5227:
45:
5297:
2357:
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,
2798:
2440:
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
2006:
1897:
2940:
2856:
2617:
3082:
152:
2369:
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
3009:
2977:
2420:
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
278:
393:
413:
307:
2182:
1917:
2350:
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
1799:
3606:
2358:
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
2346:
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
2354:, and every algebraic lattice arises as the lattice of subalgebras of some algebra. So in that regard, subalgebras behave analogously to subsets.
5754:
4281:
2327:
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
4364:
3505:
2464:
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
2871:
2809:
4678:
4836:
3400:
3624:
2494:
giving the source (start) and target (end) vertices of each edge. An algebra all of whose operations are unary is called a
5443:
5263:
4691:
4014:
918:
3411:
2793:{\displaystyle A=\{x_{1},x_{2},...\}\in {\mathsf {P}}(S),{\mathsf {P}}f(A)=\{f(x_{1}),f(x_{2}),...\}\in {\mathsf {P}}(T)}
5771:
4696:
4686:
4423:
4276:
3629:
3276:
3620:
4832:
3382:
3324:
1005:
4174:
3018:
4929:
4673:
3498:
5749:
4234:
3927:
5629:
3668:
108:
5190:
4892:
4655:
4650:
4475:
3896:
3580:
1060:
5523:
5402:
5185:
4968:
4885:
4598:
4529:
4406:
3648:
3167:
5766:
5110:
4936:
4622:
4256:
3855:
2402:
consists of two functions, one mapping vertices to vertices and the other mapping edges to edges. The set
950:
5904:
5759:
5397:
5360:
4988:
4983:
4593:
4332:
4261:
3590:
3491:
985:
925:
than the set itself (or informally, the power set must be larger than the original set). In particular,
4917:
4507:
3901:
3869:
3560:
5414:
2982:
78:
5448:
5340:
5328:
5323:
5207:
5156:
5053:
4551:
4512:
3989:
3634:
2945:
2519:
1212:
is defined in which the number in each ordered pair represents the position of the paired element of
1000:
Boolean algebras, this is no longer true, but every infinite
Boolean algebra can be represented as a
967:
252:
3663:
5256:
5048:
4978:
4517:
4369:
4352:
4075:
3555:
17:
378:
5868:
5786:
5661:
5613:
5427:
5350:
4880:
4857:
4818:
4704:
4645:
4291:
4211:
4055:
3999:
3612:
2152:
971:
1594:
is not unique, but the sort order of the enumerated set does not change its cardinality. (E.g.,
398:
5820:
5701:
5513:
5333:
5170:
4897:
4875:
4842:
4735:
4581:
4566:
4539:
4490:
4374:
4309:
4134:
4100:
4095:
3969:
3800:
3777:
3136:
2858:. The contravariant power set functor is different from the covariant version in that it sends
2511:
2362:, there is no guarantee that a class of algebras contains an algebra that can play the role of
1099:
1067:
66:
5736:
5706:
5650:
5570:
5550:
5528:
5100:
4953:
4745:
4463:
4199:
4105:
3964:
3949:
3830:
3805:
1662:
respectively as the position of binary digit sequences.) The enumeration is possible even if
1039:
283:
3204:
can be identified with, is equivalent to, or bijective to the set of all the functions from
2506:
plays for subsets. Such a class is a special case of the more general notion of elementary
5810:
5800:
5634:
5565:
5518:
5458:
5345:
5073:
5035:
4912:
4716:
4556:
4480:
4458:
4286:
4244:
4143:
4110:
3974:
3762:
3673:
3474:
3120:
2527:
2026:
1721:
1087:
1022:
852:
195:
3358:
3334:
8:
5805:
5716:
5624:
5619:
5433:
5375:
5313:
5249:
5202:
5093:
5078:
5058:
5015:
4902:
4852:
4778:
4723:
4660:
4453:
4448:
4396:
4164:
4153:
3825:
3725:
3653:
3644:
3640:
3575:
3570:
3152:
2340:
2191:
926:
1674:
is infinite), such as the set of integers or rationals, but not possible for example if
5728:
5723:
5508:
5463:
5370:
5231:
5000:
4963:
4948:
4941:
4924:
4728:
4710:
4576:
4502:
4485:
4438:
4251:
4160:
3994:
3979:
3939:
3891:
3876:
3864:
3820:
3795:
3565:
3514:
2531:
2201:
2167:
1572:
1025:(with the empty set as the identity element and each set being its own inverse), and a
963:
921:
shows that the power set of a set (whether infinite or not) always has strictly higher
649:
207:
31:
4184:
1680:
is the set of real numbers, in which case we cannot enumerate all irrational numbers.
5585:
5422:
5385:
5355:
5286:
5226:
5166:
4973:
4783:
4773:
4665:
4546:
4381:
4357:
4138:
4122:
4027:
4004:
3881:
3850:
3815:
3710:
3545:
3396:
3378:
3366:
3320:
3272:
2351:
172:
88:
5873:
5863:
5848:
5843:
5711:
5365:
5180:
5175:
5068:
5025:
4847:
4808:
4803:
4788:
4614:
4571:
4468:
4266:
4216:
3790:
3752:
3354:
3347:
3330:
3116:
1689:
2800:. Elsewhere in this article, the power set was defined as the set of functions of
5742:
5680:
5498:
5318:
5161:
5151:
5105:
5088:
5043:
5005:
4907:
4827:
4634:
4561:
4534:
4522:
4428:
4342:
4316:
4271:
4239:
4040:
3842:
3785:
3735:
3700:
3658:
3316:
3112:
2583:. The covariant functor is defined more simply. as the functor which sends a set
2515:
2347:
44:
5878:
5675:
5656:
5560:
5545:
5502:
5438:
5380:
5146:
5125:
5083:
5063:
4958:
4813:
4411:
4401:
4391:
4386:
4320:
4194:
4070:
3959:
3954:
3932:
3533:
3157:
2437:
1042:, that the power set considered together with both of these operations forms a
938:
328:
3419:
5898:
5883:
5685:
5599:
5594:
5120:
4798:
4305:
4090:
4080:
4050:
4035:
3705:
3462:
3370:
3162:
3132:
3124:
1018:
930:
5853:
3353:. The University Series in Undergraduate Mathematics. van Nostrand Company.
1623:
to the integers without changing the number of one-to-one correspondences.)
1257:
corresponding to the position of it in the sequence exists in the subset of
1251:
is at the second from the right, and 1 in the sequence means the element of
5833:
5828:
5646:
5575:
5533:
5392:
5296:
5020:
4867:
4768:
4760:
4640:
4588:
4497:
4433:
4416:
4347:
4206:
4065:
3767:
3550:
3308:
3140:
2806:
into the set with 2 elements. Formally, this defines a natural isomorphism
2386:
1043:
62:
3450:
975:
5858:
5493:
5130:
5010:
4189:
4179:
4126:
3810:
3730:
3715:
3595:
3540:
3342:
2238:
1705:
1166:
1026:
946:
934:
922:
577:
160:
2530:. Although the term "power object" is sometimes used synonymously with
1032:
when considered with the operation of intersection (with the entire set
5838:
5609:
5272:
4060:
3915:
3886:
3692:
3444:
2370:
2022:
2001:{\displaystyle \left|2^{S}\right|=2^{n}=\sum _{k=0}^{n}{\binom {n}{k}}}
1588:
to integers is arbitrary, so this representation of all the subsets of
1055:
1001:
993:
203:
5641:
5604:
5555:
5212:
5115:
4168:
4085:
4045:
3945:
3757:
3747:
3720:
3483:
2148:
1892:{\displaystyle \left|2^{S}\right|=\sum _{k=0}^{|S|}{\binom {|S|}{k}}}
942:
801:
416:
187:
2614:(here, a function between sets) to the image morphism. That is, for
5197:
4995:
4443:
4148:
3742:
2495:
3440:
98:
The power set is the set that contains all subsets of a given set.
