2946:
3016:
93:
Outside set theory, the word "class" is sometimes used synonymously with "set". This usage dates from a historical period where classes and sets were not distinguished as they are in modern set-theoretic terminology. Many discussions of "classes" in the 19th century and earlier are really referring
968:(NBG); classes are the basic objects in this theory, and a set is then defined to be a class that is an element of some other class. However, the class existence axioms of NBG are restricted so that they only quantify over sets, rather than over all classes. This causes NBG to be a
90:'s set-theoretical writing, the phrase "ultimate class" is often used instead of the phrase "proper class" emphasising that in the systems he considers, certain classes cannot be members, and are thus the final term in any membership chain to which they belong.
202:
is proper. The paradoxes do not arise with classes because there is no notion of classes containing classes. Otherwise, one could, for example, define a class of all classes that do not contain themselves, which would lead to a
Russell paradox for classes. A
978:
admits proper classes as basic objects, like NBG, but also allows quantification over all proper classes in its class existence axioms. This causes MK to be strictly stronger than both NBG and ZFC.
310:
148:
Within set theory, many collections of sets turn out to be proper classes. Examples include the class of all sets (the universal class), the class of all ordinal numbers, and the class of all
872:
1325:
529:
260:
603:
577:
553:
493:
218:
does not formalize the notion of classes, so each formula with classes must be reduced syntactically to a formula without classes. For example, one can reduce the formula
989:, the concept of "proper class" still makes sense (not all classes are sets) but the criterion of sethood is not closed under subsets. For example, any set theory with a
959:
741:
3473:
2000:
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695:
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that all its members share. Classes act as a way to have set-like collections while differing from sets so as to avoid paradoxes, especially
1343:
3162:
2982:
2410:
1733:
708:
can also be generalised to classes. A class function is not a function in the usual sense, since it is not a set; it is rather a formula
674:
Because classes do not have any formal status in the theory of ZF, the axioms of ZF do not immediately apply to classes. However, if an
3490:
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itself). In particular, one can identify the "class of all sets" with the set of all predicates equivalent to
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with the class of all ordinal numbers. This method is used, for example, in the proof that there is no
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68:, axiomatize the notion of "proper class", e.g., as entities that are not members of another entity.
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534:
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to sets, or rather perhaps take place without considering that certain classes can fail to be sets.
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835:. For example, the class function mapping each set to its powerset may be expressed as the formula
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suggests a proof that the class of all sets which do not contain themselves is proper, and the
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that "all classes are sets". With a rigorous foundation, these paradoxes instead suggest
27:
Collection of sets in mathematics that can be defined based on a property of its members
3597:
3394:
3375:
3279:
3264:
3221:
3157:
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2110:
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1913:
1789:
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1651:
1252:
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746:
335:
315:
138:
131:
80:
3617:
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3404:
3318:
3313:
2839:
2517:
2024:
1809:
1799:
1769:
1754:
1424:
1087:, Studies in Logic and the Foundations of mathematical vol. 90, ed. J. Barwise (1977)
1039:
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215:
1108:, Springer Monographs in Mathematics (third millennium ed.), Berlin, New York:
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2359:
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2216:
2152:
2135:
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1925:
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1486:
1269:
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holds; thus, the class can be described as the set of all predicates equivalent to
460:
160:
111:
1046:
106:
of a given type will usually be a proper class. Examples include the class of all
60:). The precise definition of "class" depends on foundational context. In work on
46:(or sometimes other mathematical objects) that can be unambiguously defined by a
3577:
3212:
2849:
2729:
1908:
1898:
1845:
1529:
1449:
1434:
1314:
1259:
1126:
1101:
35:
3557:
3425:
3328:
2991:
1779:
1634:
1605:
1411:
1189:
31:
190:
that certain classes are proper (i.e., that they are not sets). For example,
3360:
3323:
3274:
3172:
2931:
2834:
1887:
1804:
1764:
1728:
1664:
1476:
1466:
1439:
1194:
156:
17:
1062:
2916:
2714:
2162:
1867:
1461:
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83:, and the class of all sets, are proper classes in many formal systems.
2512:
1304:
986:
71:
A class that is not a set (informally in
ZermeloâFraenkel) is called a
64:, the notion of class is informal, whereas other set theories, such as
3385:
3207:
1202:
499:
interpreting ZF, then the object language "class-builder expression"
3257:
3024:
2056:
1402:
1247:
697:
is assumed, then the sets of smaller rank form a model of ZF (a
2964:
1206:
2960:
850:
564:
540:
480:
352:, it is necessary to be able to expand each of the formulas
141:
are a proper class of objects that have the properties of a
155:
One way to prove that a class is proper is to place it in
555:
by the collection of all the elements from the domain of
207:, on the other hand, can have proper classes as members.
