5216:
34:
5286:
2346:
However, there are two important properties of subsets that do not carry over to subalgebras in general. First, although the subsets of a set form a set (as well as a lattice), in some classes it may not be possible to organize the subalgebras of an algebra as itself an algebra in that class,
2787:
2429:
on two vertices (hence four edges, namely two self-loops and two more edges forming a cycle) augmented with a fifth edge, namely a second self-loop at one of the vertices. We can therefore organize the subgraphs of
1995:
1886:
2929:
2845:
2606:
3071:
141:
2358:
Certain classes of algebras enjoy both of these properties. The first property is more common; the case of having both is relatively rare. One class that does have both is that of
2998:
3595:
2966:
2409:
can then be organized as the graph whose vertices and edges are respectively the vertex and edge functions appearing in that set. Furthermore, the subgraphs of a multigraph
267:
382:
402:
5743:
4270:
296:
2171:
4353:
3494:
1906:
2339:
of all subsets of some set. The generalization to arbitrary algebras is that the set of subalgebras of an algebra, again ordered by inclusion, is always an
1788:
2347:
although they can always be organized as a lattice. Secondly, whereas the subsets of a set are in bijection with the functions from that set to the set
2335:
The power set of a set, when ordered by inclusion, is always a complete atomic
Boolean algebra, and every complete atomic Boolean algebra arises as the
2343:, and every algebraic lattice arises as the lattice of subalgebras of some algebra. So in that regard, subalgebras behave analogously to subsets.
4667:
2316:
A set can be regarded as an algebra having no nontrivial operations or defining equations. From this perspective, the idea of the power set of
4825:
3389:
3613:
5432:
5252:
4680:
4003:
2453:
What is special about a multigraph as an algebra is that its operations are unary. A multigraph has two sorts of elements forming a set
5760:
4265:
2860:
4685:
4675:
4412:
3618:
3265:
2798:
4163:
3609:
4821:
3371:
3313:
994:
4918:
4662:
3487:
5738:
4223:
3916:
3657:
5179:
4881:
4644:
4639:
4464:
3885:
3569:
2483:
giving the source (start) and target (end) vertices of each edge. An algebra all of whose operations are unary is called a
5512:
5391:
5174:
4957:
4874:
4587:
4518:
4395:
3637:
907:
5755:
4245:
3400:
2782:{\displaystyle A=\{x_{1},x_{2},...\}\in {\mathsf {P}}(S),{\mathsf {P}}f(A)=\{f(x_{1}),f(x_{2}),...\}\in {\mathsf {P}}(T)}
5099:
4925:
4611:
3844:
939:
5748:
4250:
5893:
5386:
5349:
4582:
4321:
3579:
3480:
974:
4977:
4972:
3007:
4906:
4496:
3890:
3858:
3549:
3623:
5437:
5329:
5317:
5312:
5196:
5145:
5042:
4540:
4501:
3978:
2508:
956:
5037:
3652:
97:
5245:
4967:
4506:
4358:
4341:
4064:
3544:
5857:
5775:
5650:
5602:
5416:
5339:
4869:
4846:
4807:
4693:
4634:
4280:
4200:
4044:
3988:
3601:
2141:
960:
2391:
consists of two functions, one mapping vertices to vertices and the other mapping edges to edges. The set
5809:
5690:
5502:
5322:
5159:
4886:
4864:
4831:
4724:
4570:
4555:
4528:
4479:
4363:
4298:
4123:
4089:
4084:
3958:
3789:
3766:
3125:
2500:
1088:
1056:
1049:
55:
5725:
5639:
5559:
5539:
5517:
5089:
4942:
4734:
4452:
4188:
4094:
3953:
3938:
3819:
3794:
1028:
914:
than the set itself (or informally, the power set must be larger than the original set). In particular,
5215:
2971:
5799:
5789:
5623:
5554:
5507:
5447:
5334:
5062:
5024:
4901:
4705:
4545:
4469:
4447:
4275:
4233:
4132:
4099:
3963:
3751:
3662:
3109:
2934:
2516:
2015:
1710:
1201:
is defined in which the number in each ordered pair represents the position of the paired element of
1076:
1011:
989:
Boolean algebras, this is no longer true, but every infinite
Boolean algebra can be represented as a
841:
241:
184:
5794:
5705:
5618:
5613:
5608:
5422:
5364:
5302:
5238:
5191:
5082:
5067:
5047:
5004:
4891:
4841:
4767:
4712:
4649:
4442:
4437:
4385:
4153:
4142:
3814:
3714:
3642:
3633:
3629:
3564:
3559:
3141:
2329:
2180:
915:
367:
33:
5717:
5712:
5497:
5452:
5359:
5220:
4989:
4952:
4937:
4930:
4913:
4699:
4565:
4491:
4474:
4427:
4240:
4149:
3983:
3968:
3928:
3880:
3865:
3853:
3809:
3784:
3554:
3503:
2520:
2190:
1561:
952:
638:
196:
20:
4717:
4173:
1583:
is not unique, but the sort order of the enumerated set does not change its cardinality. (E.g.,
387:
2847:. The contravariant power set functor is different from the covariant version in that it sends
2351:, there is no guarantee that a class of algebras contains an algebra that can play the role of
5574:
5411:
5403:
5374:
5344:
5275:
5155:
4962:
4772:
4762:
4654:
4535:
4370:
4346:
4127:
4111:
4016:
3993:
3870:
3839:
3804:
3699:
3534:
3385:
3367:
3355:
3309:
3261:
2340:
161:
77:
67:
1651:
respectively as the position of binary digit sequences.) The enumeration is possible even if
272:
5862:
5852:
5837:
5832:
5700:
5354:
5169:
5164:
5057:
5014:
4836:
4797:
4792:
4777:
4603:
4560:
4457:
4255:
4205:
3779:
3741:
3343:
3319:
3193:
can be identified with, is equivalent to, or bijective to the set of all the functions from
3105:
2495:
plays for subsets. Such a class is a special case of the more general notion of elementary
1678:
3463:
5731:
5669:
5487:
5307:
5150:
5140:
5094:
5077:
5032:
4994:
4896:
4816:
4623:
4550:
4523:
4511:
4417:
4331:
4305:
4260:
4228:
4029:
3831:
3774:
3724:
3689:
3647:
3347:
3323:
3305:
3101:
2504:
2336:
1663:
is infinite), such as the set of integers or rationals, but not possible for example if
5867:
5664:
5645:
5549:
5534:
5491:
5427:
5369:
5135:
5114:
5072:
5052:
4947:
4802:
4400:
4390:
4380:
4375:
4309:
4183:
4059:
3948:
3943:
3921:
3522:
3146:
2426:
2156:
1014:(with the empty set as the identity element and each set being its own inverse), and a
927:
910:
shows that the power set of a set (whether infinite or not) always has strictly higher
317:
1669:
is the set of real numbers, in which case we cannot enumerate all irrational numbers.
5887:
5872:
5842:
5674:
5588:
5583:
5109:
4787:
4294:
4079:
4069:
4039:
4024:
3694:
3359:
3151:
3121:
3113:
1007:
919:
5822:
5817:
5635:
5564:
5522:
5381:
5285:
5009:
4856:
4757:
4749:
4629:
4577:
4486:
4422:
4405:
4336:
4195:
4054:
3756:
3539:
3336:
3297:
3129:
2375:
1032:
51:
2789:. Elsewhere in this article, the power set was defined as the set of functions of
964:
2572:. The covariant functor is defined more simply. as the functor which sends a set
5847:
5482:
5119:
4999:
4178:
4168:
4115:
3799:
3719:
3704:
3584:
3529:
3331:
3156:
2227:
1694:
1155:
1015:
935:
923:
911:
566:
149:
1031:, that the power set considered together with both of these operations forms a
5827:
5695:
5598:
5261:
4049:
3904:
3875:
3681:
3433:
3408:
2359:
2011:
1044:
990:
982:
192:
3451:
5630:
5593:
5544:
5201:
5104:
4157:
4074:
4034:
3934:
3746:
3736:
3709:
3342:. The University Series in Undergraduate Mathematics. van Nostrand Company.
2137:
1612:
to the integers without changing the number of one-to-one correspondences.)