4793:
3585:
3128:
2935:{\displaystyle f(A)=B\subseteq T,{\overline {\mathsf {P}}}f(B)=A}
2560:
2851:{\displaystyle {\overline {\mathsf {P}}}\cong {\text{Set}}(-,2)}
2263:
or , and the set of subsets with cardinality strictly less than
1038:
as the identity element). It can hence be shown, by proving the
5666:
5488:
1626:
However, such finite binary representation is only possible if
1029:
179:
1743:
For example, the power set of a set with three elements, has:
790:
consists of all the indicator functions of all the subsets of
5538:
5305:
5241:
4337:
3683:
3528:
3478:
3285:
2507:
1668:
has an infinite cardinality (i.e., the number of elements in
996:
to the
Boolean algebra of the power set of a finite set. For
1245:
is located at the first from the right of this sequence and
3466:
3454:
1901:
Therefore, one can deduce the following identity, assuming
1736:
elements which are elements of the power set of a set with
1698:âelements combination from some set is another name for a
3248:
765:
is identified by or equivalent to the indicator function
199:
1610:
can be used to construct another injective mapping from
2559:
There is both a covariant and contravariant power set
1732:
elements; in other words it's the number of sets with
202:
axioms), the existence of the power set of any set is
27:
Mathematical set containing all subsets of a given set
3021:
2985:
2948:
2874:
2812:
2620:
2170:
1920:
1802:
1169:
with the binary representations of numbers from 0 to
401:
381:
286:
255:
111:
1774:
elements (the complements of the singleton subsets),
1049:
3236:
3227:
3225:
2426:are in bijection with the graph homomorphisms from
984:and can be viewed as the prototypical example of a
3346:
3135:functor of a function between sets; likewise, the
3076:
3003:
2971:
2934:
2850:
2792:
2176:
2000:
1891:
627:. This fact as well as the reason of the notation
407:
387:
301:
272:
146:
2498:. Every class of presheaves contains a presheaf
2226:
1992:
1979:
1883:
1858:
5896:
3365:
3291:
3222:
2286:or . Similarly, the set of non-empty subsets of
2339:generalizes naturally to the subalgebras of an
1683:
1142:This equivalence can be applied to the example
3077:{\displaystyle h^{*}:C(b,c)\rightarrow C(a,c)}
1139:, are considered identical set-theoretically.
5257:
3499:
592:(i.e., the number of all elements in the set
2768:
2712:
2665:
2627:
1263:for the sequence while 0 means it does not.
1066:is the notation representing the set of all
1021:when it is considered with the operation of
3390:
3254:
2554:
5264:
5250:
3691:
3506:
3492:
147:{\displaystyle x\in P(S)\iff x\subseteq S}
134:
130:
43:
2502:that plays the role for subalgebras that
703:, and it indicates whether an element of
652:or a characteristic function of a subset
604:), then the number of all the subsets of
257:
3190:, meaning the set of all functions from
2216:whose each element is expanded with the
1788:Using this relationship, we can compute
1177:being the number of elements in the set
3393:Theory Of Automata And Formal Languages
3313:Fundamentals of contemporary set theory
2476:of edges, and has two unary operations
2010:
1692:is closely related to the power set. A
1218:in a sequence of binary digits such as
14:
5897:
3513:
3341:
3307:
3242:
3196:to a given set of two elements (e.g.,
2905:
2816:
2776:
2692:
2673:
937:infinite. The power set of the set of
5245:
3487:
3409:
3231:
1632:can be enumerated. (In this example,
835:, the number of all the functions in
780:as the set of all the functions from
30:For the search engine developer, see
2942:. This is because a general functor
2155:whose only element is the empty set.
1004:of a power set Boolean algebra (see
3200:), is used because the powerset of
1784:elements (the original set itself).
1704:âelements subset, so the number of
198:(as developed, for example, in the
24:
1983:
1862:
962:, together with the operations of
402:
287:
49:The elements of the power set of {
25:
5916:
3434:
3266:
3092:and takes them to morphisms from
1050:Representing subsets as functions
988:. In fact, one can show that any
886:is the set of all functions from
5295:
5225:
3004:{\displaystyle h:a\rightarrow b}
1764:element (the singleton subsets),
3301:
2972:{\displaystyle {\text{C}}(-,c)}
2322:
1113:
644:are demonstrated in the below.
273:{\displaystyle \mathbb {P} (S)}
5271:
3260:
3180:
3071:
3059:
3053:
3050:
3038:
2995:
2966:
2954:
2923:
2917:
2884:
2878:
2845:
2833:
2787:
2781:
2753:
2740:
2731:
2718:
2706:
2700:
2684:
2678:
2227:Subsets of limited cardinality
2184:be any element of the set and
1873:
1865:
1849:
1841:
1724:) is a number of subsets with
1006:Stone's representation theorem
929:shows that the power set of a
296:
290:
267:
261:
131:
127:
121:
13:
1:
5186:History of mathematical logic
3375:Sheaves in Geometry and Logic
3215:
3208:to the given two-element set.
3084:, which takes morphisms from
565:
5111:Primitive recursive function
3292:Mac Lane & Moerdijk 1992
2909:
2820:
1754:elements (the empty subset),
1684:Relation to binomial theorem
1196:. First, the enumerated set
1143:
951:Cardinality of the continuum
388:{\displaystyle \varnothing }
7:
3391:Puntambekar, A. A. (2007).
3146:
2868:image morphism, so that if
1266:For the whole power set of
880:holds. Generally speaking,
506:and hence the power set of
10:
5921:
5755:von NeumannâBernaysâGödel
4175:SchröderâBernstein theorem
3902:Monadic predicate calculus
3561:Foundations of mathematics
3395:. Technical Publications.
3269:Category Theory in Context
919:Cantor's diagonal argument
408:{\displaystyle \emptyset }
364:, then all the subsets of
339:
29:
5819:
5782:
5694:
5584:
5556:One-to-one correspondence
5472:
5413:
5304:
5293:
5279:
5221:
5208:Philosophy of mathematics
5157:Automated theorem proving
5139:
5034:
4866:
4759:
4611:
4328:
4304:
4282:Von NeumannâBernaysâGödel
4227:
4121:
4025:
3923:
3914:
3841:
3776:
3682:
3604:
3521:
3123:can be understood as the
2373:. Given two multigraphs
2333:as the set of subsets of
1290:
1285:
1280:
1277:
943:one-to-one correspondence
576:is a finite set with the
102:
94:
84:
74:
42:
3173:
3131:between power sets, the
2555:Functors and quantifiers
2247:is sometimes denoted by
2194:; then the power set of
817:. Since each element in
214:is variously denoted as
4858:Self-verifying theories
4679:Tarski's axiomatization
3630:Tarski's undefinability
3625:incompleteness theorems
2438:complete directed graph
1728:elements in a set with
1299:{ }
1011:The power set of a set
956:The power set of a set
753:otherwise. Each subset
683:to the two-element set
631:denoting the power set
302:{\displaystyle \wp (S)}
5514:Constructible universe
5341:Constructibility (V=L)
5232:Mathematics portal
4843:Proof of impossibility
4491:propositional variable
3801:Propositional calculus
3137:existential quantifier
3078:
3005:
2973:
2936:
2852:
2794:
2408:of homomorphisms from
2241:less than or equal to
2231:The set of subsets of
2178:
2002:
1975:
1893:
1854:
831:under any function in
823:corresponds to either
409:
389:
303:
274:
148:
5737:Principia Mathematica
5571:Transfinite induction
5430:(i.e. set difference)
5101:Kolmogorov complexity
5054:Computably enumerable
4954:Model complete theory
4746:Principia Mathematica
3806:Propositional formula
3635:BanachâTarski paradox
3416:mathworld.wolfram.com
3079:
3011:to precomposition by
3006:
2974:
2937:
2853:
2795:
2269:is sometimes denoted
2179:
2147:The power set of the
2042:proceeds as follows:
2003:
1955:
1894:
1824:
1120:and the power set of
664:with the cardinality
410:
390:
304:
275:
149:
5811:Burali-Forti paradox
5566:Set-builder notation
5519:Continuum hypothesis
5459:Symmetric difference
5049:ChurchâTuring thesis
5036:Computability theory
4245:continuum hypothesis
3763:Square of opposition
3621:Gödel's completeness
3121:universal quantifier
3019:
2983:
2946:
2872:
2810:
2618:
2528:subobject classifier
2522:) and has an object
2292:might be denoted by
2168:
2158:For a non-empty set
2027:recursive definition
2011:Recursive definition
1918:
1800:
1722:binomial coefficient
1650:are enumerated with
1098:) is the set of all
1088:von Neumann ordinals
1082:" can be defined as
1023:symmetric difference
853:von Neumann ordinals
399:
379:
284:
253:
196:axiomatic set theory
109:
5772:TarskiâGrothendieck
5203:Mathematical object
5094:P versus NP problem
5059:Computable function
4853:Reverse mathematics
4779:Logical consequence
4656:primitive recursive
4651:elementary function
4424:Free/bound variable
4277:TarskiâGrothendieck
3796:Logical connectives
3726:Logical equivalence
3576:Logical consequence
3475:Power set Algorithm
3410:Weisstein, Eric W.