932:
912:
880:
841:
821:
789:
769:
749:
714:
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651:
631:
611:
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561:
537:
505:
477:
436:
410:
384:
358:
338:
318:
268:
224:
305:{\displaystyle \forall x(x\in A\leftrightarrow x=x)}
3538:
3501:
3413:
3303:
3191:
3132:
3023:
2998:
2858:
2753:
2585:
2478:
2330:
2023:
1946:
1840:
1744:
1633:
1560:
1495:
1410:
1401:
1323:
1240:
701:), and its subsets can be thought of as "classes".
953:
918:
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866:
827:
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396:
370:
344:
324:
304:
254:
75:, and a class that is a set is sometimes called a
993:has proper classes which are subclasses of sets.
456:into a formula without an occurrence of a class.
1051:(3rd ed.), Heldermann Verlag, pp. 9â12
1141:Smullyan, Raymond M.; Fitting, Melvin (2010),
182:can be explained in terms of the inconsistent
2976:
1218:
1083:J. R. Shoenfield, "Axioms of Set Theory". In
926:may be expressed with the shorthand notation
126:forms a proper class (or whose collection of
8:
518:
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249:
231:
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383:
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337:
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267:
223:
1002:
7:
1143:Set Theory And The Continuum Problem
66:von NeumannâBernaysâGödel set theory
867:{\displaystyle y={\mathcal {P}}(x)}
743:with the property that for any set
1040:"Sets, classes, and conglomerates"
933:
913:
822:
715:
463:, the classes can be described as
269:
130:forms a proper class) is called a
25:
964:Another approach is taken by the
874:. The fact that the ordered pair
79:. For instance, the class of all
3014:
2944:
966:von NeumannâBernaysâGödel axioms
34:and its applications throughout
1064:abeq2 â Metamath Proof Explorer
981:In other set theories, such as
524:{\displaystyle \{x\mid \phi \}}
255:{\displaystyle A=\{x\mid x=x\}}
198:suggests that the class of all
1085:Handbook of Mathematical Logic
942:
936:
893:
881:
861:
855:
802:
790:
763:there is no more than one set
730:
718:
598:{\displaystyle \lambda x\phi }
572:{\displaystyle {\mathcal {A}}}
548:{\displaystyle {\mathcal {A}}}
488:{\displaystyle {\mathcal {A}}}
299:
287:
275:
211:Classes in formal set theories
56:
1:
2905:History of mathematical logic
1067:, us.metamath.org, 1993-08-05
1015:The Principles of Mathematics
180:paradoxes of naive set theory
2830:Primitive recursive function
1038:; Strecker, George (2007),
62:ZermeloâFraenkel set theory
3640:
3474:von NeumannâBernaysâGödel
1894:SchröderâBernstein theorem
1621:Monadic predicate calculus
1280:Foundations of mathematics
1161:Introduction to Set Theory
954:{\displaystyle \Phi (x)=y}
736:{\displaystyle \Phi (x,y)}
332:and a set variable symbol
3275:One-to-one correspondence
3012:
2940:
2927:Philosophy of mathematics
2876:Automated theorem proving
2047:
2001:Von NeumannâBernaysâGödel
1642:
1163:, McGraw-Hill Book Co.,
1159:Monk, Donald J. (1969),
704:In ZF, the concept of a
2577:Self-verifying theories
2398:Tarski's axiomatization
1349:Tarski's undefinability
1344:incompleteness theorems
976:MorseâKelley set theory
690:{\displaystyle \kappa }
3233:Constructible universe
3060:Constructibility (V=L)
2951:Mathematics portal
2562:Proof of impossibility
2210:propositional variable
1520:Propositional calculus
1145:, Dover Publications,
970:conservative extension