1246:
corresponding to the position of it in the sequence exists in the subset of
1240:
is at the second from the right, and 1 in the sequence means the element of
931:
790:
405:
176:
2795:
into the set with 2 elements. Formally, this defines a natural isomorphism
3439:
5186:
4984:
4432:
4137:
3731:
2484:
2519:. Although the term "power object" is sometimes used synonymously with
1021:
when considered with the operation of intersection (with the entire set
4782:
3574:
3117:
2549:
1990:{\displaystyle \left|2^{S}\right|=2^{n}=\sum _{k=0}^{n}{\binom {n}{k}}}
1577:
to integers is arbitrary, so this representation of all the subsets of
5655:
5477:
3472:
1881:{\displaystyle \left|2^{S}\right|=\sum _{k=0}^{|S|}{\binom {|S|}{k}}}
1018:
168:
2603:(here, a function between sets) to the image morphism. That is, for
3467:
5527:
5294:
4326:
3672:
3517:
3429:
2496:
87:
The power set is the set that contains all subsets of a given set.
2924:{\displaystyle f(A)=B\subseteq T,{\overline {\mathsf {P}}}f(B)=A}
3455:
3443:
2840:{\displaystyle {\overline {\mathsf {P}}}\cong {\text{Set}}(-,2)}
2252:
or , and the set of subsets with cardinality strictly less than
1027:
as the identity element). It can hence be shown, by proving the
5234:
3476:
1615:
However, such finite binary representation is only possible if
1732:
For example, the power set of a set with three elements, has:
779:
consists of all the indicator functions of all the subsets of
188:
5230:
1657:
has an infinite cardinality (i.e., the number of elements in
985:
to the
Boolean algebra of the power set of a finite set. For
1234:
is located at the first from the right of this sequence and
1890:
Therefore, one can deduce the following identity, assuming
1725:
elements which are elements of the power set of a set with
1687:âelements combination from some set is another name for a
754:
is identified by or equivalent to the indicator function
1599:
can be used to construct another injective mapping from
2548:
There is both a covariant and contravariant power set
1721:
elements; in other words it's the number of sets with
191:
axioms), the existence of the power set of any set is
16:
Mathematical set containing all subsets of a given set
3010:
2974:
2937:
2863:
2801:
2609:
2159:
1909:
1791:
1158:
with the binary representations of numbers from 0 to
390:
370:
275:
244:
100:
1763:
elements (the complements of the singleton subsets),
5808:
5771:
5683:
5573:
5461:
5402:
5293:
5268:
5128:
5023:
4855:
4748:
4600:
4293:
4216:
4110:
4014:
3903:
3830:
3765:
3680:
3671:
3593:
3510:
3216:
3214:
2415:are in bijection with the graph homomorphisms from
973:and can be viewed as the prototypical example of a
91:
83:
73:
63:
3335:
3124:functor of a function between sets; likewise, the
3065:
2992:
2960:
2923:
2839:
2781:
2165:
1989:
1880:
616:. This fact as well as the reason of the notation
396:
376:
290:
261:
135:
2487:. Every class of presheaves contains a presheaf
1981:
1968:
1872:
1847:
3280:
2275:or . Similarly, the set of non-empty subsets of
2328:generalizes naturally to the subalgebras of an
1131:This equivalence can be applied to the example
3066:{\displaystyle h^{*}:C(b,c)\rightarrow C(a,c)}
1128:, are considered identical set-theoretically.
5246:
3488:
581:(i.e., the number of all elements in the set
8:
2757:
2701:
2654:
2616:
1252:for the sequence while 0 means it does not.
1055:is the notation representing the set of all
1010:when it is considered with the operation of
26:
3243:
5253:
5239:
5231:
4314:
3909:
3677:
3495:
3481:
3473:
136:{\displaystyle x\in P(S)\iff x\subseteq S}
123:
119:
32:
25:
3015:
3009:
2973:
2938:
2936:
2893:
2891:
2862:
2817:
2804:
2802:
2800:
2764:
2763:
2736:
2714:
2680:
2679:
2661:
2660:
2636:
2623:
2608:
2491:that plays the role for subalgebras that
2158:
1980:
1967:
1965:
1959:
1948:
1935:
1918:
1908:
1871:
1861:
1853:
1846:
1844:
1837:
1829:
1828:
1817:
1800:
1790:
692:, and it indicates whether an element of
641:or a characteristic function of a subset
593:), then the number of all the subsets of
389:
369:
274:
246:
245:
243:
99:
3179:, meaning the set of all functions from
2205:whose each element is expanded with the
1777:Using this relationship, we can compute
1263:
1166:being the number of elements in the set
3382:Theory Of Automata And Formal Languages
3302:Fundamentals of contemporary set theory
3210:
3168:
2465:of edges, and has two unary operations
1681:is closely related to the power set. A
1207:in a sequence of binary digits such as
371:
3231:
3185:to a given set of two elements (e.g.,
2894:
2805:
2765:
2681:
2662:
926:infinite. The power set of the set of
3220:
1621:can be enumerated. (In this example,
824:, the number of all the functions in
769:as the set of all the functions from
19:For the search engine developer, see
7:
2931:. This is because a general functor
2144:whose only element is the empty set.
993:of a power set Boolean algebra (see
3189:), is used because the powerset of
1773:elements (the original set itself).
1693:âelements subset, so the number of
187:(as developed, for example, in the
1972:
1851:
951:, together with the operations of
391:
276:
38:The elements of the power set of {
14:
3081:and takes them to morphisms from
1039:Representing subsets as functions
977:. In fact, one can show that any
875:is the set of all functions from
5284:
5214:
2993:{\displaystyle h:a\rightarrow b}
1753:element (the singleton subsets),
2961:{\displaystyle {\text{C}}(-,c)}
1102:
633:are demonstrated in the below.
262:{\displaystyle \mathbb {P} (S)}
3060:
3048:
3042:
3039:
3027:
2984:
2955:
2943:
2912:
2906:
2873:
2867:
2834:
2822:
2776:
2770:
2742:
2729:
2720:
2707:
2695:
2689:
2673:
2667:
2216:Subsets of limited cardinality
2173:be any element of the set and
1862:
1854:
1838:
1830:
1713:) is a number of subsets with
995:Stone's representation theorem
918:shows that the power set of a
285:
279:
256:
250:
120:
116:
110:
1:
5175:History of mathematical logic
3364:Sheaves in Geometry and Logic
3197:to the given two-element set.
3073:, which takes morphisms from
5100:Primitive recursive function
3281:Mac Lane & Moerdijk 1992
2898:
2809:
1743:elements (the empty subset),
1673:Relation to binomial theorem
1185:. First, the enumerated set
1132:
940:Cardinality of the continuum
377:{\displaystyle \varnothing }
3380:Puntambekar, A. A. (2007).
2857:image morphism, so that if
1255:For the whole power set of
869:holds. Generally speaking,
495:and hence the power set of
5912:
5744:von NeumannâBernaysâGödel
4164:SchröderâBernstein theorem
3891:Monadic predicate calculus
3550:Foundations of mathematics
3384:. Technical Publications.
3258:Category Theory in Context
908:Cantor's diagonal argument
397:{\displaystyle \emptyset }
353:, then all the subsets of
18:
5545:One-to-one correspondence
5282:
5210:
5197:Philosophy of mathematics
5146:Automated theorem proving
4317:
4271:Von NeumannâBernaysâGödel
3912:
3112:can be understood as the
2362:. Given two multigraphs
2322:as the set of subsets of
1279:
1274:
1269:
1266:
932:one-to-one correspondence
565:is a finite set with the
31:
3120:between power sets, the
2544:Functors and quantifiers
2236:is sometimes denoted by
2183:; then the power set of
806:. Since each element in
203:is variously denoted as
4847:Self-verifying theories
4668:Tarski's axiomatization
3619:Tarski's undefinability
3614:incompleteness theorems
2427:complete directed graph
1717:elements in a set with
1288:{ }
1000:The power set of a set
945:The power set of a set
742:otherwise. Each subset
672:to the two-element set
620:denoting the power set
291:{\displaystyle \wp (S)}
5503:Constructible universe
5330:Constructibility (V=L)
5221:Mathematics portal
4832:Proof of impossibility
4480:propositional variable
3790:Propositional calculus
3126:existential quantifier
3067:
2994:
2962:
2925:
2841:
2783:
2397:of homomorphisms from
2230:less than or equal to
2220:The set of subsets of
2167:
1991:
1964:
1882:
1843:
820:under any function in
812:corresponds to either
398:
378:
292:
263:
137:
5726:Principia Mathematica
5560:Transfinite induction
5419:(i.e. set difference)
5090:Kolmogorov complexity
5043:Computably enumerable
4943:Model complete theory
4735:Principia Mathematica
3795:Propositional formula
3624:BanachâTarski paradox
3405:mathworld.wolfram.com
3068:
3000:to precomposition by
2995:
2963:
2926:
2842:
2784:
2258:is sometimes denoted
2168:
2136:The power set of the
2031:proceeds as follows:
1992:
1944:
1883:
1813:
1109:and the power set of
653:with the cardinality
399:
379:
293:
264:
138:
5800:Burali-Forti paradox
5555:Set-builder notation
5508:Continuum hypothesis
5448:Symmetric difference
5038:ChurchâTuring thesis
5025:Computability theory
4234:continuum hypothesis
3752:Square of opposition
3610:Gödel's completeness
3110:universal quantifier
3008:
2972:
2935:
2861:
2799:
2607:
2517:subobject classifier
2511:) and has an object
2281:might be denoted by
2157:
2147:For a non-empty set
2016:recursive definition
2000:Recursive definition
1907:
1789:
1711:binomial coefficient
1639:are enumerated with
1087:) is the set of all
1077:von Neumann ordinals
1071:" can be defined as
1012:symmetric difference
842:von Neumann ordinals
388:
368:
273:
242:
185:axiomatic set theory
98:
5761:TarskiâGrothendieck
5192:Mathematical object
5083:P versus NP problem
5048:Computable function
4842:Reverse mathematics
4768:Logical consequence
4645:primitive recursive
4640:elementary function
4413:Free/bound variable
4266:TarskiâGrothendieck
3785:Logical connectives
3715:Logical equivalence
3565:Logical consequence
3464:Power set Algorithm
3399:Weisstein, Eric W.