3377:, Springer-Verlag,
2341:algebraic structure
2210:and a power set of
2192:relative complement
1796:using the formula:
1086:(see, for example,
992:Boolean algebra is
868:is also denoted as
851:(see, for example,
843:. Since the number
677:is a function from
210:. The powerset of
39:
5905:Operations on sets
5361:Limitation of size
5001:Transfer principle
4964:Semantics of logic
4949:Categorical theory
4925:Non-standard model
4439:Logical connective
3566:Information theory
3515:Mathematical logic
3367:Mac Lane, Saunders
3115:and the theory of
3074:
3001:
2969:
2932:
2848:
2790:
2547:is required to be
2541:, in topos theory
2532:exponential object
2447:as the multigraph
2432:to the multigraph
2204:of a power set of
2174:
1998:
1889:
931:countably infinite
847:can be defined as
796:. In other words,
650:indicator function
405:
385:
299:
270:
208:axiom of power set
178:is the set of all
144:
103:Symbolic statement
37:
32:Powerset (company)
5892:
5891:
5801:Russell's paradox
5750:ZermeloâFraenkel
5651:Dedekind-infinite
5524:Diagonal argument
5423:Cartesian product
5287:Set (mathematics)
5239:
5238:
5171:Abstract category
4974:Theories of truth
4784:Rule of inference
4774:Natural deduction
4755:
4754:
4300:
4299:
4005:Cartesian product
3910:
3909:
3816:Many-valued logic
3791:Boolean functions
3674:Russell's paradox
3649:diagonal argument
3546:First-order logic
3402:978-81-8431-193-8
2979:takes a morphism
2952:
2912:
2831:
2823:
2436:definable as the
2352:algebraic lattice
2177:{\displaystyle e}
1990:
1881:
1573:injective mapping
1569:
1568:
1283:of binary digits
1040:distributive laws
804:to the power set
800:is equivalent or
157:
156:
16:(Redirected from
5912:
5874:Bertrand Russell
5864:John von Neumann
5849:Abraham Fraenkel
5844:Richard Dedekind
5806:Suslin's problem
5717:Cantor's theorem
5434:De Morgan's laws
5299:
5266:
5259:
5252:
5243:
5242:
5230:
5229:
5181:History of logic
5176:Category of sets
5069:Decision problem
4848:Ordinal analysis
4789:Sequent calculus
4687:Boolean algebras
4627:
4626:
4601:
4572:logical/constant
4326:
4325:
4312:
4235:ZermeloâFraenkel
3986:Set operations:
3921:
3920:
3858:
3689:
3688:
3669:LöwenheimâSkolem
3556:Formal semantics
3508:
3501:
3494:
3485:
3484:
3430:
3428:
3427:
3418:. Archived from
3406:
3387:
3362:
3352:
3349:Naive set theory
3338:
3315:. Universitext.
3309:Devlin, Keith J.
3295:
3289:
3283:
3282:
3264:
3258:
3255:Puntambekar 2007
3252:
3246:
3240:
3234:
3229:
3209:
3207:
3203:
3199:
3195:
3189:
3184:
3153:Cantor's theorem
3117:elementary topoi
3083:
3081:
3080:
3075:
3031:
3030:
3015:, so a function
3010:
3008:
3007:
3002:
2978:
2976:
2975:
2970:
2953:
2950:
2941:
2939:
2938:
2933:
2913:
2908:
2903:
2863:
2857:
2855:
2854:
2849:
2832:
2829:
2824:
2819:
2814:
2805:
2799:
2797:
2796:
2791:
2780:
2779:
2752:
2751:
2730:
2729:
2696:
2695:
2677:
2676:
2652:
2651:
2639:
2638:
2613:
2599:
2593:
2588:
2582:
2580:
2579:
2578:
2571:
2569:
2568:
2550:
2546:
2540:
2539:
2525:
2520:cartesian closed
2505:
2501:
2493:
2475:
2470:of vertices and
2469:
2460:
2450:
2446:
2435:
2431:
2425:
2419:
2413:
2407:
2401:
2384:
2378:
2365:
2361:
2338:
2332:
2318:
2312:
2311:
2305:
2296:
2291:
2285:
2273:
2268:
2262:
2251:
2246:
2236:
2221:
2215:
2209:
2199:
2189:
2183:
2181:
2180:
2175:
2163:
2138:
2132:
2131:
2110:
2109:
2100:
2099:
2093:
2078:
2065:
2059:
2058:
2052:
2041:
2035:
2034:
2020:
2007:
2005:
2004:
1999:
1997:
1996:
1995:
1982:
1974:
1969:
1951:
1950:
1938:
1934:
1933:
1913:
1908:
1898:
1896:
1895:
1890:
1888:
1887:
1886:
1877:
1876:
1868:
1861:
1853:
1852:
1844:
1838:
1820:
1816:
1815:
1795:
1793:
1783:
1779:
1773:
1769:
1763:
1759:
1753:
1749:
1739:
1735:
1731:
1727:
1719:
1703:
1697:
1690:binomial theorem
1679:
1673:
1667:
1661:
1657:
1653:
1649:
1643:
1637:
1631:
1622:
1616:
1615:
1609:
1593:
1587:
1581:
1580:
1565:
1557:
1549:
1544:
1525:
1517:
1509:
1504:
1489:
1481:
1473:
1468:
1453:
1445:
1437:
1432:
1427:{
1421:
1413:
1405:
1400:
1385:
1377:
1369:
1364:
1353:
1345:
1337:
1332:
1321:
1313:
1305:
1300:
1275:
1274:
1271:
1262:
1256:
1250:
1244:
1238:
1232:
1217:
1211:
1195:
1190:
1182:
1176:
1172:
1164:
1138:
1132:
1131:
1125:
1119:
1111:
1107:
1097:
1093:
1085:
1081:
1077:
1073:
1065:
1037:
1016:
983:
961:
945:with the set of
941:can be put in a
927:Cantor's theorem
913:
911:
905:
897:
891:
885:
879:
877:
871:
867:
866:
860:
850:
846:
842:
838:
834:
830:
826:
822:
816:
810:
809:
799:
795:
789:
785:
779:
775:
764:
758:
752:
748:
732:
726:
720:
714:
708:
702:
686:
682:
676:
671:
663:
657:
643:
637:
636:
630:
626:
624:
618:
617:
609:
603:
597:
591:
586:
575:
561:
509:
502:
485:
472:
459:
446:
437:
428:
419:or the null set)
414:
412:
411:
406:
394:
392:
391:
386:
374:
367:
363:
347:
335:
325:
319:
318:
313:. Any subset of
312:
308:
306:
305:
300:
279:
277:
276:
271:
260:
248:
237:
231:
226:
220:
219:
213:
193:
186:, including the
185:
177:
153:
151:
150:
145:
65:with respect to
47:
40:
36:
21:
5920:
5919:
5915:
5914:
5913:
5911:
5910:
5909:
5895:
5894:
5893:
5888:
5815:
5794:
5778:
5743:New Foundations
5690:
5580:
5499:Cardinal number
5482:
5468:
5409:
5300:
5291:
5275:
5270:
5240:
5235:
5224:
5217:
5162:Category theory
5152:Algebraic logic
5135:
5106:Lambda calculus
5044:Church encoding
5030:
5006:Truth predicate
4862:
4828:Complete theory
4751:
4620:
4616:
4612:
4607:
4599:
4319: and
4315:
4310:
4296:
4272:New Foundations
4240:axiom of choice
4223:
4185:Gödel numbering
4125: and
4117:
4021:
3906:
3856:
3837:
3786:Boolean algebra
3772:
3736:Equiconsistency
3701:Classical logic
3678:
3659:Halting problem
3647: and
3623: and
3611: and
3610:
3605:Theorems (
3600:
3517:
3512:
3437:
3425:
3423:
3403:
3385:
3343:Halmos, Paul R.
