4049:
4085:
3597:
8664:
8358:
42:
6017:
4044:{\displaystyle {\begin{aligned}\left|A_{1}\cup A_{2}\cup A_{3}\cup \ldots \cup A_{n}\right|=&\left(\left|A_{1}\right|+\left|A_{2}\right|+\left|A_{3}\right|+\ldots \left|A_{n}\right|\right)\\&{}-\left(\left|A_{1}\cap A_{2}\right|+\left|A_{1}\cap A_{3}\right|+\ldots \left|A_{n-1}\cap A_{n}\right|\right)\\&{}+\ldots \\&{}+\left(-1\right)^{n-1}\left(\left|A_{1}\cap A_{2}\cap A_{3}\cap \ldots \cap A_{n}\right|\right).\end{aligned}}}
5941:
8676:
31:
988:
1087:
3491:
2798:
2776:
2745:
2710:
2572:
4142:, or some equivalent name, it is common, especially where the number of terms involved is finite, to regard the object in question (which is in fact a class) as defined by the enumeration of its terms, and as consisting possibly of a single term, which in that case
1665:
237:
is a set with an endless list of elements. To describe an infinite set in roster notation, an ellipsis is placed at the end of the list, or at both ends, to indicate that the list continues forever. For example, the set of
381:
3070:
3498:
The inclusion–exclusion principle is a technique for counting the elements in a union of two finite sets in terms of the sizes of the two sets and their intersection. It can be expressed symbolically as
1530:
2705:
to distinguish it from the relative complement below. Example: If the universal set is taken to be the set of integers, then the complement of the set of even integers is the set of odd integers.
3602:
2699:
3590:
1382:
5442:
1218:
There are sets of such mathematical importance, to which mathematicians refer so frequently, that they have acquired special names and notational conventions to identify them.
1935:
6737:
5415:
2256:
2234:
2200:
2163:
1857:
1835:
1757:
1735:
1576:
1554:
1432:
1410:
1314:
1292:
1267:
1241:
1213:
1188:
1163:
1138:
1113:
1793:
653:
7412:
6474:
71:
of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets. A set may have a finite number of elements or be an
3494:
The inclusion-exclusion principle for two finite sets states that the size of their union is the sum of the sizes of the sets minus the size of their intersection.
188:
in 1908. In a set, all that matters is whether each element is in it or not, so the ordering of the elements in roster notation is irrelevant (in contrast, in a
1593:
4754:
Sets: Naïve, Axiomatic and
Applied: A Basic Compendium with Exercises for Use in Set Theory for Non Logicians, Working and Teaching Mathematicians and Students
325:
8394:
7495:
6636:
4289:
In subsequent efforts to resolve these paradoxes since the time of the original formulation of naïve set theory, the properties of sets have been defined by
1054:
sets (possibly all or none), there is a zone for the elements that belong to all the selected sets and none of the others. For example, if the sets are
8631:
2463:
is uncountable. Moreover, the power set is always strictly "bigger" than the original set, in the sense that any attempt to pair up the elements of
7809:
1015:
is a graphical representation of a collection of sets; each set is depicted as a planar region enclosed by a loop, with its elements inside. If
4228:
4115:
A set is a gathering together into a whole of definite, distinct objects of our perception or our thought—which are called elements of the set.
7967:
5785:
5717:
5690:
5663:
5636:
5593:
5566:
5393:
5366:
5339:
5312:
5285:
5258:
5231:
5174:
5147:
5071:
5001:
4952:
4900:
4864:
4816:
4789:
4762:
4735:
4695:
4668:
4641:
4614:
4558:
4474:
4447:
4420:
6755:
4310:
7822:
7145:
6163:
5983:
2300:
The continuum hypothesis, formulated by Georg Cantor in 1878, is the statement that there is no set with cardinality strictly between the
7407:
6491:
7827:
7817:
7554:
6760:
7305:
6751:
3485:
3011:
2176:
Some infinite cardinalities are greater than others. Arguably one of the most significant results from set theory is that the set of
316:
Set-builder notation specifies a set as a selection from a larger set, determined by a condition on the elements. For example, a set
8472:
7963:
5923:
5900:
5877:
5851:
4531:
2309:
4212:
can be distinguished by the number of occurrences of an element; e.g. and represent different multisets, while and are equal.
