:
4910:
4907:
4905:
4904:
4900:
4899:
4897:
4894:
4893:
4891:
4887:
4883:
4880:
4878:
4875:
4874:
4873:
4869:
4865:
4862:
4861:
4860:
4856:
4852:
4849:
4847:
4844:
4842:
4839:
4837:
4834:
4832:
4829:
4827:
4824:
4823:
4821:
4817:
4816:
4814:
4809:
4803:
4798:Example
4795:
4787:
4782:
4781:
4780:
4777:
4775:
4772:
4768:
4765:
4763:
4760:
4758:
4755:
4753:
4750:
4749:
4748:
4745:
4743:
4740:
4738:
4735:
4733:
4730:
4726:
4723:
4721:
4718:
4717:
4716:
4713:
4709:
4706:
4704:
4701:
4699:
4696:
4694:
4691:
4690:
4689:
4686:
4684:
4681:
4677:
4674:
4672:
4669:
4667:
4664:
4663:
4662:
4659:
4655:
4652:
4650:
4647:
4645:
4642:
4640:
4637:
4635:
4632:
4630:
4627:
4626:
4625:
4622:
4620:
4617:
4615:
4612:
4610:
4607:
4603:
4600:
4598:
4595:
4593:
4590:
4588:
4585:
4584:
4583:
4580:
4578:
4575:
4573:
4570:
4568:
4565:
4561:
4558:
4556:
4555:by definition
4553:
4552:
4551:
4548:
4544:
4541:
4540:
4539:
4536:
4534:
4531:
4529:
4526:
4524:
4521:
4519:
4516:
4515:
4512:
4509:
4507:
4503:
4498:
4492:
4488:
4478:
4475:
4473:
4470:
4468:
4465:
4463:
4460:
4458:
4455:
4453:
4450:
4448:
4445:
4443:
4442:Kripke–Platek
4440:
4438:
4435:
4431:
4428:
4426:
4423:
4422:
4421:
4418:
4417:
4415:
4411:
4403:
4400:
4399:
4398:
4395:
4393:
4390:
4386:
4383:
4382:
4381:
4378:
4376:
4373:
4371:
4368:
4366:
4363:
4361:
4358:
4355:
4351:
4347:
4344:
4340:
4337:
4335:
4332:
4330:
4327:
4326:
4325:
4321:
4318:
4317:
4315:
4313:
4309:
4305:
4297:
4294:
4292:
4289:
4287:
4286:constructible
4284:
4283:
4282:
4279:
4277:
4274:
4272:
4269:
4267:
4264:
4262:
4259:
4257:
4254:
4252:
4249:
4247:
4244:
4242:
4239:
4237:
4234:
4232:
4229:
4227:
4224:
4222:
4219:
4218:
4216:
4214:
4209:
4201:
4198:
4196:
4193:
4191:
4188:
4186:
4183:
4181:
4178:
4176:
4173:
4172:
4170:
4166:
4163:
4161:
4158:
4157:
4156:
4153:
4151:
4148:
4146:
4143:
4141:
4138:
4136:
4132:
4128:
4126:
4123:
4119:
4116:
4115:
4114:
4111:
4110:
4107:
4104:
4102:
4098:
4088:
4085:
4083:
4080:
4078:
4075:
4073:
4070:
4068:
4065:
4063:
4060:
4056:
4053:
4052:
4051:
4048:
4044:
4039:
4038:
4037:
4034:
4033:
4031:
4029:
4025:
4017:
4014:
4012:
4009:
4007:
4004:
4003:
4002:
3999:
3997:
3994:
3992:
3989:
3987:
3984:
3982:
3979:
3977:
3974:
3972:
3969:
3968:
3966:
3964:
3963:Propositional
3960:
3954:
3951:
3949:
3946:
3944:
3941:
3939:
3936:
3934:
3931:
3929:
3926:
3922:
3919:
3918:
3917:
3914:
3912:
3909:
3907:
3904:
3902:
3899:
3897:
3894:
3892:
3891:Logical truth
3889:
3887:
3884:
3883:
3881:
3879:
3875:
3872:
3870:
3866:
3860:
3857:
3855:
3852:
3850:
3847:
3845:
3842:
3840:
3837:
3835:
3831:
3827:
3823:
3821:
3818:
3816:
3813:
3811:
3807:
3804:
3803:
3801:
3799:
3793:
3788:
3782:
3779:
3777:
3774:
3772:
3769:
3767:
3764:
3762:
3759:
3757:
3754:
3752:
3749:
3747:
3744:
3742:
3739:
3737:
3734:
3732:
3729:
3727:
3724:
3720:
3717:
3716:
3715:
3712:
3711:
3709:
3705:
3701:
3694:
3689:
3687:
3682:
3680:
3675:
3674:
3671:
3662:
3661:
3656:
3653:
3648:
3645:
3640:
3636:
3635:
3625:
3623:3-540-44085-2
3619:
3615:
3611:
3607:
3606:
3587:
3583:
3576:
3563:on 2013-01-23
3559:
3552:
3551:
3544:
3537:
3533:
3529:
3523:
3520:, p. 6,
3519:
3515:
3511:
3510:Rudin, Walter
3505:
3497:
3491:
3487:
3480:
3472:
3466:
3462:
3455:
3447:
3441:
3437:
3432:
3431:
3422:
3418:
3407:
3404:
3398:
3395:
3392:
3389:
3386:
3383:
3377:
3374:
3371:
3368:
3365:
3362:
3356:
3355:Convex subset
3353:
3352:
3332:
3326:
3320:
3314:
3291:
3288:
3285:
3277:
3258:
3247:
3243:
3239:
3220:
3217:
3214:
3203:
3202:partial order
3199:
3196:
3192:
3188:
3185:are given by
3184:
3183:join and meet
3180:
3176:
3175:partial order
3172:
3171:
3156:
3148:
3140:
3132:
3129:
3126:
3113:
3110:
3107:
3100:
3099:
3095:
3091:
3087:
3083:
3079:
3075:
3074:
3059:
3056:
3053:
3050:
3047:
3044:
3036:
3033:
3030:
3023:
3022:
3018:
3014:
3010:
3006:
3005:
2990:
2987:
2984:
2981:
2978:
2975:
2967:
2964:
2961:
2954:
2953:
2949:
2946:
2942:
2938:
2934:
2933:
2927:
2925:
2922:
2906:
2884:
2876:
2865:
2849:
2829:
2821:
2796:
2793:
2764:
2744:
2735:
2733:
2729:
2711:
2707:
2698:
2695:of which the
2682:
2677:
2669:
2666:
2663:
2652:
2634:
2630:
2609:
2606:
2603:
2583:
2579:
2573:
2569:
2565:
2562:
2559:
2554:
2550:
2546:
2541:
2537:
2532:
2528:
2525:
2505:
2502:
2499:
2476:
2473:
2470:
2459:
2455:
2434:
2426:
2423:
2415:
2411:
2407:
2388:
2382:
2366:
2353:
2350:
2347:
2344:
2336:
2333:
2330:
2307:
2277:
2274:
2271:
2263:
2260:
2257:
2234:
2216:
2215:partial order
2200:
2193:
2188:
2171:
2153:
2137:
2116:
2111:
2104:
2099:
2098:
2097:
2095:
2094:Euler diagram
2085:
2083:
2079:
2075:
2072:
2068:
2064:
2060:
2056:
2052:
2049:
2045:
2041:
2037:
2033:
2029:
2014:
2011:
2008:
2001:is true, and
1988:
1985:
1982:
1972:
1970:
1955:
1952:
1949:
1929:
1926:
1923:
1915:
1914:
1910:
1905:
1896:
1894:
1890:
1874:
1871:
1868:
1865:
1857:
1853:
1837:
1834:
1831:
1828:
1808:
1800:
1796:
1792:
1788:
1784:
1768:
1765:
1762:
1759:
1751:
1747:
1731:
1728:
1725:
1722:
1702:
1699:
1679:
1671:
1655:
1635:
1615:
1612:
1592:
1582:
1578:
1564:
1544:
1535:
1533:
1517:
1514:
1511:
1508:
1500:
1484:
1481:
1461:
1451:
1447:
1433:
1413:
1384:
1381:
1378:
1358:
1355:
1352:
1344:
1343:
1339:
1325:
1322:
1319:
1299:
1296:
1293:
1273:
1270:
1267:
1259:
1258:
1254:
1239:
1236:
1233:
1213:
1205:
1204:
1203:Irreflexivity
1200:
1199:
1196:Proper subset
1178:
1175:
1172:
1152:
1149:
1146:
1126:
1123:
1120:
1112:
1111:
1107:
1093:
1090:
1087:
1067:
1064:
1061:
1041:
1038:
1035:
1027:
1026:
1022:
1008:
1005:
1002:
982:
974:
973:
969:
968:
953:
950:
947:
944:
924:
921:
918:
898:
895:
892:
884:
875:
873:
869:
853:
850:
819:
798:
795:
792:
789:
782:, denoted by
781:
777:
773:
769:
765:
762:
747:
744:
741:
734:, denoted by
733:
729:
725:
721:
717:
714:
713:
712:
711:
710:
708:
704:
700:
696:
692:
688:
684:
663:
660:
657:
654:
647:, denoted by
646:
642:
638:
635:
620:
617:
614:
607:, denoted by
606:
602:
598:
595:
594:
593:
592:
591:
589:
585:
581:
577:
573:
563:
549:
546:
543:
540:
520:
516:
512:
509:
506:
500:
497:
494:
490:
486:
475:
456:
453:
450:
438:
435:
432:
421:
412:
408:
404:
401:
398:
394:
391:
390:
389:
376:
373:
370:
367:
359:
355:
349:
335:
332:
329:
320:
307:
303:
299:
296:
293:
287:
284:
281:
277:
273:
250:
247:
244:
235:
233:
229:
227:
222:
218:
214:
210:
206:
202:
198:
194:
190:
186:
185:proper subset
182:
178:
174:
170:
166:
162:
158:
154:
150:
146:
142:
138:
135:
115:
112:
109:
101:
97:
81:
78:
75:
67:
63:
59:
54:
53:Euler diagram
50:
44:
40:
33:
19:
6223:
6210:
6172:
6138:
6026:Georg Cantor
6021:Paul Bernays
5952:Morse–Kelley
5927:
5860:
5859:Subset
5858:
5806:hereditarily
5768:Venn diagram
5726:ordered pair
5641:Intersection
5585:Axiom schema
5408:
5206:Ultraproduct
5053:Model theory
5018:Independence
4954:Formal proof
4946:Proof theory
4929:
4902:
4859:real numbers
4831:second-order
4742:Substitution
4619:Metalanguage
4560:conservative
4533:Axiom schema
4477:Constructive
4447:Morse–Kelley
4413:Set theories
4392:Aleph number
4385:inaccessible
4291:Grothendieck
4175:intersection
4062:Higher-order
4050:Second-order
3996:Truth tables
3953:Venn diagram
3736:Formal proof
3658:
3613:
3610:Jech, Thomas
3603:Bibliography
3589:. Retrieved
3585:
3575:
3565:, retrieved
3558:the original
3549:
3543:
3513:
3504:
3485:
3479:
3460:
3454:
3429:
3421:
3406:Total subset
3275:
3245:
3187:intersection
3089:
3085:
3081:
3077:
3016:
3012:
3008:
2944:
2940:
2936:
2757:-subsets of
2736:
2731:
2696:
2653:-tuple from
2650:
2457:
2405:
2367:
2189:
2129:
2091:
2082:real numbers
2063:line segment
2047:
2043:
2040:prime number
2035:
2031:
1892:
1888:
1855:
1851:
1794:
1790:
1786:
1782:
1749:
1745:
1577:to indicate
1536:
1498:
1446:to indicate
1405:
1340:
1257:Transitivity
1255:
1201:
1110:Antisymmetry
1108:
1025:Transitivity
1023:
970:
871:
815:
779:
775:
771:
767:
763:
731:
727:
723:
719:
715:
706:
703:there exists
698:
690:
686:
682:
680:
644:
640:
636:
604:
600:
596:
587:
583:
575:
571:
569:
473:
417:
410:
406:
402:
396:
392:
357:
353:
351:
321:
236:
231:
225:
224:
220:
216:
212:
208:
204:
200:
196:
192:
188:
184:
180:
176:
172:
168:
164:
160:
156:
152:
144:
140:
136:
131:
99:
95:
65:
61:
57:
6274:existential
6051:Thomas Jech
5894:Alternative
5873:Uncountable
5827:Ultrafilter
5686:Cardinality
5590:replacement
5531:Determinacy
5316:Type theory
5264:undecidable
5196:Truth value
5083:equivalence
4762:non-logical
4375:Enumeration
4365:Isomorphism
4312:cardinality
4296:Von Neumann
4261:Ultrafilter
4226:Uncountable
4160:equivalence
4077:Quantifiers
4067:Fixed-point
4036:First-order
3916:Consistency
3901:Proposition
3878:Traditional
3849:Lindström's
3839:Compactness
3781:Type theory
3726:Cardinality