3327:
3317:Springer-Verlag
3304:
3299:
3298:
3290:
3286:
3279:
3265:
3261:
3253:
3249:
3241:
3237:
3230:
3223:
3218:
3213:
3212:
3205:
3201:
3197:
3191:
3187:
3185:
3181:
3176:
3149:
3113:category theory
3026:
3022:
3020:
3017:
3016:
2984:
2981:
2980:
2949:
2947:
2944:
2943:
2904:
2902:
2873:
2870:
2869:
2859:
2828:
2815:
2813:
2811:
2808:
2807:
2801:
2775:
2774:
2747:
2743:
2725:
2721:
2691:
2690:
2672:
2671:
2647:
2643:
2634:
2630:
2619:
2616:
2615:
2601:
2600:and a morphism
2591:
2590:
2584:
2576:
2575:
2574:
2573:
2566:
2565:
2564:
2557:
2548:
2542:
2535:
2534:
2523:
2503:
2499:
2477:
2471:
2465:
2456:
2448:
2442:
2433:
2427:
2421:
2415:
2409:
2403:
2389:
2380:
2374:
2363:
2359:
2334:
2328:
2325:
2309:
2308:
2307:
2299:
2294:
2293:
2287:
2279:
2271:
2270:
2264:
2256:
2249:
2248:
2242:
2232:
2229:
2217:
2211:
2205:
2195:
2185:
2169:
2166:
2165:
2159:
2129:
2128:
2107:
2106:
2097:
2096:
2095:
2080:
2070:
2069:Otherwise, let
2056:
2055:
2054:
2047:
2032:
2031:
2030:
2016:
2013:
1991:
1978:
1977:
1976:
1970:
1959:
1946:
1942:
1929:
1925:
1921:
1919:
1916:
1915:
1904:
1902:
1882:
1872:
1864:
1863:
1857:
1856:
1855:
1848:
1840:
1839:
1828:
1811:
1807:
1803:
1801:
1798:
1797:
1791:
1789:
1781:
1777:
1771:
1767:
1761:
1757:
1751:
1747:
1737:
1733:
1729:
1725:
1709:
1699:
1693:
1686:
1675:
1669:
1663:
1659:
1655:
1651:
1645:
1639:
1633:
1627:
1613:
1612:
1611:
1595:
1589:
1578:
1577:
1576:
1564:
1560:
1556:
1552:
1547:
1530:
1524:
1520:
1516:
1512:
1507:
1494:
1488:
1484:
1480:
1476:
1471:
1458:
1452:
1448:
1444:
1440:
1435:
1426:
1420:
1416:
1412:
1408:
1403:
1390:
1384:
1380:
1376:
1372:
1367:
1358:
1352:
1348:
1344:
1340:
1335:
1326:
1320:
1316:
1312:
1308:
1303:
1298:
1292:
1288:interpretation
1287:
1282:
1267:
1258:
1252:
1246:
1240:
1234:
1231:
1219:
1213:
1197:
1186:
1184:
1178:
1174:
1170:
1147:
1129:
1128:
1127:
1121:
1117:
1109:
1103:
1095:
1091:
1083:
1079:
1075:
1071:
1059:
1052:
1033:
1012:
986:Boolean algebra
979:
957:
939:natural numbers
907:
906:| = |
901:
899:
893:
887:
881:
875:
873:
869:
858:
857:
856:
848:
844:
840:
836:
832:
828:
824:
818:
807:
806:
805:
797:
791:
787:
781:
777:
774:
766:
760:
754:
750:
742:
734:
728:
722:
716:
710:
704:
696:
688:
684:
678:
667:
665:
659:
653:
634:
633:
632:
628:
615:
614:
613:
611:
605:
599:
593:
582:
580:
571:
568:
511:
507:
488:
475:
462:
449:
440:
431:
422:
400:
397:
396:
380:
377:
376:
372:
365:
349:
345:
342:
333:
316:
315:
314:
310:
285:
282:
281:
256:
254:
251:
250:
239:
229:
228:
217:
216:
215:
211:
191:
183:
175:
110:
107:
106:
70:
35:
28:
23:
22:
15:
12:
11:
5:
5918:
5908:
5907:
5890:
5889:
5887:
5886:
5881:
5879:Thoralf Skolem
5876:
5871:
5866:
5861:
5856:
5851:
5846:
5841:
5836:
5831:
5825:
5823:
5817:
5816:
5814:
5813:
5808:
5803:
5797:
5795:
5793:
5792:
5789:
5783:
5780:
5779:
5777:
5776:
5775:
5774:
5769:
5764:
5763:
5762:
5747:
5746:
5745:
5733:
5732:
5731:
5720:
5719:
5714:
5709:
5704:
5698:
5696:
5692:
5691:
5689:
5688:
5683:
5678:
5673:
5664:
5659:
5654:
5644:
5639:
5638:
5637:
5632:
5627:
5617:
5607:
5602:
5597:
5591:
5589:
5582:
5581:
5579:
5578:
5573:
5568:
5563:
5561:Ordinal number
5558:
5553:
5548:
5543:
5542:
5541:
5536:
5526:
5521:
5516:
5511:
5506:
5496:
5491:
5485:
5483:
5481:
5480:
5477:
5473:
5470:
5469:
5467:
5466:
5461:
5456:
5451:
5446:
5441:
5439:Disjoint union
5436:
5431:
5425:
5419:
5417:
5411:
5410:
5408:
5407:
5406:
5405:
5400:
5389:
5388:
5386:Martin's axiom
5383:
5378:
5373:
5368:
5363:
5358:
5353:
5351:Extensionality
5348:
5343:
5338:
5337:
5336:
5331:
5326:
5316:
5310:
5308:
5302:
5301:
5294:
5292:
5290:
5289:
5283:
5281:
5277:
5276:
5269:
5268:
5261:
5254:
5246:
5237:
5236:
5222:
5219:
5218:
5216:
5215:
5210:
5205:
5200:
5195:
5194:
5193:
5183:
5178:
5173:
5164:
5159:
5154:
5149:
5147:Abstract logic
5143:
5141:
5137:
5136:
5134:
5133:
5128:
5126:Turing machine
5123:
5118:
5113:
5108:
5103:
5098:
5097:
5096:
5091:
5086:
5081:
5076:
5066:
5064:Computable set
5061:
5056:
5051:
5046:
5040:
5038:
5032:
5031:
5029:
5028:
5023:
5018:
5013:
5008:
5003:
4998:
4993:
4992:
4991:
4986:
4981:
4971:
4966:
4961:
4959:Satisfiability
4956:
4951:
4946:
4945:
4944:
4934:
4933:
4932:
4922:
4921:
4920:
4915:
4910:
4905:
4900:
4890:
4889:
4888:
4883:
4876:Interpretation
4872:
4870:
4864:
4863:
4861:
4860:
4855:
4850:
4845:
4840:
4830:
4825:
4824:
4823:
4822:
4821:
4811:
4806:
4796:
4791:
4786:
4781:
4776:
4771:
4765:
4763:
4757:
4756:
4753:
4752:
4750:
4749:
4741:
4740:
4739:
4738:
4733:
4732:
4731:
4726:
4721:
4701:
4700:
4699:
4697:minimal axioms
4694:
4683:
4682:
4681:
4670:
4669:
4668:
4663:
4658:
4653:
4648:
4643:
4630:
4628:
4609:
4608:
4606:
4605:
4604:
4603:
4591:
4586:
4585:
4584:
4579:
4574:
4569:
4559:
4554:
4549:
4544:
4543:
4542:
4537:
4527:
4526:
4525:
4520:
4515:
4510:
4500:
4495:
4494:
4493:
4488:
4483:
4473:
4472:
4471:
4466:
4461:
4456:
4451:
4446:
4436:
4431:
4426:
4421:
4420:
4419:
4414:
4409:
4404:
4394:
4389:
4387:Formation rule
4384:
4379:
4378:
4377:
4372:
4362:
4361:
4360:
4350:
4345:
4340:
4335:
4329:
4323:
4306:Formal systems
4302:
4301:
4298:
4297:
4295:
4294:
4289:
4284:
4279:
4274:
4269:
4264:
4259:
4254:
4249:
4248:
4247:
4242:
4231:
4229:
4225:
4224:
4222:
4221:
4220:
4219:
4209:
4204:
4203:
4202:
4195:Large cardinal
4192:
4187:
4182:
4177:
4172:
4158:
4157:
4156:
4151:
4146:
4131:
4129:
4119:
4118:
4116:
4115:
4114:
4113:
4108:
4103:
4093:
4088:
4083:
4078:
4073:
4068:
4063:
4058:
4053:
4048:
4043:
4038:
4032:
4030:
4023:
4022:
4020:
4019:
4018:
4017:
4012:
4007:
4002:
3997:
3992:
3984:
3983:
3982:
3977:
3967:
3962:
3960:Extensionality
3957:
3955:Ordinal number
3952:
3942:
3937:
3936:
3935:
3924:
3918:
3912:
3911:
3908:
3907:
3905:
3904:
3899:
3894:
3889:
3884:
3879:
3874:
3873:
3872:
3862:
3861:
3860:
3847:
3845:
3839:
3838:
3836:
3835:
3834:
3833:
3828:
3823:
3813:
3808:
3803:
3798:
3793:
3788:
3782:
3780:
3774:
3773:
3771:
3770:
3765:
3760:
3755:
3750:
3745:
3740:
3739:
3738:
3728:
3723:
3718:
3713:
3708:
3703:
3697:
3695:
3686:
3680:
3679:
3677:
3676:
3671:
3666:
3661:
3656:
3651:
3639:Cantor's
3637:
3632:
3627:
3617:
3615:
3602:
3601:
3599:
3598:
3593:
3588:
3583:
3578:
3573:
3568:
3563:
3558:
3553:
3548:
3543:
3538:
3537:
3536:
3525:
3523:
3519:
3518:
3511:
3510:
3503:
3496:
3488:
3482:
3481:
3472:
3460:
3448:
3436:
3435:External links
3433:
3432:
3431:
3407:
3401:
3388:
3383:
3371:Moerdijk, Ieke
3363:
3339:
3325:
3303:
3300:
3297:
3296:
3284:
3278:978-0486809038
3277:
3267:Riehl, Emily.
3259:
3257:, pp. 1â2
3247:
3235:
3220:
3219:
3217:
3214:
3211:
3210:
3178:
3177:
3175:
3172:
3171:
3170:
3165:
3160:
3158:Family of sets
3155:
3148:
3145:
3073:
3070:
3067:
3064:
3061:
3058:
3055:
3052:
3049:
3046:
3043:
3040:
3037:
3034:
3029:
3025:
3000:
2997:
2994:
2991:
2988:
2968:
2965:
2962:
2959:
2956:
2931:
2928:
2925:
2922:
2919:
2916:
2911:
2907:
2901:
2898:
2895:
2892:
2889:
2886:
2883:
2880:
2877:
2847:
2844:
2841:
2838:
2835:
2827:
2822:
2818:
2789:
2786:
2783:
2778:
2773:
2770:
2767:
2764:
2761:
2758:
2755:
2750:
2746:
2742:
2739:
2736:
2733:
2728:
2724:
2720:
2717:
2714:
2711:
2708:
2705:
2702:
2699:
2694:
2689:
2686:
2683:
2680:
2675:
2670:
2667:
2664:
2661:
2658:
2655:
2650:
2646:
2642:
2637:
2633:
2629:
2626:
2623:
2556:
2553:
2518:(and moreover
2324:
2321:
2297:
2274:
2252:
2228:
2225:
2224:
2223:
2173:
2156:
2141:
2140:
2067:
2012:
2009:
1994:
1989:
1986:
1981:
1973:
1968:
1965:
1962:
1958:
1954:
1949:
1945:
1941:
1937:
1932:
1928:
1924:
1885:
1880:
1875:
1871:
1867:
1860:
1851:
1847:
1843:
1837:
1834:
1831:
1827:
1823:
1819:
1814:
1810:
1806:
1786:
1785:
1775:
1765:
1755:
1685:
1682:
1567:
1566:
1562:
1558:
1554:
1550:
1545:
1527:
1526:
1522:
1518:
1514:
1510:
1505:
1491:
1490:
1486:
1482:
1478:
1474:
1469:
1455:
1454:
1450:
1446:
1442:
1438:
1433:
1423:
1422:
1418:
1414:
1410:
1406:
1401:
1387:
1386:
1382:
1378:
1374:
1370:
1365:
1355:
1354:
1350:
1346:
1342:
1338:
1333:
1323:
1322:
1318:
1314:
1310:
1306:
1301:
1295:
1294:
1289:
1284:
1279:
1229:
1051:
1048:
916:
915:
770:
738:
692:
567:
564:
504:
503:
486:
473:
460:
447:
438:
429:
420:
404:
384:
375:(also denoted
341:
338:
329:family of sets
298:
295:
292:
289:
269:
266:
263:
259:
155:
154:
143:
140:
137:
133:
129:
126:
123:
120:
117:
114:
104:
100:
99:
96:
92:
91:
86:
82:
81:
76:
72:
71:
48:
26:
9:
6:
4:
3:
2:
5917:
5906:
5903:
5902:
5900:
5885:
5884:Ernst Zermelo
5882:
5880:
5877:
5875:
5872:
5870:
5869:Willard Quine
5867:
5865:
5862:
5860:
5857:
5855:
5852:
5850:
5847:
5845:
5842:
5840:
5837:
5835:
5832:
5830:
5827:
5826:
5824:
5822:
5821:Set theorists
5818:
5812:
5809:
5807:
5804:
5802:
5799:
5798:
5796:
5790:
5788:
5785:
5784:
5781:
5773:
5770:
5768:
5767:KripkeâPlatek
5765:
5761:
5758:
5757:
5756:
5753:
5752:
5751:
5748:
5744:
5741:
5740:
5739:
5738:
5734:
5730:
5727:
5726:
5725:
5722:
5721:
5718:
5715:
5713:
5710:
5708:
5705:
5703:
5700:
5699:
5697:
5693:
5687:
5684:
5682:
5679:
5677:
5674:
5672:
5670:
5665:
5663:
5660:
5658:
5655:
5652:
5648:
5645:
5643:
5640:
5636:
5633:
5631:
5628:
5626:
5623:
5622:
5621:
5618:
5615:
5611:
5608:
5606:
5603:
5601:
5598:
5596:
5593:
5592:
5590:
5587:
5583:
5577:
5574:
5572:
5569:
5567:
5564:
5562:
5559:
5557:
5554:
5552:
5549:
5547:
5544:
5540:
5537:
5535:
5532:
5531:
5530:
5527:
5525:
5522:
5520:
5517:
5515:
5512:
5510:
5507:
5504:
5500:
5497:
5495:
5492:
5490:
5487:
5486:
5484:
5478:
5475:
5474:
5471:
5465:
5462:
5460:
5457:
5455:
5452:
5450:
5447:
5445:
5442:
5440:
5437:
5435:
5432:
5429:
5426:
5424:
5421:
5420:
5418:
5416:
5412:
5404:
5403:specification
5401:
5399:
5396:
5395:
5394:
5391:
5390:
5387:
5384:
5382:
5379:
5377:
5374:
5372:
5369:
5367:
5364:
5362:
5359:
5357:
5354:
5352:
5349:
5347:
5344:
5342:
5339:
5335:
5332:
5330:
5327:
5325:
5322:
5321:
5320:
5317:
5315:
5312:
5311:
5309:
5307:
5303:
5298:
5288:
5285:
5284:
5282:
5278:
5274:
5267:
5262:
5260:
5255:
5253:
5248:
5247:
5244:
5234:
5233:
5228:
5220:
5214:
5211:
5209:
5206:
5204:
5201:
5199:
5196:
5192:
5189:
5188:
5187:
5184:
5182:
5179:
5177:
5174:
5172:
5168:
5165:
5163:
5160:
5158:
5155:
5153:
5150:
5148:
5145:
5144:
5142:
5138:
5132:
5129:
5127:
5124:
5122:
5121:Recursive set
5119:
5117:
5114:
5112:
5109:
5107:
5104:
5102:
5099:
5095:
5092:
5090:
5087:
5085:
5082:
5080:
5077:
5075:
5072:
5071:
5070:
5067:
5065:
5062:
5060:
5057:
5055:
5052:
5050:
5047:
5045:
5042:
5041:
5039:
5037:
5033:
5027:
5024:
5022:
5019:
5017:
5014:
5012:
5009:
5007:
5004:
5002:
4999:
4997:
4994:
4990:
4987:
4985:
4982:
4980:
4977:
4976:
4975:
4972:
4970:
4967:
4965:
4962:
4960:
4957:
4955:
4952:
4950:
4947:
4943:
4940:
4939:
4938:
4935:
4931:
4930:of arithmetic
4928:
4927:
4926:
4923:
4919:
4916:
4914:
4911:
4909:
4906:
4904:
4901:
4899:
4896:
4895:
4894:
4891:
4887:
4884:
4882:
4879:
4878:
4877:
4874:
4873:
4871:
4869:
4865:
4859:
4856:
4854:
4851:
4849:
4846:
4844:
4841:
4838:
4837:from ZFC
4834:
4831:
4829:
4826:
4820:
4817:
4816:
4815:
4812:
4810:
4807:
4805:
4802:
4801:
4800:
4797:
4795:
4792:
4790:
4787:
4785:
4782:
4780:
4777:
4775:
4772:
4770:
4767:
4766:
4764:
4762:
4758:
4748:
4747:
4743:
4742:
4737:
4736:non-Euclidean
4734:
4730:
4727:
4725:
4722:
4720:
4719:
4715:
4714:
4712:
4709:
4708:
4706:
4702:
4698:
4695:
4693:
4690:
4689:
4688:
4684:
4680:
4677:
4676:
4675:
4671:
4667:
4664:
4662:
4659:
4657:
4654:
4652:
4649:
4647:
4644:
4642:
4639:
4638:
4636:
4632:
4631:
4629:
4624:
4618:
4613:Example
4610:
4602:
4597:
4596:
4595:
4592:
4590:
4587:
4583:
4580:
4578:
4575:
4573:
4570:
4568:
4565:
4564:
4563:
4560:
4558:
4555:
4553:
4550:
4548:
4545:
4541:
4538:
4536:
4533:
4532:
4531:
4528:
4524:
4521:
4519:
4516:
4514:
4511:
4509:
4506:
4505:
4504:
4501:
4499:
4496:
4492:
4489:
4487:
4484:
4482:
4479:
4478:
4477:
4474:
4470:
4467:
4465:
4462:
4460:
4457:
4455:
4452:
4450:
4447:
4445:
4442:
4441:
4440:
4437:
4435:
4432:
4430:
4427:
4425:
4422:
4418:
4415:
4413:
4410:
4408:
4405:
4403:
4400:
4399:
4398:
4395:
4393:
4390:
4388:
4385:
4383:
4380:
4376:
4373:
4371:
4370:by definition
4368:
4367:
4366:
4363:
4359:
4356:
4355:
4354:
4351:
4349:
4346:
4344:
4341:
4339:
4336:
4334:
4331:
4330:
4327:
4324:
4322:
4318:
4313:
4307:
4303:
4293:
4290:
4288:
4285:
4283:
4280:
4278:
4275:
4273:
4270:
4268:
4265:
4263:
4260:
4258:
4257:KripkeâPlatek
4255:
4253:
4250:
4246:
4243:
4241:
4238:
4237:
4236:
4233:
4232:
4230:
4226:
4218:
4215:
4214:
4213:
4210:
4208:
4205:
4201:
4198:
4197:
4196:
4193:
4191:
4188:
4186:
4183:
4181:
4178:
4176:
4173:
4170:
4166:
4162:
4159:
4155:
4152:
4150:
4147:
4145:
4142:
4141:
4140:
4136:
4133:
4132:
4130:
4128:
4124:
4120:
4112:
4109:
4107:
4104:
4102:
4101:constructible
4099:
4098:
4097:
4094:
4092:
4089:
4087:
4084:
4082:
4079:
4077:
4074:
4072:
4069:
4067:
4064:
4062:
4059:
4057:
4054:
4052:
4049:
4047:
4044:
4042:
4039:
4037:
4034:
4033:
4031:
4029:
4024:
4016:
4013:
4011:
4008:
4006:
4003:
4001:
3998:
3996:
3993:
3991:
3988:
3987:
3985:
3981:
3978:
3976:
3973:
3972:
3971:
3968:
3966:
3963:
3961:
3958:
3956:
3953:
3951:
3947:
3943:
3941:
3938:
3934:
3931:
3930:
3929:
3926:
3925:
3922:
3919:
3917:
3913:
3903:
3900:
3898:
3895:
3893:
3890:
3888:
3885:
3883:
3880:
3878:
3875:
3871:
3868:
3867:
3866:
3863:
3859:
3854:
3853:
3852:
3849:
3848:
3846:
3844:
3840:
3832:
3829:
3827:
3824:
3822:
3819:
3818:
3817:
3814:
3812:
3809:
3807:
3804:
3802:
3799:
3797:
3794:
3792:
3789:
3787:
3784:
3783:
3781:
3779:
3778:Propositional
3775:
3769:
3766:
3764:
3761:
3759:
3756:
3754:
3751:
3749:
3746:
3744:
3741:
3737:
3734:
3733:
3732:
3729:
3727:
3724:
3722:
3719:
3717:
3714:
3712:
3709:
3707:
3706:Logical truth
3704:
3702:
3699:
3698:
3696:
3694:
3690:
3687:
3685:
3681:
3675:
3672:
3670:
3667:
3665:
3662:
3660:
3657:
3655:
3652:
3650:
3646:
3642:
3638:
3636:
3633:
3631:
3628:
3626:
3622:
3619:
3618:
3616:
3614:
3608:
3603:
3597:
3594:
3592:
3589:
3587:
3584:
3582:
3579:
3577:
3574:
3572:
3569:
3567:
3564:
3562:
3559:
3557:
3554:
3552:
3549:
3547:
3544:
3542:
3539:
3535:
3532:
3531:
3530:
3527:
3526:
3524:
3520:
3516:
3509:
3504:
3502:
3497:
3495:
3490:
3489:
3486:
3480:
3476:
3473:
3471:
3469:
3464:
3461:
3459:
3457:
3452:
3449:
3446:
3442:
3439:
3438:
3422:on 2023-04-06
3421:
3417:
3413:
3408:
3404:
3398:
3394:
3389:
3386:
3384:0-387-97710-4
3380:
3376:
3372:
3368:
3364:
3360:
3356:
3351:
3350:
3344:
3340:
3336:
3332:
3328:
3326:0-387-90441-7
3322:
3318:
3314:
3310:
3306:
3305:
3293:
3288:
3280:
3274:
3270:
3263:
3256:
3251:
3244:
3239:
3233:
3228:
3226:
3221:
3194:
3186:The notation
3183:
3179:
3169:
3166:
3164:
3163:Field of sets
3161:
3159:
3156:
3154:
3151:
3150:
3144:
3142:
3138:
3134:
3133:inverse image
3130:
3126:
3125:right adjoint
3122:
3118:
3114:
3109:
3107:
3103:
3099:
3095:
3091:
3087:
3068:
3065:
3062:
3056:
3047:
3044:
3041:
3035:
3032:
3027:
3023:
3014:
2998:
2992:
2989:
2986:
2963:
2960:
2957:
2929:
2926:
2920:
2914:
2899:
2896:
2893:
2890:
2887:
2881:
2875:
2867:
2862:
2842:
2839:
2836:
2825:
2804:
2784:
2771:
2765:
2762:
2759:
2756:
2748:
2744:
2737:
2734:
2726:
2722:
2715:
2709:
2703:
2697:
2687:
2681:
2668:
2662:
2659:
2656:
2653:
2648:
2644:
2640:
2635:
2631:
2624:
2621:
2612:
2608:
2604:
2597:
2587:
2562:
2552:
2545:
2538:
2533:
2529:
2521:
2517:
2513:
2509:
2497:
2492:
2488:
2484:
2480:
2474:
2468:
2462:
2459:
2454:
2451:, called the
2445:
2439:
2430:
2424:
2418:
2412:
2406:
2400:
2396:
2392:
2388:
2383:
2377:
2372:
2367:
2366:in this way.