3502:
8387:
8060:
7804:
6629:
8510:
7365:
7058:
6799:
6469:
2313:
102:
8641:
8321:
8023:
7786:
7781:
7606:
7027:
6711:
4313:
however, it is not possible to use first-order logic to prove any such particular axiomatic set theory is free from paradox.
211:
For sets with many elements, especially those following an implicit pattern, the list of members can be abbreviated using an
1441:
1908:
Sets of positive or negative numbers are sometimes denoted by superscript plus and minus signs, respectively. For example,
8701:
8316:
8099:
8016:
7729:
7660:
7537:
6779:
6243:
6122:
5760:
8654:
7387:
6486:
8241:
8067:
7753:
6986:
4084:
7392:
6479:
2647:
8636:
8462:
8380:
7724:
7463:
6721:
6622:
6117:
6080:
2169:
is infinite. In fact, all the special sets of numbers mentioned in the section above are infinite. Infinite sets have
86:
Sets are uniquely characterized by their elements; this means that two sets that have precisely the same elements are
8119:
8114:
1905:
Each of the above sets of numbers has an infinite number of elements. Each is a subset of the sets listed below it.
5410:
4301:. The purpose of the axioms is to provide a basic framework from which to deduce the truth or falsity of particular
8696:
8568:
8404:
8048:
7638:
7032:
7000:
6691:
5915:
106:
16:
This article is about what mathematicians call "intuitive" or "naive" set theory. For a more detailed account, see
6765:
2236:); some authors use "countable" to mean "countably infinite". Sets with cardinality strictly greater than that of
2028:(or a one-to-one correspondence) if the function is both injective and surjective — in this case, each element of
1323:
8535:
8338:
8287:
8184:
7682:
7643:
7120:
6168:
6060:
6048:
6043:
4302:
4077:
2855:
1806:
8179:
6794:
5953:
8109:
7648:
7500:
7483:
7206:
6686:
5976:
5843:
3264:
are finite. When one or both are infinite, multiplication of cardinal numbers is defined to make this true.)
5802:
4388:
By an 'aggregate' (Menge) we are to understand any collection into a whole (Zusammenfassung zu einem Ganzen)
2180:
has greater cardinality than the set of natural numbers. Sets with cardinality less than or equal to that of
8515:
8011:
7988:
7949:
7835:
7776:
7422:
7342:
7186:
7130:
6743:
6588:
6506:
6381:
6333:
6147:
6070:
2941:
2612:
2280:
672:
91:
80:
8616:
8601:
8558:
8520:
8425:
8301:
8028:
8006:
7973:
7866:
7712:
7697:
7670:
7621:
7505:
7440:
7265:
7231:
7226:
7100:
6931:
6908:
6540:
6421:
6233:
6053:
4328:
4166:
3459:
3307:
3271:
with symmetric difference as the addition of the ring and intersection as the multiplication of the ring.
2593:
2210:
1974:
1947:
1796:
768:
445:
435:
422:
87:
8578:
8553:
8231:
8084:
7876:
7594:
7330:
7236:
7095:
7080:
6961:
6936:
6456:
6370:
6290:
6270:
6248:
5436:
4358:
3300:
701:
mean different things; Halmos draws the analogy that a box containing a hat is not the same as the hat.
472:
219:'. For instance, the set of the first thousand positive integers may be specified in roster notation as
59:
8663:
8357:
5734:
1911:
430:
to determine membership. Semantic definitions and definitions using set-builder notation are examples.