3518:McGraw-Hill
3094:cardinality
2921:transfinite
2454:cardinality
2250:defined by
2217:on the set
2076:The set of
2071:cardinality
2053:The set of
972:Reflexivity
197:containment
6046:Kurt Gödel
6031:Paul Cohen
5868:Transitive
5636:Identities
5620:Complement
5607:Operations
5568:Regularity
5536:projective
5499:Adjunction
5458:Set theory
5127:elementary
4820:arithmetic
4688:Quantifier
4666:functional
4538:Expression
4256:Transitive
4200:identities
4185:complement
4118:hereditary
4101:Set theory
3614:Set Theory
3591:2020-08-23
3567:2012-09-07
3413:References
3238:isomorphic
2864:set theory
2492:for which
1793:less than
1670:inequality
820:, written
709:), then:
566:Definition
234:elements.
163:is then a
6351:therefore
6339:therefore
6294:tautology
6258:universal
5979:Paradoxes
5899:Axiomatic
5878:Universal
5854:Singleton
5849:Recursive
5792:Countable
5787:Amorphous
5646:Power set
5563:Power set
5514:dependent
5509:countable
5398:Supertask
5301:Recursion
5259:decidable
5093:saturated
5071:of models
4994:deductive
4989:axiomatic
4909:Hilbert's
4896:Euclidean
4877:canonical
4800:axiomatic
4732:Signature
4661:Predicate
4550:Extension
4472:Ackermann
4397:Operation
4276:Universal
4266:Recursive
4241:Singleton
4236:Inhabited
4221:Countable
4211:Types of
4195:power set
4165:partition
4082:Predicate
4028:Predicate
3943:Syllogism
3933:Soundness
3906:Inference
3896:Tautology
3798:paradoxes
3660:MathWorld
3370:Mereology
3321:⊆
3289:≤
3221:⪯
3130:∩
3111:⊆
3048:∪
3034:⊆
2979:∩
2965:⊆
2607:⊆
2563:…
2404:of a set
2383:
2348:⊆
2334:≤
2275:⊆
2268:⟺
2261:≤
2201:⊆
2152:power set
2126:Power set
2030:The set {
2012:⊊
1986:⊆
1968:are true.
1953:⊊
1927:⊆
1869:⊂
1858:, but if
1832:⊆
1809:⊂
1752:, but if
1726:≤
1680:≤
1656:⊂
1636:⊆
1613:⊋
1593:⊊
1565:⊃
1545:⊂
1512:⊂
1482:⊇
1462:⊆
1434:⊃
1414:⊂
1397:is False.
1382:⊊
1356:⊊
1342:Asymmetry
1323:⊊
1297:⊊
1271:⊊
1252:is False.
1237:⊊
1150:⊆
1124:⊆
1091:⊆
1065:⊆
1039:⊆
1006:⊆
948:⊆
922:⊆
896:⊆
868:vacuously
851:∅
818:empty set
793:⊋
745:⊊
658:⊇
618:⊆
544:⊆
510:∈
504:⇒
498:∈
484:∀
454:∈
445:⇒
436:∈
371:⊆
352:Let sets
333:⊆
297:∈
291:⇒
285:∈
271:∀
248:⊆
193:inclusion
143:of a set
113:⊇
102:(denoted
79:⊆
68:(denoted
6408:Category
6337:entails,
6323:entails,
6211:superset
5983:Problems
5887:Theories
5863:Superset
5839:Infinite
5668:Concepts
5548:Infinity
5465:Overview
5383:Logicism
5376:timeline
5352:Concrete
5211:Validity
5181:T-schema
5174:Kripke's
5169:Tarski's
5164:semantic
5154:Strength
5103:submodel
5098:spectrum
5066:function
4914:Tarski's
4903:Elements
4890:geometry
4846:Robinson
4767:variable
4752:function
4725:spectrum
4715:Sentence
4671:variable
4614:Language
4567:Relation
4528:Automata
4518:Alphabet
4502:language
4356:-jection
4334:codomain
4320:Function
4281:Universe
4251:Infinite
4155:Relation
3938:Validity
3928:Argument
3826:theorem,
3655:"Subset"
3612:(2002).