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920:
900:
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450:
424:
423:{\displaystyle A\in x}
398:
372:
371:{\displaystyle x\in A}
346:
326:
306:
256:
114:, and many others. In
102:The collection of all
3456:Principia Mathematica
3290:Transfinite induction
3149:(i.e. set difference)
2820:Kolmogorov complexity
2773:Computably enumerable
2673:Model complete theory
2465:Principia Mathematica
1525:Propositional formula
1354:BanachâTarski paradox
956:
921:
919:{\displaystyle \Phi }
901:
899:{\displaystyle (x,y)}
869:
830:
828:{\displaystyle \Phi }
810:
808:{\displaystyle (x,y)}
778:
758:
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699:Grothendieck universe
692:
676:inaccessible cardinal
666:
640:
638:{\displaystyle \phi }
620:
618:{\displaystyle \phi }
600:
574:
550:
526:
490:
451:
425:
399:
373:
347:
327:
307:
257:
3530:Burali-Forti paradox
3285:Set-builder notation
3238:Continuum hypothesis
3178:Symmetric difference
2768:ChurchâTuring thesis
2755:Computability theory
1964:continuum hypothesis
1482:Square of opposition
1340:Gödel's completeness
1133:, Berlin, New York:
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910:
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196:Burali-Forti paradox
122:whose collection of
104:algebraic structures
3491:TarskiâGrothendieck
2922:Mathematical object
2813:P versus NP problem
2778:Computable function
2572:Reverse mathematics
2498:Logical consequence
2375:primitive recursive
2370:elementary function
2143:Free/bound variable
1996:TarskiâGrothendieck
1515:Logical connectives
1445:Logical equivalence
1295:Logical consequence
1020:Chapter VI: Classes
783:such that the pair
664:{\displaystyle x=x}
465:equivalence classes
459:Semantically, in a
449:{\displaystyle A=x}
397:{\displaystyle x=A}
110:, the class of all
42:is a collection of
3080:Limitation of size
2720:Transfer principle
2683:Semantics of logic
2668:Categorical theory
2644:Non-standard model
2158:Logical connective
1285:Information theory
1234:Mathematical logic
1186:Weisstein, Eric W.
951:
916:
896:
864:
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687:
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531:is interpreted in
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446:
420:
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368:
342:
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302:
252:
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3610:
3520:Russell's paradox
3469:ZermeloâFraenkel
3370:Dedekind-infinite
3243:Diagonal argument
3142:Cartesian product
3006:Set (mathematics)
2958:
2957:
2890:Abstract category
2693:Theories of truth
2503:Rule of inference
2493:Natural deduction
2474:
2473:
2019:
2018:
1724:Cartesian product
1629:
1628:
1535:Many-valued logic
1510:Boolean functions
1393:Russell's paradox
1368:diagonal argument
1265:First-order logic
1152:978-0-486-47484-7
1119:978-3-540-44085-7
985:or the theory of
776:{\displaystyle y}
756:{\displaystyle x}
345:{\displaystyle x}
325:{\displaystyle A}
192:Russell's paradox
166:on three or more
52:Russell's paradox
16:(Redirected from
3631:
3593:Bertrand Russell
3583:John von Neumann
3568:Abraham Fraenkel
3563:Richard Dedekind
3525:Suslin's problem
3436:Cantor's theorem
3153:De Morgan's laws
3018:
2985:
2978:
2971:
2962:
2949:
2948:
2900:History of logic
2895:Category of sets
2788:Decision problem
2567:Ordinal analysis
2508:Sequent calculus