8573:
8530:
8204:
8166:
8043:
7847:
7687:
7611:
7589:
7417:
7375:
7274:
7241:
7105:
6893:
6804:
6530:
6520:
6354:
6285:
6238:
6178:
6065:
5469:
4498:
4294:
4235:
4127:
4067:
The concept of a set emerged in mathematics at the end of the 19th century. The German word for set,
3311:
2988:
2295:
454:
311:
5612:
5542:
5526:
8706:
8583:
8333:
8224:
8209:
8189:
8146:
8033:
7983:
7909:
7854:
7791:
7584:
7579:
7527:
7295:
7284:
6956:
6856:
6784:
6775:
6771:
6706:
6701:
6525:
6436:
6349:
6344:
6339:
6153:
6095:
6033:
5969:
5196:
5120:
5047:
4974:
4835:
4261:
4123:
3346:
3292:
3288:
3280:
3191:
2276:
402:
239:
68:
2239:
2217:
2183:
2146:
1840:
1818:
1740:
1718:
1559:
1537:
1415:
1393:
1297:
1275:
1250:
1224:
1196:
1171:
1146:
1121:
1096:
8668:
8505:
8452:
8412:
8362:
8131:
8094:
8079:
8072:
8055:
7841:
7707:
7633:
7616:
7569:
7382:
7291:
7125:
7110:
7070:
7022:
7007:
6995:
6951:
6926:
6696:
6645:
6448:
6443:
6228:
6183:
6090:
5892:
5487:
4338:
3296:
2831:
2550:
2527:
2521:
1774:
7859:
7315:
638:
4464:
3594:
A more general form of the principle gives the cardinality of any finite union of finite sets:
8680:
8445:
8297:
8104:
7914:
7904:
7796:
7677:
7512:
7488:
7269:
7253:
7135:
7012:
6981:
6946:
6841:
6676:
6142:
6105:
6075:
5919:
5896:
5873:
5847:
5781:
5713:
5707:
5686:
5680:
5659:
5632:
5589:
5562:
5556:
5505:
5389:
5383:
5362:
5335:
5329:
5308:
5281:
5254:
5227:
5221:
5170:
5143:
5067:
4997:
4948:
4896:
4890:
4860:
4854:
4812:
4785:
4758:
4731:
4691:
4664:
4637:
4631:
4610:
4554:
4527:
4523:
4470:
4443:
4416:
4306:
3076:
2546:
1953:
1764:
1587:
877:
A third pair of operators ⊂ and ⊃ are used differently by different authors: some authors use
41:
8372:
5653:
5137:
5061:
4806:
4725:
4685:
4658:
4604:
4107:, one of the founders of set theory, gave the following definition at the beginning of his
2126:
More formally, two sets share the same cardinality if there exists a bijection between them.
8611:
8495:
8435:
8311:
8306:
8199:
8156:
7978:
7939:
7934:
7919:
7745:
7702:
7599:
7397:
7347:
6921:
6883:
6593:
6583:
6568:
6563:
6431:
6085:
5755:
5495:
5477:
5424:
5302:
5275:
5164:
4991:
4779:
4752:
4548:
4437:
4410:
4333:
4298:
4268:
4156:
4119:
3284:
1768:
1141:
17:
4088:
Passage with a translation of the original set definition of Georg Cantor. The German word
3432:, because the set of all squares is subset of the set of all real numbers. Since for every
8593:
8292:
8282:
8236:
8219:
8174:
8136:
8038:
7958:
7765:
7692:
7665:
7653:
7559:
7473:
7447:
7402:
7370:
7171:
6973:
6916:
6866:
6831:
6789:
6462:
6400:
6218:
6038:
4323:
4072:
2566:
2317:
2283:
2260:
1660:{\displaystyle \mathbf {Q} =\left\{{\frac {a}{b}}\mid a,b\in \mathbf {Z} ,b\neq 0\right\}}
1583:
1579:
1244:
1191:
1091:
118:
8621:
4579:
5473:
8563:
8440:
8277:
8256:
8214:
8194:
8089:
7944:
7542:
7532:
7522:
7517:
7451:
7325:
7201:
7090:
7085:
7063:
6664:
6598:
6395:
6376:
6280:
6265:
6222:
6158:
6100:
5836:
4516:
4343:
4195:
2902:
2166:
1860:
1317:
292:
5940:
5500:
5457:
4126:(a set is a class, but some classes, such as the class of all sets, are not sets; see
8690:
8430:
8251:
7929:
7436:
7221:
7211:
7181:
7166:
6836:
6603:
6573:
6405:
6319:
6314:
5831:
3089:
2268:
2204:
1012:
185:
161:
35:
4714:", p.278. Bulletin of Symbolic Logic vol. 9, no. 3, (2003). Accessed 21 August 2023.