2355:
2353:
2349:
2344:
2342:
2337:
2331:
2320:
2316:
2303:
2290:
2283:
2278:
2267:
2260:
2255:
2245:
2240:
2235:
2220:
2214:
2208:
2203:
2198:
2193:
2188:
2171:
2162:
2157:
2154:
2150:
2146:
2145:
2144:
2136:
2126:
2122:
2118:
2114:
2104:
2091:
2087:
2083:
2077:
2073:
2068:
2063:
2050:
2045:
2044:
2043:
2039:
2028:
2024:
2019:
2008:
1987:
1984:
1971:
1966:
1963:
1960:
1956:
1952:
1947:
1943:
1939:
1935:
1930:
1926:
1922:
1912:
1907:
1899:
1878:
1869:
1845:
1835:
1832:
1829:
1825:
1821:
1817:
1812:
1808:
1804:
1776:
1770:subsets with
1766:
1760:subsets with
1756:
1746:
1745:
1744:
1741:
1723:
1720:(also called
1717:
1713:
1708:, denoted as
1707:
1702:
1696:
1691:
1681:
1678:
1672:
1666:
1648:
1642:
1636:
1630:
1624:
1620:
1607:
1603:
1599:
1592:
1585:
1574:
1559:
1551:
1546:
1542:
1538:
1534:
1529:
1528:
1519:
1511:
1506:
1502:
1498:
1493:
1492:
1483:
1475:
1470:
1466:
1462:
1457:
1456:
1447:
1439:
1434:
1430:
1425:
1424:
1415:
1407:
1402:
1398:
1394:
1389:
1388:
1379:
1371:
1366:
1362:
1357:
1356:
1347:
1339:
1334:
1330:
1325:
1324:
1315:
1307:
1302:
1297:
1296:
1276:
1273:
1270:
1264:
1261:
1255:
1249:
1243:
1237:
1227:
1223:
1216:
1209:
1205:
1201:
1194:
1189:
1181:
1168:
1165:, to get the
1162:
1158:
1154:
1150:
1145:
1140:
1136:
1124:
1115:
1106:
1101:
1089:
1069:
1064:
1063:
1057:
1047:
1045:
1041:
1036:
1031:
1028:
1024:
1020:
1019:abelian group
1015:
1009:
1007:
1003:
999:
995:
991:
987:
982:
977:
973:
969:
965:
960:
954:
952:
948:
944:
940:
936:
932:
928:
924:
920:
910:
904:
896:
890:
884:
864:
854:
821:
814:
803:
794:
784:
773:
769:
763:
757:
746:
741:
737:
731:
725:
719:
713:
707:
700:
695:
691:
687:, denoted as
681:
675:
670:
662:
656:
651:
647:
646:
645:
641:
622:
608:
602:
596:
590:
585:
579:
574:
563:
559:
555:
551:
547:
543:
539:
535:
531:
527:
523:
519:
515:
500:
496:
492:
487:
483:
479:
474:
470:
466:
461:
457:
453:
448:
444:
439:
435:
430:
426:
421:
418:
382:
371:
370:
369:
361:
357:
353:
337:
331:
330:
323:
293:
264:
246:
242:
235:
224:
209:
205:
201:
197:
189:
181:
174:
170:
166:
162:
141:
138:
135:
124:
118:
115:
112:
105:
101:
97:
93:
90:
87:
83:
80:
79:Set operation
77:
73:
68:
64:
60:
56:
52:
46:
41:
33:
19:
5834:Georg Cantor
5829:Paul Bernays
5760:MorseâKelley
5735:
5668:
5667:Subset
5614:hereditarily
5576:Venn diagram
5534:ordered pair
5453:
5449:Intersection
5393:Axiom schema
5223:
5021:Ultraproduct
4868:Model theory
4833:Independence
4769:Formal proof
4761:Proof theory
4744:
4717:
4674:real numbers
4646:second-order
4557:Substitution
4434:Metalanguage
4375:conservative
4348:Axiom schema
4292:Constructive
4262:MorseâKelley
4228:Set theories
4207:Aleph number
4200:inaccessible
4106:Grothendieck
4009:
3990:intersection
3877:Higher-order
3865:Second-order
3811:Truth tables
3768:Venn diagram
3551:Formal proof
3467:
3463:Power object
3455:
3424:. Retrieved
3420:the original
3415:
3392:
3374:
3348:
3312:
3302:Bibliography
3294:, p. 58
3287:
3268:
3262:
3250:
3245:, p. 50
3238:
3192:
3182:
3141:left adjoint
3110:
3105:
3101:
3097:
3093:
3089:
3085:
3012:
2865:
2860:
2802:
2610:
2606:
2602:
2595:
2585:
2581:: Set â Set
2558:
2543:
2536:
2490:
2486:
2482:
2478:
2472:
2466:
2463:
2457:
2453:power object
2452:
2443:
2428:
2422:
2416:
2410:
2404:
2398:
2394:
2390:
2387:homomorphism
2381:
2375:
2368:
2356:
2345:
2343:or algebra.
2335:
2329:
2326:
2323:Power object
2314:
2301:
2288:
2281:
2276:
2265:
2258:
2253:
2243:
2233:
2230:
2218:
2212:
2206:
2196:
2186:
2160:
2142:
2134:
2124:
2120:
2116:
2112:
2102:
2089:
2085:
2081:
2075:
2071:
2061:
2048:
2037:
2017:
2014:
1910:
1905:
1900:
1787:
1780:subset with
1750:subset with
1742:
1715:
1711:
1706:combinations
1700:
1694:
1687:
1676:
1670:
1664:
1646:
1640:
1634:
1628:
1625:
1618:
1605:
1601:
1597:
1590:
1583:
1570:
1540:
1536:
1532:
1500:
1496:
1464:
1460:
1428:
1396:
1392:
1360:
1328:
1268:
1265:
1259:
1253:
1247:
1241:
1235:
1225:
1221:
1214:
1207:
1203:
1199:
1192:
1187:
1179:
1160:
1156:
1152:
1148:
1141:
1134:
1122:
1104:
1061:
1053:
1044:Boolean ring
1034:
1013:
1010:
997:
989:
980:
968:intersection
958:
955:
947:real numbers
917:
908:
902:
894:
888:
882:
872:. Obviously
862:
819:
812:
792:
782:
771:
767:
761:
755:
744:
739:
735:
729:
723:
717:
711:
705:
698:
693:
689:
679:
673:
668:
660:
654:
639:
620:
606:
600:
594:
588:
583:
572:
569:
557:
553:
549:
545:
541:
537:
533:
529:
525:
521:
517:
513:
505:
498:
494:
490:
481:
477:
468:
464:
455:
451:
442:
433:
424:
359:
355:
351:
343:
327:
326:is called a
321:
244:
240:
233:
222:
168:
164:
158:
58:
54:
50:
5859:Thomas Jech
5702:Alternative
5681:Uncountable
5635:Ultrafilter
5494:Cardinality
5398:replacement
5346:Determinacy
5131:Type theory
5079:undecidable
5011:Truth value
4898:equivalence
4577:non-logical
4190:Enumeration
4180:Isomorphism
4127:cardinality
4111:Von Neumann
4076:Ultrafilter
4041:Uncountable
3975:equivalence
3892:Quantifiers
3882:Fixed-point
3851:First-order
3731:Consistency
3716:Proposition
3693:Traditional
3664:Lindström's
3654:Compactness
3596:Type theory
3541:Cardinality
3412:"Power Set"
3243:Devlin 1979
3168:Combination
2570:: Set â Set
2526:, called a
2371:multigraphs
2239:cardinality
2143:In words:
1778:C(3, 3) = 1
1768:C(3, 2) = 3
1758:C(3, 1) = 3
1748:C(3, 0) = 1
1293:equivalent
1167:isomorphism
1146:, in which
1114:shown above
1027:commutative
935:uncountably
923:cardinality
727:belongs to
715:or not; If
709:belongs to
578:cardinality
348:is the set
194:itself. In
161:mathematics
5854:Kurt Gödel
5839:Paul Cohen
5676:Transitive
5444:Identities
5428:Complement
5415:Operations
5376:Regularity
5314:Adjunction
5273:Set theory
4942:elementary
4635:arithmetic
4503:Quantifier
4481:functional
4353:Expression
4071:Transitive
4015:identities
4000:complement
3933:hereditary
3916:Set theory
3445:PlanetMath
3426:2020-09-05
3359:0087.04403
3335:0407.04003
3216:References
3100:, through
2360:{0, 1} = 2
2064:) = { {} }
2023:finite set
1740:elements.