2406:Boolean algebras
2346:
2345:
2320:
2291:logical/constant
2045:
2031:
1954:ZermeloâFraenkel
1705:Set operations:
1640:
1577:
1408:
1388:LöwenheimâSkolem
1275:Formal semantics
1227:
1220:
1213:
1204:
1199:
1198:
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1131:Basic Set Theory
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1024:Internet Archive
1010:Bertrand Russell
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625:(which includes
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484:
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469:logical formulas
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184:tacit assumption
164:complete lattice
150:cardinal numbers
57:§ Paradoxes
21:
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3607:
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3513:
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3462:New Foundations
3409:
3299:
3218:Cardinal number
3201:
3187:
3128:
3019:
3010:
2994:
2989:
2959:
2954:
2943:
2936:
2881:Category theory
2871:Algebraic logic
2854:
2825:Lambda calculus
2763:Church encoding
2749:
2725:Truth predicate
2581:
2547:Complete theory
2470:
2339:
2335:
2331:
2326:
2318:
2038: and
2034:
2029:
2015:
1991:New Foundations
1959:axiom of choice
1942:
1904:Gödel numbering
1844: and
1836:
1740:
1625:
1575:
1556:
1505:Boolean algebra
1491:
1455:Equiconsistency
1420:Classical logic
1397:
1378:Halting problem
1366: and
1342: and
1330: and
1329:
1324:Theorems (
1319:
1236:
1231:
1184:
1183:
1180:
1171:
1158:
1153:
1140:
1135:Springer-Verlag
1125:
1120:
1110:Springer-Verlag
1100:
1097:
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1091:
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1070:
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1061:
1060:
1056:
1048:Category theory
1042:
1036:Herrlich, Horst
1034:
1033:
1029:
1008:
1004:
999:
983:New Foundations
928:
927:
908:
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875:
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219:
213:
200:ordinal numbers
176:
139:surreal numbers
116:category theory
100:
81:ordinal numbers
28:
23:
22:
15:
12:
11:
5:
3637:
3635:
3627:
3626:
3616:
3615:
3609:
3608:
3606:
3605:
3600:
3598:Thoralf Skolem
3595:
3590:
3585:
3580:
3575:
3570:
3565:
3560:
3555:
3550:
3544:
3542:
3536:
3535:
3533:
3532:
3527:
3522:
3516:
3514:
3512:
3511:
3508:
3502:
3499:
3498:
3496:
3495:
3494:
3493:
3488:
3483:
3482:
3481:
3466:
3465:
3464:
3452:
3451:
3450:
3439:
3438:
3433:
3428:
3423:
3417:
3415:
3411:
3410:
3408:
3407:
3402:
3397:
3392:
3383:
3378:
3373:
3363:
3358:
3357:
3356:
3351:
3346:
3336:
3326:
3321:
3316:
3310:
3308:
3301:
3300:
3298:
3297:
3292:
3287:
3282:
3280:Ordinal number
3277:
3272:
3267:
3262:
3261:
3260:
3255:
3245:
3240:
3235:
3230:
3225:
3215:
3210:
3204:
3202:
3200:
3199:
3196:
3192:
3189:
3188:
3186:
3185:
3180:
3175:
3170:
3165:
3160:
3158:Disjoint union
3155:
3150:
3144:
3138:
3136:
3130:
3129:
3127:
3126:
3125:
3124:
3119:
3108:
3107:
3105:Martin's axiom
3102:
3097:
3092:
3087:
3082:
3077:
3072:
3070:Extensionality
3067:
3062:
3057:
3056:
3055:
3050:
3045:
3035:
3029:
3027:
3021:
3020:
3013:
3011:
3009:
3008:
3002:
3000:
2996:
2995:
2990:
2988:
2987:
2980:
2973:
2965:
2956:
2955:
2941:
2938:
2937:
2935:
2934:
2929:
2924:
2919:
2914:
2913:
2912:
2902:
2897:
2892:
2883:
2878:
2873:
2868:
2866:Abstract logic
2862:
2860:
2856:
2855:
2853:
2852:
2847:
2845:Turing machine
2842:
2837:
2832:
2827:
2822:
2817:
2816:
2815:
2810:
2805:
2800:
2795:
2785:
2783:Computable set
2780:
2775:
2770:
2765:
2759:
2757:
2751:
2750:
2748:
2747:
2742:
2737:
2732:
2727:
2722:
2717:
2712:
2711:
2710:
2705:
2700:
2690:
2685:
2680:
2678:Satisfiability
2675:
2670:
2665:
2664:
2663:
2653:
2652:
2651:
2641:
2640:
2639:
2634:
2629:
2624:
2619:
2609:
2608:
2607:
2602:
2595:Interpretation
2591:
2589:
2583:
2582:
2580:
2579:
2574:
2569:
2564:
2559:
2549:
2544:
2543:
2542:
2541:
2540:
2530:
2525:
2515:
2510:
2505:
2500:
2495:
2490:
2484:
2482:
2476:
2475:
2472:
2471:
2469:
2468:
2460:
2459:
2458:
2457:
2452:
2451:
2450:
2445:
2440:
2420:
2419:
2418:
2416:minimal axioms
2413:
2402:
2401:
2400:
2389:
2388:
2387:
2382:
2377:
2372:
2367:
2362:
2349:
2347:
2328:
2327:
2325:
2324:
2323:
2322:
2310:
2305:
2304:
2303:
2298:
2293:
2288:
2278:
2273:
2268:
2263:
2262:
2261:
2256:
2246:
2245:
2244:
2239:
2234:
2229:
2219:
2214:
2213:
2212:
2207:
2202:
2192:
2191:
2190:
2185:
2180:
2175:
2170:
2165:
2155:
2150:
2145:
2140:
2139:
2138:
2133:
2128:
2123:
2113:
2108:
2106:Formation rule
2103:
2098:
2097:
2096:
2091:
2081:
2080:
2079:
2069:
2064:
2059:
2054:
2048:
2042:
2025:Formal systems
2021:
2020:
2017:
2016:
2014:
2013:
2008:
2003:
1998:
1993:
1988:
1983:
1978:
1973:
1968:
1967:
1966:
1961:
1950:
1948:
1944:
1943:
1941:
1940:
1939:
1938:
1928:
1923:
1922:
1921:
1914:Large cardinal
1911:
1906:
1901:
1896:
1891:
1877:
1876:
1875:
1870:
1865:
1850:
1848:
1838:
1837:
1835:
1834:
1833:
1832:
1827:
1822:
1812:
1807:
1802:
1797:
1792:
1787:
1782:
1777:
1772:
1767:
1762:
1757:
1751:
1749:
1742:
1741:
1739:
1738:
1737:
1736:
1731:
1726:
1721:
1716:
1711:
1703:
1702:
1701:
1696:
1686:
1681:
1679:Extensionality
1676:
1674:Ordinal number
1671:
1661:
1656:
1655:
1654:
1643:
1637:
1631:
1630:
1627:
1626:
1624:
1623:
1618:
1613:
1608:
1603:
1598:
1593:
1592:
1591:
1581:
1580:
1579:
1566:
1564:
1558:
1557:
1555:
1554:
1553:
1552:
1547:
1542:
1532:
1527:
1522:
1517:
1512:
1507:
1501:
1499:
1493:
1492:
1490:
1489:
1484:
1479:
1474:
1469:
1464:
1459:
1458:
1457:
1447:
1442:
1437:
1432:
1427:
1422:
1416:
1414:
1405:
1399:
1398:
1396:
1395:
1390:
1385:
1380:
1375:
1370:
1358:Cantor's
1356:
1351:
1346:
1336:
1334:
1321:
1320:
1318:
1317:
1312:
1307:
1302:
1297:
1292:
1287:
1282:
1277:
1272:
1267:
1262:
1257:
1256:
1255:
1244:
1242:
1238:
1237:
1232:
1230:
1229:
1222:
1215:
1207:
1201:
1200:
1179:
1178:External links
1176:
1175:
1174:
1169:
1156:
1151:
1138:
1123:
1118:
1096:
1093:
1090:
1089:
1076:
1054:
1027:
1001:
1000:
998:
995:
950:
947:
944:
941:
938:
935:
915:
895:
892:
889:
886:
883:
863:
860:
857:
852:
847:
844:
824:
804:
801:
798:
795:
792:
772:
752:
732:
729:
726:
723:
720:
717:
686:
660:
657:
654:
634:
614:
594:
591:
588:
566:
542:
520:
517:
514:
511:
508:
482:
445:
442:
439:
419:
416:
413:
393:
390:
387:
367:
364:
361:
341:
321:
312:. For a class
301:
298:
295:
292:
289:
286:
283:
280:
277:
274:
271:
251:
248:
245:
242:
239:
236:
233:
230:
227:
212:
209:
175:
172:
132:large category
99:
96:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3636:
3625:
3622:
3621:
3619:
3604:
3603:Ernst Zermelo
3601:
3599:
3596:
3594:
3591:
3589:
3588:Willard Quine
3586:
3584:
3581:
3579:
3576:
3574:
3571:
3569:
3566:
3564:
3561:
3559:
3556:
3554:
3551:
3549:
3546:
3545:
3543:
3541:
3540:Set theorists
3537:
3531:
3528:
3526:
3523:
3521:
3518:
3517:
3515:
3509:
3507:
3504:
3503:
3500:
3492:
3489:
3487:
3486:KripkeâPlatek
3484:
3480:
3477:
3476:
3475:
3472:
3471:
3470:
3467:
3463:
3460:
3459:
3458:
3457:
3453:
3449:
3446:
3445:
3444:
3441:
3440:
3437:
3434:
3432:
3429:
3427:
3424:
3422:
3419:
3418:
3416:
3412:
3406:
3403:
3401:
3398:
3396:
3393:
3391:
3389:
3384:
3382:
3379:
3377:
3374:
3371:
3367:
3364:
3362:
3359:
3355:
3352:
3350:
3347:
3345:
3342:
3341:
3340:
3337:
3334:
3330:
3327:
3325:
3322:
3320:
3317:
3315:
3312:
3311:
3309:
3306:
3302:
3296:
3293:
3291:
3288:
3286:
3283:
3281:
3278:
3276:
3273:
3271:
3268:
3266:
3263:
3259:
3256:
3254:
3251:
3250:
3249:
3246:
3244:
3241:
3239:
3236:
3234:
3231:
3229:
3226:
3223:
3219:
3216:
3214:
3211:
3209:
3206:
3205:
3203:
3197:
3194:
3193:
3190:
3184:
3181:
3179:
3176:
3174:
3171:
3169:
3166:
3164:
3161:
3159:
3156:
3154:
3151:
3148:
3145:
3143:
3140:
3139:
3137:
3135:
3131:
3123:
3122:specification
3120:
3118:
3115:
3114:
3113:
3110:
3109:
3106:
3103:
3101:
3098:
3096:
3093:
3091:
3088:
3086:
3083:
3081:
3078:
3076:
3073:
3071:
3068:
3066:
3063:
3061:
3058:
3054:
3051:
3049:
3046:
3044:
3041:
3040:
3039:
3036:
3034:
3031:
3030:
3028:
3026:
3022:
3017:
3007:
3004:
3003:
3001:
2997:
2993:
2986:
2981:
2979:
2974:
2972:
2967:
2966:
2963:
2953:
2952:
2947:
2939:
2933:
2930:
2928:
2925:
2923:
2920:
2918:
2915:
2911:
2908:
2907:
2906:
2903:
2901:
2898:
2896:
2893:
2891:
2887:
2884:
2882:
2879:
2877:
2874:
2872:
2869:
2867:
2864:
2863:
2861:
2857:
2851:
2848:
2846:
2843:
2841:
2840:Recursive set
2838:
2836:
2833:
2831:
2828:
2826:
2823:
2821:
2818:
2814:
2811:
2809:
2806:
2804:
2801:
2799:
2796:
2794:
2791:
2790:
2789:
2786:
2784:
2781:
2779:
2776:
2774:
2771:
2769:
2766:
2764:
2761:
2760:
2758:
2756:
2752:
2746:
2743:
2741:
2738:
2736:
2733:
2731:
2728:
2726:
2723:
2721:
2718:
2716:
2713:
2709:
2706:
2704:
2701:
2699:
2696:
2695:
2694:
2691:
2689:
2686:
2684:
2681:
2679:
2676:
2674:
2671:
2669:
2666:
2662:
2659:
2658:
2657:
2654:
2650:
2649:of arithmetic
2647:
2646:
2645:
2642:
2638:
2635:
2633:
2630:
2628:
2625:
2623:
2620:
2618:
2615:
2614:
2613:
2610:
2606:
2603:
2601:
2598:
2597:
2596:
2593:
2592:
2590:
2588:
2584:
2578:
2575:
2573:
2570:
2568:
2565:
2563:
2560:
2557:
2556:from ZFC
2553:
2550:
2548:
2545:
2539:
2536:
2535:
2534:
2531:
2529:
2526:
2524:
2521:
2520:
2519:
2516:
2514:
2511:
2509:
2506:
2504:
2501:
2499:
2496:
2494:
2491:
2489:
2486:
2485:
2483:
2481:
2477:
2467:
2466:
2462:
2461:
2456:
2455:non-Euclidean
2453:
2449:
2446:
2444:
2441:
2439:
2438:
2434:
2433:
2431:
2428:
2427:
2425:
2421:
2417:
2414:
2412:
2409:
2408:
2407:
2403:
2399:
2396:
2395:
2394:
2390:
2386:
2383:
2381:
2378:
2376:
2373:
2371:
2368:
2366:
2363:
2361:
2358:
2357:
2355:
2351:
2350:
2348:
2343:
2337:
2332:Example
2329:
2321:
2316:
2315:
2314:
2311:
2309:
2306:
2302:
2299:
2297:
2294:
2292:
2289:
2287:
2284:
2283:
2282:
2279:
2277:
2274:
2272:
2269:
2267:
2264:
2260:
2257:
2255:
2252:
2251:
2250:
2247:
2243:
2240:
2238:
2235:
2233:
2230:
2228:
2225:
2224:
2223:
2220:
2218:
2215:
2211:
2208:
2206:
2203:
2201:
2198:
2197:
2196:
2193:
2189:
2186:
2184:
2181:
2179:
2176:
2174:
2171:
2169:
2166:
2164:
2161:
2160:
2159:
2156:
2154:
2151:
2149:
2146:
2144:
2141:
2137:
2134:
2132:
2129:
2127:
2124:
2122:
2119:
2118:
2117:
2114:
2112:
2109:
2107:
2104:
2102:
2099:
2095:
2092:
2090:
2089:by definition
2087:
2086:
2085:
2082:
2078:
2075:
2074:
2073:
2070:
2068:
2065:
2063:
2060:
2058:
2055:
2053:
2050:
2049:
2046:
2043:
2041:
2037:
2032:
2026:
2022:
2012:
2009:
2007:
2004:
2002:
1999:
1997:
1994:
1992:
1989:
1987:
1984:
1982:
1979:
1977:
1976:KripkeâPlatek
1974:
1972:
1969:
1965:
1962:
1960:
1957:
1956:
1955:
1952:
1951:
1949:
1945:
1937:
1934:
1933:
1932:
1929:
1927:
1924:
1920:
1917:
1916:
1915:
1912:
1910:
1907:
1905:
1902:
1900:
1897:
1895:
1892:
1889:
1885:
1881:
1878:
1874:
1871:
1869:
1866:
1864:
1861:
1860:
1859:
1855:
1852:
1851:
1849:
1847:
1843:
1839:
1831:
1828:
1826:
1823:
1821:
1820:constructible
1818:
1817:
1816:
1813:
1811:
1808:
1806:
1803:
1801:
1798:
1796:
1793:
1791:
1788:
1786:
1783:
1781:
1778:
1776:
1773:
1771:
1768:
1766:
1763:
1761:
1758:
1756:
1753:
1752:
1750:
1748:
1743:
1735:
1732:
1730:
1727:
1725:
1722:
1720:
1717:
1715:
1712:
1710:
1707:
1706:
1704:
1700:
1697:
1695:
1692:
1691:
1690:
1687:
1685:
1682:
1680:
1677:
1675:
1672:
1670:
1666:
1662:
1660:
1657:
1653:
1650:
1649:
1648:
1645:
1644:
1641:
1638:
1636:
1632:
1622:
1619:
1617:
1614:
1612:
1609:
1607:
1604:
1602:
1599:
1597:
1594:
1590:
1587:
1586:
1585:
1582:
1578:
1573:
1572:
1571:
1568:
1567:
1565:
1563:
1559:
1551:
1548:
1546:
1543:
1541:
1538:
1537:
1536:
1533:
1531:
1528:
1526:
1523:
1521:
1518:
1516:
1513:
1511:
1508:
1506:
1503:
1502:
1500:
1498:
1497:Propositional
1494:
1488:
1485:
1483:
1480:
1478:
1475:
1473:
1470:
1468:
1465:
1463:
1460:
1456:
1453:
1452:
1451:
1448:
1446:
1443:
1441:
1438:
1436:
1433:
1431:
1428:
1426:
1425:Logical truth
1423:
1421:
1418:
1417:
1415:
1413:
1409:
1406:
1404:
1400:
1394:
1391:
1389:
1386:
1384:
1381:
1379:
1376:
1374:
1371:
1369:
1365:
1361:
1357:
1355:
1352:
1350:
1347:
1345:
1341:
1338:
1337:
1335:
1333:
1327:
1322:
1316:
1313:
1311:
1308:
1306:
1303:
1301:
1298:
1296:
1293:
1291:
1288:
1286:
1283:
1281:
1278:
1276:
1273:
1271:
1268:
1266:
1263:
1261:
1258:
1254:
1251:
1250:
1249:
1246:
1245:
1243:
1239:
1235:
1228:
1223:
1221:
1216:
1214:
1209:
1208:
1205:
1197:
1196:
1191:
1187:
1182:
1181:
1177:
1172:
1170:9780070427150
1166:
1162:
1157:
1154:
1148:
1144:
1139:
1136:
1132:
1128:
1124:
1121:
1115:
1111:
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3553:Georg Cantor
3548:Paul Bernays
3479:MorseâKelley
3454:
3387:
3386:Subset
3333:hereditarily
3295:Venn diagram
3253:ordered pair
3227:
3168:Intersection
3112:Axiom schema
2942:
2740:Ultraproduct
2587:Model theory
2552:Independence
2488:Formal proof
2480:Proof theory
2463:
2436:
2393:real numbers
2365:second-order
2276:Substitution
2153:Metalanguage
2094:conservative
2067:Axiom schema
2011:Constructive
1981:MorseâKelley
1947:Set theories
1926:Aleph number
1919:inaccessible
1825:Grothendieck
1709:intersection
1658:
1596:Higher-order
1584:Second-order
1530:Truth tables
1487:Venn diagram
1270:Formal proof
1193:
1160:
1142:
1130:
1105:
1102:Jech, Thomas
1084:
1079:
1069:, retrieved
1063:
1057:
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461:metalanguage
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101:
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73:proper class
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39:
29:
18:Proper class
3578:Thomas Jech
3421:Alternative
3400:Uncountable
3354:Ultrafilter
3213:Cardinality
3117:replacement
3065:Determinacy
2850:Type theory
2798:undecidable
2730:Truth value
2617:equivalence
2296:non-logical
1909:Enumeration
1899:Isomorphism
1846:cardinality
1830:Von Neumann
1795:Ultrafilter
1760:Uncountable
1694:equivalence
1611:Quantifiers
1601:Fixed-point
1570:First-order
1450:Consistency
1435:Proposition
1412:Traditional
1383:Lindström's
1373:Compactness
1315:Type theory
1260:Cardinality
1190:"Set Class"
77:small class
36:mathematics