1221:
Many of these important sets are represented in mathematical texts using bold (e.g.
8457:
8420:
8151:
7998:
7899:
7891:
7771:
7719:
7628:
7564:
7547:
7478:
7337:
7196:
6898:
6681:
6553:
6548:
6366:
6295:
6253:
6112:
6016:
5866:
4277:
collection of distinct elements, but problems arise from the vagueness of the term
4273:
4104:
3268:
2301:
2272:
2139:
1166:
1035:
386:
234:
72:
8626:
5775:
5626:
5583:
5356:
5248:
4942:
273:
Another way to define a set is to use a rule to determine what the elements are:
8545:
8525:
8467:
8261:
8141:
7320:
7310:
7257:
6941:
6861:
6846:
6726:
6671:
6578:
6213:
5861:
5777:
Labyrinth of
Thought: A History of Set Theory and Its Role in Modern Mathematics
5628:
Labyrinth of
Thought: A History of Set Theory and Its Role in Modern Mathematics
2449:
2177:
2073:
1760:
197:
122:
50:
5462:
Proceedings of the
National Academy of Sciences of the United States of America
5102:
Sourendra Nath, De (January 2015). "Unit-1 Sets and
Functions: 1. Set Theory".
5017:
4463:
Thomas H. Cormen; Charles E Leiserson; Ronald L Rivest; Clifford Stein (2001).
30:
8500:
8482:
7191:
7046:
7017:
6823:
6558:
6426:
6329:
5992:
5682:
From Kant to
Hilbert Volume 1: A Source Book in the Foundations of Mathematics
4062:
2549:(meaning any two sets of the partition contain no element in common), and the
2305:
2011:
987:
413:
98:
45:
This set equals the one depicted above since both have the very same elements.
21:
5428:
4227:
The simple concept of a set has proved enormously useful in mathematics, but
2271:(i.e., the number of points on a line) is the same as the cardinality of any
956:
The empty set is a subset of every set, and every set is a subset of itself:
8343:
8246:
7299:
7216:
7176:
7140:
7076:
6888:
6878:
6851:
6361:
6324:
6275:
6173:
4353:
4348:
2492:
2445:
2347:
2329:
2025:
1993:
618:
76:
5509:
5482:
5091:. Vol. 1. Arya Publications (Avichal Publishing Company). p. A=3.
3306:
One of the main applications of naive set theory is in the construction of
1086:
109:
for all branches of mathematics since the first half of the 20th century.
8328:
8126:
7574:
7279:
6873:
4917:
4209:
4161:
The foremost property of a set is that it can have elements, also called
3320:
2599:(a set containing all elements being discussed) has been fixed, and that
1116:
376:{\displaystyle F=\{n\mid n{\text{ is an integer, and }}0\leq n\leq 19\}.}
212:
189:
5945:
7924:
6716:
4711:
1435:
1031:. If two sets have no elements in common, the regions do not overlap.