3366:, Springer-Verlag,
2330:algebraic structure
2199:and a power set of
2181:relative complement
1785:using the formula:
1075:(see, for example,
981:Boolean algebra is
857:is also denoted as
840:(see, for example,
832:. Since the number
666:is a function from
199:. The powerset of
28:
5894:Operations on sets
5350:Limitation of size
4990:Transfer principle
4953:Semantics of logic
4938:Categorical theory
4914:Non-standard model
4428:Logical connective
3555:Information theory
3504:Mathematical logic
3356:Mac Lane, Saunders
3104:and the theory of
3063:
2990:
2958:
2921:
2837:
2779:
2536:is required to be
2530:, in topos theory
2521:exponential object
2436:as the multigraph
2421:to the multigraph
2193:of a power set of
2163:
1987:
1878:
920:countably infinite
836:can be defined as
785:. In other words,
639:indicator function
394:
374:
288:
259:
197:axiom of power set
167:is the set of all
133:
92:Symbolic statement
21:Powerset (company)
5881:
5880:
5790:Russell's paradox
5739:ZermeloâFraenkel
5640:Dedekind-infinite
5513:Diagonal argument
5412:Cartesian product
5276:Set (mathematics)
5228:
5227:
5160:Abstract category
4963:Theories of truth
4773:Rule of inference
4763:Natural deduction
4744:
4743:
4289:
4288:
3994:Cartesian product
3899:
3898:
3805:Many-valued logic
3780:Boolean functions
3663:Russell's paradox
3638:diagonal argument
3535:First-order logic
3391:978-81-8431-193-8
2968:takes a morphism
2941:
2901:
2820:
2812:
2425:definable as the
2341:algebraic lattice
2166:{\displaystyle e}
1979:
1870:
1562:injective mapping
1558:
1557:
1272:of binary digits
1029:distributive laws
793:to the power set
789:is equivalent or
146:
145:
5901:
5863:Bertrand Russell
5853:John von Neumann
5838:Abraham Fraenkel
5833:Richard Dedekind
5795:Suslin's problem
5706:Cantor's theorem
5423:De Morgan's laws
5288:
5255:
5248:
5241:
5232:
5219:
5218:
5170:History of logic
5165:Category of sets
5058:Decision problem
4837:Ordinal analysis
4778:Sequent calculus
4676:Boolean algebras
4616:
4615:
4590:
4561:logical/constant
4315:
4301:
4224:ZermeloâFraenkel
3975:Set operations:
3910:
3847:
3678:
3658:LöwenheimâSkolem
3545:Formal semantics
3497:
3490:
3483:
3474:
3419:
3417:
3416:
3407:. Archived from
3395:
3376:
3351:
3341:
3338:Naive set theory
3327:
3304:. Universitext.
3298:Devlin, Keith J.
3284:
3278:
3272:
3271:
3253:
3247:
3244:Puntambekar 2007
3241:
3235:
3229:
3223:
3218:
3198:
3196:
3192:
3188:
3184:
3178:
3173:
3142:Cantor's theorem
3106:elementary topoi
3072:
3070:
3069:
3064:
3020:
3019:
3004:, so a function
2999:
2997:
2996:
2991:
2967:
2965:
2964:
2959:
2942:
2939:
2930:
2928:
2927:
2922:
2902:
2897:
2892:
2852:
2846:
2844:
2843:
2838:
2821:
2818:
2813:
2808:
2803:
2794:
2788:
2786:
2785:
2780:
2769:
2768:
2741:
2740:
2719:
2718:
2685:
2684:
2666:
2665:
2641:
2640:
2628:
2627:
2602:
2588:
2582:
2577:
2571:
2569:
2568:
2567:
2560:
2558:
2557:
2539:
2535:
2529:
2528:
2514:
2509:cartesian closed
2494:
2490:
2482:
2464:
2459:of vertices and
2458:
2449:
2439:
2435:
2424:
2420:
2414:
2408:
2402:
2396:
2390:
2373:
2367:
2354:
2350:
2327:
2321:
2307:
2301:
2300:
2294:
2285:
2280:
2274:
2262:
2257:
2251:
2240:
2235:
2225:
2210:
2204:
2198:
2188:
2178:
2172:
2170:
2169:
2164:
2152:
2127:
2121:
2120:
2099:
2098:
2089:
2088:
2082:
2067:
2054:
2048:
2047:
2041:
2030:
2024:
2023:
2009:
1996:
1994:
1993:
1988:
1986:
1985:
1984:
1971:
1963:
1958:
1940:
1939:
1927:
1923:
1922:
1902:
1897:
1887:
1885:
1884:
1879:
1877:
1876:
1875:
1866:
1865:
1857:
1850:
1842:
1841:
1833:
1827:
1809:
1805:
1804:
1784:
1782:
1772:
1768:
1762:
1758:
1752:
1748:
1742:
1738:
1728:
1724:
1720:
1716:
1708:
1692:
1686:
1679:binomial theorem
1668:
1662:
1656:
1650:
1646:
1642:
1638:
1632:
1626:
1620:
1611:
1605:
1604:
1598:
1582:
1576:
1570:
1569:
1554:
1546:
1538:
1533:
1514:
1506:
1498:
1493:
1478:
1470:
1462:
1457:
1442:
1434:
1426:
1421:
1416:{
1410:
1402:
1394:
1389:
1374:
1366:
1358:
1353:
1342:
1334:
1326:
1321:
1310:
1302:
1294:
1289:
1264:
1260:
1251:
1245:
1239:
1233:
1227:
1221:
1206:
1200:
1184:
1179:
1171:
1165:
1161:
1153:
1127:
1121:
1120:
1114:
1108:
1100:
1096:
1086:
1082:
1074:
1070:
1066:
1062:
1054:
1026:
1005:
972:
950:
934:with the set of
930:can be put in a
916:Cantor's theorem
902:
900:
894:
886:
880:
874:
868:
866:
860:
856:
855:
849:
839:
835:
831:
827:
823:
819:
815:
811:
805:
799:
798:
788:
784:
778:
774:
768:
764:
753:
747:
741:
737:
721:
715:
709:
703:
697:
691:
675:
671:
665:
660:
652:
646:
632:
626:
625:
619:
615:
613:
607:
606:
598:
592:
586:
580:
575:
564:
550:
498:
491:
474:
461:
448:
435:
426:
417:
408:or the null set)
403:
401:
400:
395:
383:
381:
380:
375:
363:
356:
352:
336:
324:
314:
308:
307:
302:. Any subset of
301:
297:
295:
294:
289:
268:
266:
265:
260:
249:
237:
226:
220:
215:
209:
208:
202:
182:
175:, including the
174:
166:
142:
140:
139:
134:
54:with respect to
36:
29:
5911:
5910:
5904:
5903:
5902:
5900:
5899:
5898:
5884:
5883:
5882:
5877:
5804:
5783:
5767:
5732:New Foundations
5679:
5569:
5488:Cardinal number
5471:
5457:
5398:
5289:
5280:
5264:
5259:
5229:
5224:
5213:
5206:
5151:Category theory
5141:Algebraic logic
5124:
5095:Lambda calculus
5033:Church encoding
5019:
4995:Truth predicate
4851:
4817:Complete theory
4740:
4609:
4605:
4601:
4596:
4588:
4308: and
4304:
4299:
4285:
4261:New Foundations
4229:axiom of choice
4212:
4174:Gödel numbering
4114: and
4106:
4010:
3895:
3845:
3826:
3775:Boolean algebra
3761:
3725:Equiconsistency
3690:Classical logic
3667:
3648:Halting problem
3636: and
3612: and
3600: and
3599:
3594:Theorems (
3589:
3506:
3501:
3426:
3414:
3412:
3398:
3392:
3379:
3374:
3354:
3332:Halmos, Paul R.