3624:Set theory
3573:Kurt Gödel
3558:Paul Cohen
3395:Transitive
3163:Identities
3147:Complement
3134:Operations
3095:Regularity
3033:Adjunction
2992:Set theory
2661:elementary
2354:arithmetic
2222:Quantifier
2200:functional
2072:Expression
1790:Transitive
1734:identities
1719:complement
1652:hereditary
1635:Set theory
1106:Set Theory
1095:References
1071:2016-03-09
906:satisfies
815:satisfies
168:generators
32:set theory
3506:Paradoxes
3426:Axiomatic
3405:Universal
3381:Singleton
3376:Recursive
3319:Countable
3314:Amorphous
3173:Power set
3090:Power set
3048:dependent
3043:countable
2932:Supertask
2835:Recursion
2793:decidable
2627:saturated
2605:of models
2528:deductive
2523:axiomatic
2443:Hilbert's
2430:Euclidean
2411:canonical
2334:axiomatic
2266:Signature
2195:Predicate
2084:Extension
2006:Ackermann
1931:Operation
1810:Universal
1800:Recursive
1775:Singleton
1770:Inhabited
1755:Countable
1745:Types of
1729:power set
1699:partition
1616:Predicate
1562:Predicate
1477:Syllogism
1467:Soundness
1440:Inference
1430:Tautology
1332:paradoxes
1195:MathWorld
934:Φ
914:Φ
823:Φ
716:Φ
685:κ
633:ϕ
613:ϕ
593:ϕ
587:λ
579:on which
516:ϕ
513:∣
497:structure
415:∈
363:∈
288:↔
282:∈
270:∀
238:∣
174:Paradoxes
157:bijection
128:morphisms
3618:Category
3510:Problems
3414:Theories
3390:Superset
3366:Infinite
3195:Concepts
3075:Infinity
2999:Overview
2917:Logicism
2910:timeline
2886:Concrete
2745:Validity
2715:T-schema
2708:Kripke's
2703:Tarski's
2698:semantic
2688:Strength
2637:submodel
2632:spectrum
2600:function
2448:Tarski's
2437:Elements
2424:geometry
2380:Robinson
2301:variable
2286:function
2259:spectrum
2249:Sentence
2205:variable
2148:Language
2101:Relation
2062:Automata
2052:Alphabet
2036:language
1890:-jection
1868:codomain
1854:Function
1815:Universe
1785:Infinite
1689:Relation
1472:Validity
1462:Argument
1360:theorem,
1129:(1979),
1127:Levy, A.
1104:(2003),
1012:(1903).
987:semisets
972:of ZFC.
706:function
120:category
98:Examples
48:property
3448:General
3443:Zermelo
3349:subbase
3331: (
3270:Forcing
3248:Element
3220: (
3198:Methods
3085:Pairing
2859:Related
2656:Diagram
2554: (
2533:Hilbert
2518:Systems
2513:Theorem
2391:of the
2336:systems
2116:Formula
2111:Grammar
2027: (
1971:General
1684:Forcing
1669:Element
1589:Monadic
1364:paradox
1305:Theorem
1241:General
124:objects
3339:Filter
3329:Finite
3265:Family
3208:Almost
3053:global
3038:Choice
3025:Axioms
2622:finite
2385:Skolem
2338:
2313:Theory
2281:Symbol
2271:String
2254:atomic
2131:ground
2126:closed
2121:atomic
2077:ground
2040:syntax
1936:binary
1863:domain
1780:Finite
1545:finite
1403:Logics
1362:
1310:Theory
1167:
1149:
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1022:, via
430:, and
188:proofs
108:groups
3431:Naive
3361:Fuzzy
3324:Empty
3307:types
3258:tuple
3228:Class
3222:large
3183:Union
3100:Union
2612:Model
2360:Peano
2217:Proof
2057:Arity
1986:Naive
1873:image
1805:Fuzzy
1765:Empty
1714:union
1659:Class
1300:Model
1290:Lemma
1248:Axiom
1043:(PDF)
997:Notes
495:is a
471:: If
143:field
88:Quine
54:(see
40:class
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2735:Type
2538:list
2342:list
2319:list
2308:Term
2242:rank
2136:open
2030:list
1842:Maps
1747:sets
1606:Free
1576:list
1326:list
1253:list
1165:ISBN
1147:ISBN
1114:ISBN
178:The
161:free
137:The
118:, a
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3305:Set
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1858:Map
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