401:
is an integer in the range from 0 to 19 inclusive". Some authors use a
281:
254:
5166:
Mathematical and
Computer Programming Techniques for Computer Graphics
3490:
6614:
6386:
6208:
5491:
4290:
710:
4231:
arise if no restrictions are placed on how sets can be constructed:
4134:
When mathematicians deal with what they call a manifold, aggregate,
2797:
2775:
2744:
2709:
2571:
2320:. (ZFC is the most widely-studied version of axiomatic set theory.)
3190:
The operations above satisfy many identities. For example, one of
940:
The set of all humans is a proper subset of the set of all mammals.
681:
is a set with exactly one element; such a set may also be called a
8606:
7468:
6814:
6659:
6258:
6025:
4384:
Contributions to the founding of the theory of transfinite numbers
4213:
4083:
3489:
2796:
2774:
2743:
2708:
2570:
1085:
389:"|" means "such that", and the description can be interpreted as "
193:
40:
29:
4856:
Hegel's Rabble: An
Investigation into Hegel's Philosophy of Right
462:
of elements; a roster involving an ellipsis would be an example.
94:. In particular, this implies that there is only one empty set.
8376:
6618:
5965:
5954:
Cantor's "Beiträge zur Begründung der transfiniten
Mengenlehre"
3065:{\displaystyle A\,\Delta \,B=(A\setminus B)\cup (B\setminus A)}
5277:
Formal Models of
Computation: The Ultimate Limits of Computing
4657:
Ignacio Bello; Anton Kaul; Jack R. Britton (29 January 2013).
4630:
David Johnson; David B. Johnson; Thomas A. Mowry (June 2004).
5961:
5558:
The Heart of Mathematics: An invitation to effective thinking
4778:
Alfred Basta; Stephan DeLong; Nadine Basta (1 January 2013).
4633:
Finite Mathematics: Practical Applications (Docutech Version)
2545:
is in exactly one of these subsets. That is, the subsets are
57:
is a collection of different things; these things are called
5838:
Georg Cantor: His Mathematics and Philosophy of the Infinite
5358:
The Real Numbers: An Introduction to Set Theory and Analysis
905:(and not necessarily a proper subset), while others reserve
1800:
200:
of a set, the ordering of the terms matters). For example,
5735:"Beiträge zur Begründung der transfiniten Mengenlehre (1)"
4386:. New York Dover Publications (1954 English translation).
4382:
Cantor, Georg; Jourdain, Philip E.B. (Translator) (1915).
3462:. In functional notation, this relation can be written as
2115:. Repeated members in roster notation are not counted, so
1066:, there should be a zone for the elements that are inside
1996:(or one-to-one) if it maps any two different elements of
20:. For a rigorous modern axiomatic treatment of sets, see
3279:
Sets are ubiquitous in modern mathematics. For example,
4169:
when they have the same elements. More precisely, sets
631:) is the unique set that has no members. It is denoted
5561:. Springer Science & Business Media. p. 183.
5388:. Springer Science & Business Media. p. 211.
5307:. The Mathematical Association of America. p. 7.
5223:
Lectures in Logic and Set Theory: Volume 2, Set Theory
4724:
Stephen B. Maurer; Anthony Ralston (21 January 2005).
1525:{\displaystyle \mathbf {Z} =\{...,-2,-1,0,1,2,3,...\}}
1050:
zones such that for each way of selecting some of the
416:
uses specific terms to classify types of definitions:
8652:
5555:
Edward B. Burger; Michael Starbird (18 August 2004).
5188:
5186:
5039:
5037:
4966:
4964:
4751:
D. Van Dalen; H. C. Doets; H. De Swart (9 May 2014).
3600:
3505:
3014:
2650:
2374:{∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}
2242:
2220:
2186:
2149:
1914:
1843:
1821:
1777:
1743:
1721:
1596:
1562:
1540:
1444:
1418:
1396:
1326:
1300:
1278:
1253:
1227:
1199:
1174:
1149:
1124:
1099:
641:
328:
280:
be the set whose members are the first four positive
75:. There is a unique set with no elements, called the
5169:. Springer Science & Business Media. p. 7.