3330:
3316:
3306:Springer-Verlag
3296:
3293:
3288:
3287:
3279:
3275:
3268:
3255:
3254:
3250:
3242:
3238:
3230:
3226:
3219:
3212:
3207:
3202:
3201:
3194:
3190:
3186:
3180:
3176:
3174:
3170:
3165:
3138:
3102:category theory
3011:
3006:
3005:
2970:
2969:
2933:
2932:
2859:
2858:
2848:
2797:
2796:
2790:
2732:
2710:
2632:
2619:
2605:
2604:
2590:
2589:and a morphism
2580:
2579:
2573:
2565:
2564:
2563:
2562:
2555:
2554:
2553:
2546:
2537:
2531:
2524:
2523:
2512:
2492:
2488:
2466:
2460:
2454:
2445:
2437:
2431:
2422:
2416:
2410:
2404:
2398:
2392:
2378:
2369:
2363:
2352:
2348:
2323:
2317:
2314:
2298:
2297:
2296:
2288:
2283:
2282:
2276:
2268:
2260:
2259:
2253:
2245:
2238:
2237:
2231:
2221:
2218:
2206:
2200:
2194:
2184:
2174:
2155:
2154:
2148:
2118:
2117:
2096:
2095:
2086:
2085:
2084:
2069:
2059:
2058:Otherwise, let
2045:
2044:
2043:
2036:
2021:
2020:
2019:
2005:
2002:
1966:
1931:
1914:
1910:
1905:
1904:
1893:
1891:
1852:
1845:
1796:
1792:
1787:
1786:
1780:
1778:
1770:
1766:
1760:
1756:
1750:
1746:
1740:
1736:
1726:
1722:
1718:
1714:
1698:
1688:
1682:
1675:
1664:
1658:
1652:
1648:
1644:
1640:
1634:
1628:
1622:
1616:
1602:
1601:
1600:
1584:
1578:
1567:
1566:
1565:
1553:
1549:
1545:
1541:
1536:
1519:
1513:
1509:
1505:
1501:
1496:
1483:
1477:
1473:
1469:
1465:
1460:
1447:
1441:
1437:
1433:
1429:
1424:
1415:
1409:
1405:
1401:
1397:
1392:
1379:
1373:
1369:
1365:
1361:
1356:
1347:
1341:
1337:
1333:
1329:
1324:
1315:
1309:
1305:
1301:
1297:
1292:
1287:
1281:
1277:interpretation
1276:
1271:
1256:
1247:
1241:
1235:
1229:
1223:
1220:
1208:
1202:
1186:
1175:
1173:
1167:
1163:
1159:
1136:
1118:
1117:
1116:
1110:
1106:
1098:
1092:
1084:
1080:
1072:
1068:
1064:
1060:
1048:
1041:
1022:
1001:
975:Boolean algebra
968:
946:
928:natural numbers
896:
895:| = |
890:
888:
882:
876:
870:
864:
862:
858:
847:
846:
845:
837:
833:
829:
825:
821:
817:
813:
807:
796:
795:
794:
786:
780:
776:
770:
766:
763:
755:
749:
743:
739:
731:
723:
717:
711:
705:
699:
693:
685:
677:
673:
667:
656:
654:
648:
642:
623:
622:
621:
617:
604:
603:
602:
600:
594:
588:
582:
571:
569:
560:
557:
500:
496:
477:
464:
451:
438:
429:
420:
411:
386:
385:
366:
365:
361:
354:
338:
334:
331:
322:
305:
304:
303:
299:
271:
270:
240:
239:
228:
218:
217:
206:
205:
204:
200:
180:
172:
164:
96:
95:
59:
24:
17:
12:
11:
5:
5909:
5908:
5905:
5897:
5896:
5886:
5885:
5879:
5878:
5876:
5875:
5870:
5868:Thoralf Skolem
5865:
5860:
5855:
5850:
5845:
5840:
5835:
5830:
5825:
5820:
5814:
5812:
5806:
5805:
5803:
5802:
5797:
5792:
5786:
5784:
5782:
5781:
5778:
5772:
5769:
5768:
5766:
5765:
5764:
5763:
5758:
5753:
5752:
5751:
5736:
5735:
5734:
5722:
5721:
5720:
5709:
5708:
5703:
5698:
5693:
5687:
5685:
5681:
5680:
5678:
5677:
5672:
5667:
5662:
5653:
5648:
5643:
5633:
5628:
5627:
5626:
5621:
5616:
5606:
5596:
5591:
5586:
5580:
5578:
5571:
5570:
5568:
5567:
5562:
5557:
5552:
5550:Ordinal number
5547:
5542:
5537:
5532:
5531:
5530:
5525:
5515:
5510:
5505:
5500:
5495:
5485:
5480:
5474:
5472:
5470:
5469:
5466:
5462:
5459:
5458:
5456:
5455:
5450:
5445:
5440:
5435:
5430:
5428:Disjoint union
5425:
5420:
5414:
5408:
5406:
5400:
5399:
5397:
5396:
5395:
5394:
5389:
5378:
5377:
5375:Martin's axiom
5372:
5367:
5362:
5357:
5352:
5347:
5342:
5340:Extensionality
5337:
5332:
5327:
5326:
5325:
5320:
5315:
5305:
5299:
5297:
5291:
5290:
5283:
5281:
5279:
5278:
5272:
5270:
5266:
5265:
5260:
5258:
5257:
5250:
5243:
5235:
5226:
5225:
5211:
5208:
5207:
5205:
5204:
5199:
5194:
5189:
5184:
5183:
5182:
5172:
5167:
5162:
5153:
5148:
5143:
5138:
5136:Abstract logic
5132:
5130:
5126:
5125:
5123:
5122:
5117:
5115:Turing machine
5112:
5107:
5102:
5097:
5092:
5087:
5086:
5085:
5080:
5075:
5070:
5065:
5055:
5053:Computable set
5050:
5045:
5040:
5035:
5029:
5027:
5021:
5020:
5018:
5017:
5012:
5007:
5002:
4997:
4992:
4987:
4982:
4981:
4980:
4975:
4970:
4960:
4955:
4950:
4948:Satisfiability
4945:
4940:
4935:
4934:
4933:
4923:
4922:
4921:
4911:
4910:
4909:
4904:
4899:
4894:
4889:
4879:
4878:
4877:
4872:
4865:Interpretation
4861:
4859:
4853:
4852:
4850:
4849:
4844:
4839:
4834:
4829:
4819:
4814:
4813:
4812:
4811:
4810:
4800:
4795:
4785:
4780:
4775:
4770:
4765:
4760:
4754:
4752:
4746:
4745:
4742:
4741:
4739:
4738:
4730:
4729:
4728:
4727:
4722:
4721:
4720:
4715:
4710:
4690:
4689:
4688:
4686:minimal axioms
4683:
4672:
4671:
4670:
4659:
4658:
4657:
4652:
4647:
4642:
4637:
4632:
4619:
4617:
4598:
4597:
4595:
4594:
4593:
4592:
4580:
4575:
4574:
4573:
4568:
4563:
4558:
4548:
4543:
4538:
4533:
4532:
4531:
4526:
4516:
4515:
4514:
4509:
4504:
4499:
4489:
4484:
4483:
4482:
4477:
4472:
4462:
4461:
4460:
4455:
4450:
4445:
4440:
4435:
4425:
4420:
4415:
4410:
4409:
4408:
4403:
4398:
4393:
4383:
4378:
4376:Formation rule
4373:
4368:
4367:
4366:
4361:
4351:
4350:
4349:
4339:
4334:
4329:
4324:
4318:
4312:
4295:Formal systems
4291:
4290:
4287:
4286:
4284:
4283:
4278:
4273:
4268:
4263:
4258:
4253:
4248:
4243:
4238:
4237:
4236:
4231:
4220:
4218:
4214:
4213:
4211:
4210:
4209:
4208:
4198:
4193:
4192:
4191:
4184:Large cardinal
4181:
4176:
4171:
4166:
4161:
4147:
4146:
4145:
4140:
4135:
4120:
4118:
4108:
4107:
4105:
4104:
4103:
4102:
4097:
4092:
4082:
4077:
4072:
4067:
4062:
4057:
4052:
4047:
4042:
4037:
4032:
4027:
4021:
4019:
4012:
4011:
4009:
4008:
4007:
4006:
4001:
3996:
3991:
3986:
3981:
3973:
3972:
3971:
3966:
3956:
3951:
3949:Extensionality
3946:
3944:Ordinal number
3941:
3931:
3926:
3925:
3924:
3913:
3907:
3901:
3900:
3897:
3896:
3894:
3893:
3888:
3883:
3878:
3873:
3868:
3863:
3862:
3861:
3851:
3850:
3849:
3836:
3834:
3828:
3827:
3825:
3824:
3823:
3822:
3817:
3812:
3802:
3797:
3792:
3787:
3782:
3777:
3771:
3769:
3763:
3762:
3760:
3759:
3754:
3749:
3744:
3739:
3734:
3729:
3728:
3727:
3717:
3712:
3707:
3702:
3697:
3692:
3686:
3684:
3675:
3669:
3668:
3666:
3665:
3660:
3655:
3650:
3645:
3640:
3628:Cantor's
3626:
3621:
3616:
3606:
3604:
3591:
3590:
3588:
3587:
3582:
3577:
3572:
3567:
3562:
3557:
3552:
3547:
3542:
3537:
3532:
3527:
3526:
3525:
3514:
3512:
3508:
3507:
3502:
3500:
3499:
3492:
3485:
3477:
3471:
3470:
3461:
3449:
3437:
3425:
3424:External links
3422:
3421:
3420:
3396:
3390:
3377:
3372:
3360:Moerdijk, Ieke
3352:
3328:
3314:
3292:
3289:
3286:
3285:
3273:
3267:978-0486809038
3266:
3256:Riehl, Emily.