1272:, we get:
1056:set theory
1002:subalgebra
994:isomorphic
972:complement
878:| = 2
625:| = 2
566:Properties
204:postulated
89:Set theory
5787:Paradoxes
5707:Axiomatic
5686:Universal
5662:Singleton
5657:Recursive
5600:Countable
5595:Amorphous
5454:Power set
5371:Power set
5329:dependent
5324:countable
5213:Supertask
5116:Recursion
5074:decidable
4908:saturated
4886:of models
4809:deductive
4804:axiomatic
4724:Hilbert's
4711:Euclidean
4692:canonical
4615:axiomatic
4547:Signature
4476:Predicate
4365:Extension
4287:Ackermann
4212:Operation
4091:Universal
4081:Recursive
4056:Singleton
4051:Inhabited
4036:Countable
4026:Types of
4010:power set
3980:partition
3897:Predicate
3843:Predicate
3758:Syllogism
3748:Soundness
3721:Inference
3711:Tautology
3613:paradoxes
3451:Power set
3441:Power set
3232:Weisstein
3054:→
3028:∗
2996:→
2958:−
2910:¯
2894:⊆
2837:−
2826:≅
2821:¯
2772:∈
2669:∈
2153:singleton
2149:empty set
2123:} :
2088:∖ {
2025:, then a
1957:∑
1909:| =
1826:∑
1495:{
1463:,
1359:{
1191:| =
1100:functions
1068:functions
1017:forms an
976:ÎŁ-algebra
802:bijective
672:| =
658:of a set
587:| =
417:empty set
403:∅
383:∅
288:℘
188:empty set
165:power set
139:⊆
132:⟺
116:∈
95:Statement
67:inclusion
38:Power set
5899:Category
5791:Problems
5695:Theories
5671:Superset
5647:Infinite
5476:Concepts
5356:Infinity
5280:Overview
5198:Logicism
5191:timeline
5167:Concrete
5026:Validity
4996:T-schema
4989:Kripke's
4984:Tarski's
4979:semantic
4969:Strength
4918:submodel
4913:spectrum
4881:function
4729:Tarski's
4718:Elements
4705:geometry
4661:Robinson
4582:variable
4567:function
4540:spectrum
4530:Sentence
4486:variable
4429:Language
4382:Relation
4343:Automata
4333:Alphabet
4317:language
4171:-jection
4149:codomain
4135:Function
4096:Universe
4066:Infinite
3970:Relation
3753:Validity
3743:Argument
3641:theorem,
3373:(1992),
3345:(1960).
3311:(1979).
3147:See also
2514:that is
2512:category
2496:presheaf
2485: :
2393: :
2222:element.
1571:Such an
1281:Sequence
998:infinite
701:â {0, 1}
697: :
169:powerset
18:Powerset
5729:General
5724:Zermelo
5630:subbase
5612: (
5551:Forcing
5529:Element
5501: (
5479:Methods
5366:Pairing
5140:Related
4937:Diagram
4835: (
4814:Hilbert
4799:Systems
4794:Theorem
4672:of the
4617:systems
4397:Formula
4392:Grammar
4308: (
4252:General
3965:Forcing
3950:Element
3870:Monadic
3645:paradox
3586:Theorem
3522:General
3465:at the
3453:at the
3139:is the
3129:functor
2864:to the
2561:functor
2348:lattice
2094:; then
2053:, then
1604:, 2), (
1600:, 1), (
1548:1, 1, 1
1508:1, 1, 0
1472:1, 0, 1
1436:1, 0, 0
1404:0, 1, 1
1368:0, 1, 0
1336:0, 0, 1
1304:0, 0, 0
1291:Decimal
1278:Subset
1228:} = 011
1206:, 2), (
1202:, 1), (
1173:, with
1094:(i.e.,
1078:. As "
974:, is a
933:set is
855:), the
733:, then
340:Example
206:by the
180:subsets
171:) of a
63:ordered
5620:Filter
5610:Finite
5546:Family
5489:Almost
5334:global
5319:Choice
5306:Axioms
4903:finite
4666:Skolem
4619:
4594:Theory
4562:Symbol
4552:String
4535:atomic
4412:ground
4407:closed
4402:atomic
4358:ground
4321:syntax
4217:binary
4144:domain
4061:Finite
3826:finite
3684:Logics
3643:
3591:Theory
3399:
3381:
3357:
3333:
3323:
3275:
3198:{0, 1}
3119:, the
2516:closed
2164:, let
1903:|
1794:|
1790:|
1658:, and
1644:, and
1608:, 3) }
1286:Binary
1210:, 3) }
1185:|
1110:{0, 1}
1096:{0, 1}
1084:{0, 1}
1030:monoid
990:finite
912:|
900:|
874:|
849:{0, 1}
837:{0, 1}
833:{0, 1}
798:{0, 1}
788:{0, 1}
776:, and
749:, and
685:{0, 1}
666:|
612:|
581:|
512:{{}, {
415:, the
163:, the
5712:Naive
5642:Fuzzy
5605:Empty
5588:types
5539:tuple
5509:Class
5503:large
5464:Union
5381:Union
4893:Model
4641:Peano
4498:Proof
4338:Arity
4267:Naive
4154:image
4086:Fuzzy
4046:Empty
3995:union
3940:Class
3581:Model
3571:Lemma
3529:Axiom
3174:Notes
3127:of a
2510:as a
2508:topos
2202:union
2200:is a
2151:is a
2115:) âȘ {
2021:is a
1575:from
1171:2 â 1
1144:above
1112:. As
1102:from
1070:from
978:over
964:union
949:(see
778:{0,1}
747:) = 1
332:over
309:, or
85:Field
5625:base
5016:Type
4819:list
4623:list
4600:list
4589:Term
4523:rank
4417:open
4311:list
4123:Maps
4028:sets
3887:Free
3857:list
3607:list
3534:list
3397:ISBN
3379:ISBN
3321:ISBN
3273:ISBN
3104:via
2572:and
2385:, a
2379:and
2275:<
2190:its
2105:) =
2079:and
2051:= {}
1688:The
1563:(10)
1523:(10)
1487:(10)
1451:(10)
1419:(10)
1383:(10)
1351:(10)
1319:(10)
970:and
898:and
548:}, {
540:}, {
532:}, {
524:}, {
520:}, {
516:}, {
368:are
190:and
167:(or
75:Type
5586:Set
4703:of
4685:of
4633:of
4165:Sur
4139:Map
3946:Ur-
3928:Set
3479:C++
3477:in
3470:Lab
3458:Lab
3443:at
3355:Zbl
3331:Zbl
3111:In
3108:.
3096:to
3088:to
2866:pre
2830:Set
2589:to
2455:of
2414:to
2306:or
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2119:âȘ {
2046:If
2029:of
2015:If
1596:{ (
1555:(2)
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1198:{ (
1183:or
1151:= {
1108:to
1090:),
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1008:).
953:).
892:to
839:is
827:or
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759:of
721:in
648:An
610:is
598:is
570:If
510:is
395:or
344:If
200:ZFC
182:of
173:set
159:In
61:}
5901::
5089:NP
4713::
4707::
4637::
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4169:Bi
4161:In
3414:.
3369:;
3329:.
3319:.
3271:.
3224:^
3143:.
2609:â
2605::
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2551:.
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1914::
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1654:,
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772:A
768:I
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69:.
59:z
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34:.
20:)
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