4996:. Springer Science & Business Media. p. 2.
4490:
4488:
4486:
4109:
Beiträge zur Begründung der transfiniten Mengenlehre
145:, especially when its elements are themselves sets.
8592:
8544:
8481:
8411:
8270:
8165:
7997:
7890:
7742:
7435:
7358:
7252:
7156:
7045:
6972:
6907:
6822:
6813:
6735:
6652:
6539:
6502:
6414:
6304:
6192:
6133:
6024:
5999:
4985:
4983:
3366:= {(scissors,paper), (paper,rock), (rock,scissors)}
2267:However, it can be shown that the cardinality of a
97:Sets are ubiquitous in modern mathematics. Indeed,
5865:
5835:
5801:Raatikainen, Panu (2022). Zalta, Edward N. (ed.).
4712:The Empty Set, the Singleton, and the Ordered Pair
4515:
4043:
3584:
3064:
2867:is the set of all things that are members of both
2693:
2250:
2228:
2194:
2157:
1965:is a rule that assigns to each "input" element of
1929:
1851:
1829:
1795:that cannot be rewritten as fractions, as well as
1787:
1751:
1729:
1659:
1570:
1548:
1524:
1426:
1404:
1376:
1308:
1286:
1261:
1235:
1207:
1182:
1157:
1132:
1107:
647:
375:
5441:: CS1 maint: DOI inactive as of September 2024 (
4606:Introduction to Mathematical Proofs: A Transition
4216:can even be distinguished by element order; e.g.
2694:{\displaystyle A^{\text{c}}=\{a\in U:a\notin A\}}
2304:and the cardinality of a straight line. In 1963,
2137:The list of elements of some sets is endless, or
1973:; more formally, a function is a special kind of
1937:represents the set of positive rational numbers.
779:. Two sets are equal if they contain each other:
90:(they are the same set). This property is called
5215:
5213:
5211:
5209:
5207:
5205:
5131:
5129:
1038:, in contrast, is a graphical representation of
105:, has been the standard way to provide rigorous
5809:. Metaphysics Research Lab, Stanford University
5416:Journal für die Reine und Angewandte Mathematik
5060:K.T. Leung; Doris Lai-chue Chen (1 July 1992).
4132:
4122:introduced the distinction between a set and a
4113:
5458:"The Independence of the Continuum Hypothesis"
4547:Seymor Lipschutz; Marc Lipson (22 June 1997).
4409:P. K. Jain; Khalil Ahmad; Om P. Ahuja (1995).
4264:shows that "the set of all sets" cannot exist.
160:defines a set by listing its elements between
8388:
6630:
5977:
4805:Laura Bracken; Ed Miller (15 February 2013).
2843:is the set of all things that are members of
1027:is completely inside the region representing
8:
5706:Paul Rusnock; Jan Sebestík (25 April 2019).
4884:
4882:
4880:
4878:
4876:
3585:{\displaystyle |A\cup B|=|A|+|B|-|A\cap B|.}
3349:of the same name, the relation "beats" from
3216:(that is, the elements outside the union of
2688:
2664:
1519:
1453:
1377:{\displaystyle \mathbf {N} =\{0,1,2,3,...\}}
1371:
1335:
367:
335:
5679:William Ewald; William Bragg Ewald (1996).
5226:. Cambridge University Press. p. 137.
4848:
4846:
4844:
117:Mathematical texts commonly denote sets by
8395:
8381:
8373:
7456:
7051:
6819:
6637:
6623:
6615:
5984:
5970:
5962:
5066:. Hong Kong University Press. p. 27.
2924:) is the set of all things that belong to
2431:has three elements, and its power set has
2129:The cardinality of the empty set is zero.
5780:. Springer Science & Business Media.
5655:The Mathematical Works of Bernard Bolzano
5499:
5481:
5361:. Springer Science & Business Media.
5253:. Springer Science & Business Media.