3248:
3246:, pp. 1â2
3236:
3224:
3209:
3208:
3206:
3203:
3200:
3199:
3167:
3166:
3164:
3161:
3160:
3159:
3154:
3149:
3147:Family of sets
3144:
3137:
3134:
3062:
3059:
3056:
3053:
3050:
3047:
3044:
3041:
3038:
3035:
3032:
3029:
3026:
3023:
3018:
3014:
2989:
2986:
2983:
2980:
2977:
2957:
2954:
2951:
2948:
2945:
2920:
2917:
2914:
2911:
2908:
2905:
2900:
2896:
2890:
2887:
2884:
2881:
2878:
2875:
2872:
2869:
2866:
2836:
2833:
2830:
2827:
2824:
2816:
2811:
2807:
2778:
2775:
2772:
2767:
2762:
2759:
2756:
2753:
2750:
2747:
2744:
2739:
2735:
2731:
2728:
2725:
2722:
2717:
2713:
2709:
2706:
2703:
2700:
2697:
2694:
2691:
2688:
2683:
2678:
2675:
2672:
2669:
2664:
2659:
2656:
2653:
2650:
2647:
2644:
2639:
2635:
2631:
2626:
2622:
2618:
2615:
2612:
2545:
2542:
2507:(and moreover
2313:
2310:
2286:
2263:
2241:
2217:
2214:
2213:
2212:
2162:
2145:
2130:
2129:
2056:
2001:
1998:
1983:
1978:
1975:
1970:
1962:
1957:
1954:
1951:
1947:
1943:
1938:
1934:
1930:
1926:
1921:
1917:
1913:
1874:
1869:
1864:
1860:
1856:
1849:
1840:
1836:
1832:
1826:
1823:
1820:
1816:
1812:
1808:
1803:
1799:
1795:
1775:
1774:
1764:
1754:
1744:
1674:
1671:
1556:
1555:
1551:
1547:
1543:
1539:
1534:
1516:
1515:
1511:
1507:
1503:
1499:
1494:
1480:
1479:
1475:
1471:
1467:
1463:
1458:
1444:
1443:
1439:
1435:
1431:
1427:
1422:
1412:
1411:
1407:
1403:
1399:
1395:
1390:
1376:
1375:
1371:
1367:
1363:
1359:
1354:
1344:
1343:
1339:
1335:
1331:
1327:
1322:
1312:
1311:
1307:
1303:
1299:
1295:
1290:
1284:
1283:
1278:
1273:
1268:
1218:
1040:
1037:
905:
904:
759:
727:
681:
556:
553:
493:
492:
475:
462:
449:
436:
427:
418:
409:
393:
373:
364:(also denoted
330:
327:
318:family of sets
287:
284:
281:
278:
258:
255:
252:
248:
144:
143:
132:
129:
126:
122:
118:
115:
112:
109:
106:
103:
93:
89:
88:
85:
81:
80:
75:
71:
70:
65:
61:
60:
37:
15:
13:
10:
9:
6:
4:
3:
2:
5907:
5906:
5895:
5892:
5891:
5889:
5874:
5873:Ernst Zermelo
5871:
5869:
5866:
5864:
5861:
5859:
5858:Willard Quine
5856:
5854:
5851:
5849:
5846:
5844:
5841:
5839:
5836:
5834:
5831:
5829:
5826:
5824:
5821:
5819:
5816:
5815:
5813:
5811:
5810:Set theorists
5807:
5801:
5798:
5796:
5793:
5791:
5788:
5787:
5785:
5779:
5777:
5774:
5773:
5770:
5762:
5759:
5757:
5756:KripkeâPlatek
5754:
5750:
5747:
5746:
5745:
5742:
5741:
5740:
5737:
5733:
5730:
5729:
5728:
5727:
5723:
5719:
5716:
5715:
5714:
5711:
5710:
5707:
5704:
5702:
5699:
5697:
5694:
5692:
5689:
5688:
5686:
5682:
5676:
5673:
5671:
5668:
5666:
5663:
5661:
5659:
5654:
5652:
5649:
5647:
5644:
5641:
5637:
5634:
5632:
5629:
5625:
5622:
5620:
5617:
5615:
5612:
5611:
5610:
5607:
5604:
5600:
5597:
5595:
5592:
5590:
5587:
5585:
5582:
5581:
5579:
5576:
5572:
5566:
5563:
5561:
5558:
5556:
5553:
5551:
5548:
5546:
5543:
5541:
5538:
5536:
5533:
5529:
5526:
5524:
5521:
5520:
5519:
5516:
5514:
5511:
5509:
5506:
5504:
5501:
5499:
5496:
5493:
5489:
5486:
5484:
5481:
5479:
5476:
5475:
5473:
5467:
5464:
5463:
5460:
5454:
5451:
5449:
5446:
5444:
5441:
5439:
5436:
5434:
5431:
5429:
5426:
5424:
5421:
5418:
5415:
5413:
5410:
5409:
5407:
5405:
5401:
5393:
5392:specification
5390:
5388:
5385:
5384:
5383:
5380:
5379:
5376:
5373:
5371:
5368:
5366:
5363:
5361:
5358:
5356:
5353:
5351:
5348:
5346:
5343:
5341:
5338:
5336:
5333:
5331:
5328:
5324:
5321:
5319:
5316:
5314:
5311:
5310:
5309:
5306:
5304:
5301:
5300:
5298:
5296:
5292:
5287:
5277:
5274:
5273:
5271:
5267:
5263:
5256:
5251:
5249:
5244:
5242:
5237:
5236:
5233:
5223:
5222:
5217:
5209:
5203:
5200:
5198:
5195:
5193:
5190:
5188:
5185:
5181:
5178:
5177:
5176:
5173:
5171:
5168:
5166:
5163:
5161:
5157:
5154:
5152:
5149:
5147:
5144:
5142:
5139:
5137:
5134:
5133:
5131:
5127:
5121:
5118:
5116:
5113:
5111:
5110:Recursive set
5108:
5106:
5103:
5101:
5098:
5096:
5093:
5091:
5088:
5084:
5081:
5079:
5076:
5074:
5071:
5069:
5066:
5064:
5061:
5060:
5059:
5056:
5054:
5051:
5049:
5046:
5044:
5041:
5039:
5036:
5034:
5031:
5030:
5028:
5026:
5022:
5016:
5013:
5011:
5008:
5006:
5003:
5001:
4998:
4996:
4993:
4991:
4988:
4986:
4983:
4979:
4976:
4974:
4971:
4969:
4966:
4965:
4964:
4961:
4959:
4956:
4954:
4951:
4949:
4946:
4944:
4941:
4939:
4936:
4932:
4929:
4928:
4927:
4924:
4920:
4919:of arithmetic
4917:
4916:
4915:
4912:
4908:
4905:
4903:
4900:
4898:
4895:
4893:
4890:
4888:
4885:
4884:
4883:
4880:
4876:
4873:
4871:
4868:
4867:
4866:
4863:
4862:
4860:
4858:
4854:
4848:
4845:
4843:
4840:
4838:
4835:
4833:
4830:
4827:
4826:from ZFC
4823:
4820:
4818:
4815:
4809:
4806:
4805:
4804:
4801:
4799:
4796:
4794:
4791:
4790:
4789:
4786:
4784:
4781:
4779:
4776:
4774:
4771:
4769:
4766:
4764:
4761:
4759:
4756:
4755:
4753:
4751:
4747:
4737:
4736:
4732:
4731:
4726:
4725:non-Euclidean
4723:
4719:
4716:
4714:
4711:
4709:
4708:
4704:
4703:
4701:
4698:
4697:
4695:
4691:
4687:
4684:
4682:
4679:
4678:
4677:
4673:
4669:
4666:
4665:
4664:
4660:
4656:
4653:
4651:
4648:
4646:
4643:
4641:
4638:
4636:
4633:
4631:
4628:
4627:
4625:
4621:
4620:
4618:
4613:
4607:
4602:Example
4599:
4591:
4586:
4585:
4584:
4581:
4579:
4576:
4572:
4569:
4567:
4564:
4562:
4559:
4557:
4554:
4553:
4552:
4549:
4547:
4544:
4542:
4539:
4537:
4534:
4530:
4527:
4525:
4522:
4521:
4520:
4517:
4513:
4510:
4508:
4505:
4503:
4500:
4498:
4495:
4494:
4493:
4490:
4488:
4485:
4481:
4478:
4476:
4473:
4471:
4468:
4467:
4466:
4463:
4459:
4456:
4454:
4451:
4449:
4446:
4444:
4441:
4439:
4436:
4434:
4431:
4430:
4429:
4426:
4424:
4421:
4419:
4416:
4414:
4411:
4407:
4404:
4402:
4399:
4397:
4394:
4392:
4389:
4388:
4387:
4384:
4382:
4379:
4377:
4374:
4372:
4369:
4365:
4362:
4360:
4359:by definition
4357:
4356:
4355:
4352:
4348:
4345:
4344:
4343:
4340:
4338:
4335:
4333:
4330:
4328:
4325:
4323:
4320:
4319:
4316:
4313:
4311:
4307:
4302:
4296:
4292:
4282:
4279:
4277:
4274:
4272:
4269:
4267:
4264:
4262:
4259:
4257:
4254:
4252:
4249:
4247:
4246:KripkeâPlatek
4244:
4242:
4239:
4235:
4232:
4230:
4227:
4226:
4225:
4222:
4221:
4219:
4215:
4207:
4204:
4203:
4202:
4199:
4197:
4194:
4190:
4187:
4186:
4185:
4182:
4180:
4177:
4175:
4172:
4170:
4167:
4165:
4162:
4159:
4155:
4151:
4148:
4144:
4141:
4139:
4136:
4134:
4131:
4130:
4129:
4125:
4122:
4121:
4119:
4117:
4113:
4109:
4101:
4098:
4096:
4093:
4091:
4090:constructible
4088:
4087:
4086:
4083:
4081:
4078:
4076:
4073:
4071:
4068:
4066:
4063:
4061:
4058:
4056:
4053:
4051:
4048:
4046:
4043:
4041:
4038:
4036:
4033:
4031:
4028:
4026:
4023:
4022:
4020:
4018:
4013:
4005:
4002:
4000:
3997:
3995:
3992:
3990:
3987:
3985:
3982:
3980:
3977:
3976:
3974:
3970:
3967:
3965:
3962:
3961:
3960:
3957:
3955:
3952:
3950:
3947:
3945:
3942:
3940:
3936:
3932:
3930:
3927:
3923:
3920:
3919:
3918:
3915:
3914:
3911:
3908:
3906:
3902:
3892:
3889:
3887:
3884:
3882:
3879:
3877:
3874:
3872:
3869:
3867:
3864:
3860:
3857:
3856:
3855:
3852:
3848:
3843:
3842:
3841:
3838:
3837:
3835:
3833:
3829:
3821:
3818:
3816:
3813:
3811:
3808:
3807:
3806:
3803:
3801:
3798:
3796:
3793:
3791:
3788:
3786:
3783:
3781:
3778:
3776:
3773:
3772:
3770:
3768:
3767:Propositional
3764:
3758:
3755:
3753:
3750:
3748:
3745:
3743:
3740:
3738:
3735:
3733:
3730:
3726:
3723:
3722:
3721:
3718:
3716:
3713:
3711:
3708:
3706:
3703:
3701:
3698:
3696:
3695:Logical truth
3693:
3691:
3688:
3687:
3685:
3683:
3679:
3676:
3674:
3670:
3664:
3661:
3659:
3656:
3654:
3651:
3649:
3646:
3644:
3641:
3639:
3635:
3631:
3627:
3625:
3622:
3620:
3617:
3615:
3611:
3608:
3607:
3605:
3603:
3597:
3592:
3586:
3583:
3581:
3578:
3576:
3573:
3571:
3568:
3566:
3563:
3561:
3558:
3556:
3553:
3551:
3548:
3546:
3543:
3541:
3538:
3536:
3533:
3531:
3528:
3524:
3521:
3520:
3519:
3516:
3515:
3513:
3509:
3505:
3498:
3493:
3491:
3486:
3484:
3479:
3478:
3475:
3469:
3465:
3462:
3460:
3458:
3453:
3450:
3448:
3446:
3441:
3438:
3435:
3431:
3428:
3427:
3423:
3411:on 2023-04-06
3410:
3406:
3402:
3397:
3393:
3387:
3383:
3378:
3375:
3373:0-387-97710-4
3369:
3365:
3361:
3357:
3353:
3349:
3345:
3340:
3339:
3333:
3329:
3325:
3321:
3317:
3315:0-387-90441-7
3311:
3307:
3303:
3299:
3295:
3294:
3290:
3282:
3277:
3274:
3269:
3263:
3259:
3252:
3249:
3245:
3240:
3237:
3233:
3228:
3225:
3222:
3217:
3215:
3211:
3204:
3183:
3175:The notation
3172:
3169:
3162:
3158:
3155:
3153:
3152:Field of sets
3150:
3148:
3145:
3143:
3140:
3139:
3135:
3133:
3131:
3127:
3123:
3122:inverse image
3119:
3115:
3114:right adjoint
3111:
3107:
3103:
3098:
3096:
3092:
3088:
3084:
3080:
3076:
3057:
3054:
3051:
3045:
3036:
3033:
3030:
3024:
3021:
3016:
3012:
3003:
2987:
2981:
2978:
2975:
2952:
2949:
2946:
2918:
2915:
2909:
2903:
2888:
2885:
2882:
2879:
2876:
2870:
2864:
2856:
2851:
2831:
2828:
2825:
2814:
2793:
2773:
2760:
2754:
2751:
2748:
2745:
2737:
2733:
2726:
2723:
2715:
2711:
2704:
2698:
2692:
2686:
2676:
2670:
2657:
2651:
2648:
2645:
2642:
2637:
2633:
2629:
2624:
2620:
2613:
2610:
2601:
2597:
2593:
2586:
2576:
2551:
2543:
2541:
2534:
2527:
2522:
2518:
2510:
2506:
2502:
2498:
2486:
2481:
2477:
2473:
2469:
2463:
2457:
2451:
2448:
2443:
2440:, called the
2434:
2428:
2419:
2413:
2407:
2401:
2395:
2389:
2385:
2381:
2377:
2372:
2366:
2361:
2356:
2355:in this way.