3582:"Subset"
3512:(1987),
3397:Subspace
3349:See also
2192:relation
1672:symbols
1452:superset
937:implies
776:superset
641:superset
590:, then:
165:superset
149:elements
18:Superset
6363:because
6227:
6206:implies
6194:implies
6176:
6142:
6117:Common
5921:General
5916:Zermelo
5822:subbase
5804: (
5743:Forcing
5721:Element
5693: (
5671:Methods
5558:Pairing
5325:Related
5122:Diagram
5020: (
4999:Hilbert
4984:Systems
4979:Theorem
4857:of the
4802:systems
4582:Formula
4577:Grammar
4493: (
4437:General
4150:Forcing
4135:Element
4055:Monadic
3830:paradox
3771:Theorem
3707:General
3644:Subsets
3536:0924157
3278:, then
3007:A set
2935:A set
1312:, then
1165:, then
1080:, then
693:is not
580:element
393:suppose
228:-subset
147:if all
55:showing
6325:proves
6221:
6170:
6136:
5812:Filter
5802:Finite
5738:Family
5681:Almost
5519:global
5504:Choice
5491:Axioms
5088:finite
4851:Skolem
4804:
4779:Theory
4747:Symbol
4737:String
4720:atomic
4597:ground
4592:closed
4587:atomic
4543:ground
4506:syntax
4402:binary
4329:domain
4246:Finite
4011:finite
3869:Logics
3828:
3776:Theory
3620:
3534:
3524:
3492:
3467:
3442:
3376:Region
3086:subset
3078:finite
3013:subset
2941:subset
2728:member
2649:) the
1789:, and
1583:proper
1579:proper
1448:subset
772:strict
768:proper
728:subset
724:strict
720:proper
701:(i.e.
689:, but
601:subset
141:subset
62:subset
6306:false
6144:&
5904:Naive
5834:Fuzzy
5797:Empty
5780:types
5731:tuple
5701:Class
5695:large
5656:Union
5573:Union
5078:Model
4826:Peano
4683:Proof
4523:Arity
4452:Naive
4339:image
4271:Fuzzy
4231:Empty
4180:union
4125:Class
3766:Model
3756:Lemma
3714:Axiom
3561:(PDF)
3554:(PDF)
3191:union
3084:is a
3011:is a
2939:is a
2919:is a
2726:is a
2452:(the
2412:—the
2213:is a
2086:power
2038:is a
1887:then
1850:then
1781:then
1744:then
1501:that
1371:then
1345:: If
1260:: If
1113:: If
1028:: If
766:is a
718:is a
695:equal
639:is a
599:is a
405:that
395:that
183:is a
139:is a
60:is a
6289:true
6245:nand
5817:base
5201:Type
5004:list
4808:list
4785:list
4774:Term
4708:rank
4602:open
4496:list
4308:Maps
4213:sets
4072:Free
4042:list
3792:list
3719:list
3618:ISBN
3522:ISBN
3490:ISBN
3465:ISBN
3440:ISBN
3189:and
3080:set
2503:<
2067:line
1942:and
1907:The
1797:(an
1763:<
1700:<
1692:and
1648:and
1605:and
1557:and
1474:and
1450:and
1426:and
1286:and
1139:and
1054:and
911:and
874:.
816:The
770:(or
722:(or
574:and
403:show
356:and
223:. A
175:and
6233:iff
6182:not
6148:and
5778:Set
4888:of
4870:of
4818:of
4350:Sur
4324:Map
4131:Ur-
4113:Set
3436:119
3236:is
3088:of
3015:of
2943:of
2730:of
2456:of
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