2344:
2342:
2338:
2333:
2331:
2326:
2320:
2311:
2309:
2305:
2292:
2279:
2272:
2267:
2256:
2249:
2244:
2234:
2229:
2224:
2215:
2209:
2203:
2197:
2192:
2187:
2182:
2177:
2160:
2151:
2146:
2143:
2139:
2135:
2134:
2133:
2125:
2115:
2111:
2107:
2103:
2093:
2080:
2076:
2072:
2066:
2062:
2057:
2052:
2039:
2034:
2033:
2032:
2028:
2017:
2013:
2008:
1999:
1997:
1976:
1973:
1960:
1955:
1952:
1949:
1945:
1941:
1936:
1932:
1928:
1924:
1919:
1915:
1911:
1901:
1896:
1888:
1867:
1858:
1834:
1824:
1821:
1818:
1814:
1810:
1806:
1801:
1797:
1793:
1765:
1759:subsets with
1755:
1749:subsets with
1745:
1735:
1734:
1733:
1730:
1712:
1709:(also called
1706:
1702:
1697:, denoted as
1696:
1691:
1685:
1680:
1672:
1670:
1667:
1661:
1655:
1637:
1631:
1625:
1619:
1613:
1609:
1596:
1592:
1588:
1581:
1574:
1563:
1548:
1540:
1535:
1531:
1527:
1523:
1518:
1517:
1508:
1500:
1495:
1491:
1487:
1482:
1481:
1472:
1464:
1459:
1455:
1451:
1446:
1445:
1436:
1428:
1423:
1419:
1414:
1413:
1404:
1396:
1391:
1387:
1383:
1378:
1377:
1368:
1360:
1355:
1351:
1346:
1345:
1336:
1328:
1323:
1319:
1314:
1313:
1304:
1296:
1291:
1286:
1285:
1265:
1262:
1259:
1253:
1250:
1244:
1238:
1232:
1226:
1216:
1212:
1205:
1198:
1194:
1190:
1183:
1178:
1170:
1157:
1154:, to get the
1151:
1147:
1143:
1139:
1134:
1129:
1125:
1113:
1104:
1095:
1090:
1078:
1058:
1053:
1052:
1046:
1038:
1036:
1034:
1030:
1025:
1020:
1017:
1013:
1009:
1008:abelian group
1004:
998:
996:
992:
988:
984:
980:
976:
971:
966:
962:
958:
954:
949:
943:
941:
937:
933:
929:
925:
921:
917:
913:
909:
899:
893:
885:
879:
873:
853:
843:
810:
803:
792:
783:
773:
762:
758:
752:
746:
735:
730:
726:
720:
714:
708:
702:
696:
689:
684:
680:
676:, denoted as
670:
664:
659:
651:
645:
640:
636:
635:
634:
630:
611:
597:
591:
585:
579:
574:
568:
563:
554:
552:
548:
544:
540:
536:
532:
528:
524:
520:
516:
512:
508:
504:
489:
485:
481:
476:
472:
468:
463:
459:
455:
450:
446:
442:
437:
433:
428:
424:
419:
415:
410:
407:
360:
359:
358:
350:
346:
342:
328:
326:
320:
319:
312:
282:
253:
235:
231:
224:
213:
198:
194:
190:
186:
178:
170:
163:
159:
155:
151:
130:
127:
124:
113:
107:
104:
101:
94:
90:
86:
82:
79:
76:
72:
69:
68:Set operation
66:
62:
57:
53:
49:
45:
41:
35:
30:
22:
5823:Georg Cantor
5818:Paul Bernays
5749:MorseâKelley
5724:
5657:
5656:Subset
5603:hereditarily
5565:Venn diagram
5523:ordered pair
5442:
5438:Intersection
5382:Axiom schema
5212:
5010:Ultraproduct
4857:Model theory
4822:Independence
4758:Formal proof
4750:Proof theory
4733:
4706:
4663:real numbers
4635:second-order
4546:Substitution
4423:Metalanguage
4364:conservative
4337:Axiom schema
4281:Constructive
4251:MorseâKelley
4217:Set theories
4196:Aleph number
4189:inaccessible
4095:Grothendieck
3998:
3979:intersection
3866:Higher-order
3854:Second-order
3800:Truth tables
3757:Venn diagram
3540:Formal proof
3456:
3452:Power object
3444:
3413:. Retrieved
3409:the original
3404:
3381:
3363:
3337:
3301:
3291:Bibliography
3283:, p. 58
3276:
3257:
3251:
3239:
3234:, p. 50
3227:
3181:
3171:
3130:left adjoint
3099:
3094:
3090:
3086:
3082:
3078:
3074:
3001:
2854:
2849:
2791:
2599:
2595:
2591:
2584:
2574:
2570:: Set â Set
2547:
2532:
2525:
2479:
2475:
2471:
2467:
2461:
2455:
2452:
2446:
2442:power object
2441:
2432:
2417:
2411:
2405:
2399:
2393:
2387:
2383:
2379:
2376:homomorphism
2370:
2364:
2357:
2345:
2334:
2332:or algebra.
2324:
2318:
2315:
2312:Power object
2303:
2290:
2277:
2270:
2265:
2254:
2247:
2242:
2232:
2222:
2219:
2207:
2201:
2195:
2185:
2175:
2149:
2131:
2123:
2113:
2109:
2105:
2101:
2091:
2078:
2074:
2070:
2064:
2060:
2050:
2037:
2026:
2006:
2003:
1899:
1894:
1889:
1776:
1769:subset with
1739:subset with
1731:
1704:
1700:
1695:combinations
1689:
1683:
1676:
1665:
1659:
1653:
1635:
1629:
1623:
1617:
1614:
1607:
1594:
1590:
1586:
1579:
1572:
1559:
1529:
1525:
1521:
1489:
1485:
1453:
1449:
1417:
1385:
1381:
1349:
1317:
1257:
1254:
1248:
1242:
1236:
1230:
1224:
1214:
1210:
1203:
1196:
1192:
1188:
1181:
1176:
1168:
1149:
1145:
1141:
1137:
1130:
1123:
1111:
1093:
1050:
1042:
1033:Boolean ring
1023:
1002:
999:
986:
978:
969:
957:intersection
947:
944:
936:real numbers
906:
897:
891:
883:
877:
871:
861:. Obviously
851:
808:
801:
781:
771:
760:
756:
750:
744:
733:
728:
724:
718:
712:
706:
700:
694:
687:
682:
678:
668:
662:
657:
649:
643:
628:
609:
595:
589:
583:
577:
572:
561:
558:
546:
542:
538:
534:
530:
526:
522:
518:
514:
510:
506:
502:
494:
487:
483:
479:
470:
466:
457:
453:
444:
440:
431:
422:
413:
348:
344:
340:
332:
316:
315:is called a
310:
233:
229:
222:
211:
157:
153:
147:
47:
43:
39:
5848:Thomas Jech
5691:Alternative
5670:Uncountable
5624:Ultrafilter
5483:Cardinality
5387:replacement
5335:Determinacy
5120:Type theory
5068:undecidable
5000:Truth value
4887:equivalence
4566:non-logical
4179:Enumeration
4169:Isomorphism
4116:cardinality
4100:Von Neumann
4065:Ultrafilter
4030:Uncountable
3964:equivalence
3881:Quantifiers
3871:Fixed-point
3840:First-order
3720:Consistency
3705:Proposition
3682:Traditional
3653:Lindström's
3643:Compactness
3585:Type theory
3530:Cardinality
3401:"Power Set"
3232:Devlin 1979
3157:Combination
2559:: Set â Set
2515:, called a
2360:multigraphs
2228:cardinality
2132:In words:
1767:C(3, 3) = 1
1757:C(3, 2) = 3
1747:C(3, 1) = 3
1737:C(3, 0) = 1
1282:equivalent
1156:isomorphism
1135:, in which
1103:shown above
1016:commutative
924:uncountably
912:cardinality
716:belongs to
704:or not; If
698:belongs to
567:cardinality
337:is the set
183:itself. In
150:mathematics
5843:Kurt Gödel
5828:Paul Cohen
5665:Transitive
5433:Identities
5417:Complement
5404:Operations
5365:Regularity
5303:Adjunction
5262:Set theory
4931:elementary
4624:arithmetic
4492:Quantifier
4470:functional
4342:Expression
4060:Transitive
4004:identities
3989:complement
3922:hereditary
3905:Set theory
3434:PlanetMath
3415:2020-09-05
3348:0087.04403
3324:0407.04003
3205:References
3089:, through
2349:{0, 1} = 2
2053:) = { {} }
2012:finite set
1729:elements.
1261:, we get:
1045:set theory
991:subalgebra
983:isomorphic
961:complement
867:| = 2
614:| = 2
555:Properties
193:postulated
78:Set theory
5776:Paradoxes
5696:Axiomatic
5675:Universal
5651:Singleton
5646:Recursive
5589:Countable
5584:Amorphous
5443:Power set
5360:Power set
5318:dependent
5313:countable
5202:Supertask
5105:Recursion
5063:decidable
4897:saturated
4875:of models
4798:deductive
4793:axiomatic
4713:Hilbert's
4700:Euclidean
4681:canonical
4604:axiomatic
4536:Signature
4465:Predicate
4354:Extension
4276:Ackermann
4201:Operation
4080:Universal
4070:Recursive
4045:Singleton
4040:Inhabited
4025:Countable
4015:Types of
3999:power set
3969:partition
3886:Predicate
3832:Predicate
3747:Syllogism
3737:Soundness
3710:Inference
3700:Tautology
3602:paradoxes
3440:Power set
3430:Power set
3221:Weisstein
3043:→
3017:∗
2985:→
2947:−
2899:¯
2883:⊆
2826:−
2815:≅
2810:¯
2761:∈
2658:∈
2142:singleton
2138:empty set
2112:} :
2077:∖ {
2014:, then a
1946:∑
1898:| =
1815:∑
1484:{
1452:,
1348:{
1180:| =
1089:functions
1057:functions
1006:forms an
965:ÎŁ-algebra
791:bijective
661:| =
647:of a set
576:| =
406:empty set
392:∅
372:∅
277:℘
177:empty set
154:power set
128:⊆
121:⟺
105:∈
84:Statement
56:inclusion
27:Power set
5888:Category
5780:Problems
5684:Theories
5660:Superset
5636:Infinite
5465:Concepts
5345:Infinity
5269:Overview
5187:Logicism
5180:timeline
5156:Concrete
5015:Validity
4985:T-schema
4978:Kripke's
4973:Tarski's
4968:semantic
4958:Strength
4907:submodel
4902:spectrum
4870:function
4718:Tarski's
4707:Elements
4694:geometry
4650:Robinson
4571:variable
4556:function
4529:spectrum
4519:Sentence
4475:variable
4418:Language
4371:Relation
4332:Automata
4322:Alphabet
4306:language
4160:-jection
4138:codomain
4124:Function
4085:Universe
4055:Infinite
3959:Relation
3742:Validity
3732:Argument
3630:theorem,
3362:(1992),
3334:(1960).
3300:(1979).
3136:See also
2503:that is
2501:category
2485:presheaf
2474: :
2382: :
2211:element.
1560:Such an
1270:Sequence
987:infinite
690:â {0, 1}
686: :
158:powerset
5718:General
5713:Zermelo
5619:subbase
5601: (
5540:Forcing
5518:Element
5490: (
5468:Methods
5355:Pairing
5129:Related
4926:Diagram
4824: (
4803:Hilbert
4788:Systems
4783:Theorem
4661:of the
4606:systems
4386:Formula
4381:Grammar
4297: (
4241:General
3954:Forcing
3939:Element
3859:Monadic
3634:paradox
3575:Theorem
3511:General
3454:at the
3442:at the
3128:is the
3118:functor
2853:to the
2550:functor
2337:lattice
2083:; then
2042:, then
1593:, 2), (
1589:, 1), (
1537:1, 1, 1
1497:1, 1, 0
1461:1, 0, 1
1425:1, 0, 0
1393:0, 1, 1
1357:0, 1, 0
1325:0, 0, 1
1293:0, 0, 0
1280:Decimal
1267:Subset
1217:} = 011
1195:, 2), (
1191:, 1), (
1162:, with
1083:(i.e.,
1067:. As "
963:, is a
922:set is
844:), the
722:, then
329:Example
195:by the
169:subsets
160:) of a
52:ordered
5609:Filter
5599:Finite
5535:Family
5478:Almost
5323:global
5308:Choice
5295:Axioms
4892:finite
4655:Skolem
4608:
4583:Theory
4551:Symbol
4541:String
4524:atomic
4401:ground
4396:closed
4391:atomic
4347:ground
4310:syntax
4206:binary
4133:domain
4050:Finite
3815:finite
3673:Logics
3632:
3580:Theory
3388:
3370:
3346:
3322:
3312:
3264:
3187:{0, 1}
3108:, the
2505:closed
2153:, let
1892:|
1783:|
1779:|
1647:, and
1633:, and
1597:, 3) }
1275:Binary
1199:, 3) }
1174:|
1099:{0, 1}
1085:{0, 1}
1073:{0, 1}
1019:monoid
979:finite
901:|
889:|
863:|
838:{0, 1}
826:{0, 1}
822:{0, 1}
787:{0, 1}
777:{0, 1}
765:, and
738:, and
674:{0, 1}
655:|
601:|
570:|
501:{{}, {
404:, the
152:, the
5701:Naive
5631:Fuzzy
5594:Empty
5577:types
5528:tuple
5498:Class
5492:large
5453:Union
5370:Union
4882:Model
4630:Peano
4487:Proof
4327:Arity
4256:Naive
4143:image
4075:Fuzzy
4035:Empty
3984:union
3929:Class
3570:Model
3560:Lemma
3518:Axiom
3163:Notes
3116:of a
2499:as a
2497:topos
2191:union
2189:is a
2140:is a
2104:) âȘ {
2010:is a
1564:from
1160:2 â 1
1133:above
1101:. As
1091:from
1059:from
967:over
953:union
938:(see
767:{0,1}
736:) = 1
321:over
298:, or
74:Field
5614:base
5005:Type
4808:list
4612:list
4589:list
4578:Term
4512:rank
4406:open
4300:list
4112:Maps
4017:sets
3876:Free
3846:list
3596:list
3523:list
3386:ISBN
3368:ISBN
3310:ISBN
3262:ISBN
3093:via
2561:and
2374:, a
2368:and
2264:<
2179:its
2094:) =
2068:and
2040:= {}
1677:The
1552:(10)
1512:(10)
1476:(10)
1440:(10)
1408:(10)
1372:(10)
1340:(10)
1308:(10)
959:and
887:and
537:}, {
529:}, {
521:}, {
513:}, {
509:}, {
505:}, {
357:are
179:and
156:(or
64:Type
5575:Set
4692:of
4674:of
4622:of
4154:Sur
4128:Map
3935:Ur-
3917:Set
3468:C++
3466:in
3459:Lab
3447:Lab
3432:at
3344:Zbl
3320:Zbl
3100:In
3097:.
3085:to
3077:to
2855:pre
2819:Set
2578:to
2444:of
2403:to
2295:or
2226:of
2108:âȘ {
2035:If
2018:of
2004:If
1585:{ (
1544:(2)
1542:111
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1398:011
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1362:010
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1187:{ (
1172:or
1140:= {
1097:to
1079:),
1063:to
1043:In
997:).
942:).
881:to
828:is
816:or
775:to
748:of
710:in
637:An
599:is
587:is
559:If
499:is
384:or
333:If
189:ZFC
171:of
162:set
148:In
50:}
5890::
5078:NP
4702::
4696::
4626::
4303:),
4158:Bi
4150:In
3403:.
3358:;
3318:.
3308:.
3260:.
3213:^
3132:.
2598:â
2594::
2552:,
2540:.
2478:â
2470:,
2450:.
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2308:.
2287:â„1
2126:)}
2116:â
2073:=
2063:â
1903::
1703:,
1699:C(
1643:,
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1528:,
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1520:{
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1448:{
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549:}}
545:,
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362:{}
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343:,
325:.
269:,
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4507:â
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2916:=
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725:I
719:A
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679:I
669:S
663:n
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650:S
644:A
631:)
629:S
627:(
624:P
618:2
612:)
610:S
608:(
605:P
596:S
590:n
584:S
578:n
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562:S
547:z
543:y
539:x
535:z
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497:S
490:}
488:z
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478:{
473:}
471:z
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465:{
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454:x
452:{
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439:{
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432:z
430:{
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421:{
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355:S
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349:z
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339:{
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323:S
313:)
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309:(
306:P
300:2
286:)
283:S
280:(
257:)
254:S
251:(
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236:)
234:S
232:(
230:P
225:)
223:S
221:(
214:)
212:S
210:(
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181:S
173:S
165:S
131:S
125:x
117:)
114:S
111:(
108:P
102:x
58:.
48:z
44:y
40:x
23:.
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