5142:. American Mathematical Soc. p. 30.
4895:. Rowman & Littlefield. p. 108.
4019:
4000:
3987:
3974:
3949:
3927:
3912:
3889:
3870:
3844:
3831:
3808:
3795:
3776:
3754:
3730:
3709:
3688:
3659:
3640:
3627:
3614:
3601:
3599:
3574:
3560:
3552:
3544:
3536:
3528:
3520:
3506:
3504:
3022:
3018:
3013:
2701:. The complement may also be called the
2655:
2649:
2244:
2243:
2241:
2222:
2221:
2219:
2188:
2187:
2185:
2151:
2150:
2148:
2044:, so that there are no unpaired elements.
1921:
1916:
1913:
1845:
1844:
1842:
1822:
1820:
1778:
1776:
1763:, including all rational numbers and all
1745:
1744:
1742:
1722:
1720:
1635:
1610:
1597:
1595:
1564:
1563:
1561:
1541:
1539:
1445:
1443:
1420:
1419:
1417:
1397:
1395:
1327:
1325:
1302:
1301:
1299:
1279:
1277:
1255:
1254:
1252:
1228:
1226:
1201:
1200:
1198:
1176:
1175:
1173:
1151:
1150:
1148:
1126:
1125:
1123:
1101:
1100:
1098:
640:
347:
327:
5411:"Ein Beitrag zur Mannigfaltigkeitslehre"
5304:The Lebesgue Integral for Undergraduates
4550:Schaum's Outline of Discrete Mathematics
4377:
4375:
3000:is the set of all things that belong to
2356:itself are elements of the power set of
2308:proved that the continuum hypothesis is
986:
771:between sets established by ⊆ is called
8659:
4553:. McGraw Hill Professional. p. 1.
4509:
4507:
4371:
3248:is the product of the cardinalities of
3053:
3035:
2553:of all the subsets of the partition is
2056:, and a bijective function is called a
5912:How To Prove It: A Structured Approach
5608:
5538:
5522:
5434:
5301:William Johnston (25 September 2015).
5192:
5116:
5089:Understanding ISC Mathematics Class XI
5043:
4990:Marek Capinski; Peter E. Kopp (2004).
4970:
4859:. Bloomsbury Publishing. p. 151.
4831:
4781:Mathematics for Information Technology
4687:Discrete Mathematics with Applications
4522:. W. H. Freeman and Company. pp.
4494:
3422:is real. This relation is a subset of
3128:{1, 2, 3} ∪ {3, 4, 5} = {1, 2, 3, 4, 5
2312:of the axiom system ZFC consisting of
2133:Infinite sets and infinite cardinality
1438:(whether positive, negative or zero):
1082:Special sets of numbers in mathematics
1078:(even if such elements do not exist).
544:For example, with respect to the sets
458:is one that describes a set by giving
6134:
5220:George Tourlakis (13 February 2003).
4238:shows that the "set of all sets that
4208:represent the same set. Unlike sets,
3328:is a subset of the Cartesian product
7:
5456:Cohen, Paul J. (December 15, 1963).
5106:. Scholar Books Pvt. Ltd. p. 5.
4892:Introduction to Abstract Mathematics
4574:
4572:
4570:
4415:. New Age International. p. 1.
3480:Principle of inclusion and exclusion
2362:, because these are both subsets of
2052:, a surjective function is called a
5807:Stanford Encyclopedia of Philosophy
3338:. For example, considering the set
3267:The power set of any set becomes a
3256:. (This is an elementary fact when
3146:{1, 2, 3} Δ {3, 4, 5} = {1, 2, 4, 5
2048:An injective function is called an
2040:is paired with a unique element of
2032:is paired with a unique element of
2018:, there is at least one element of
1977:, one that relates each element of
443:. Such definitions are also called
79:; a set with a single element is a
5774:Jose Ferreiros (1 November 2001).
5709:Bernard Bolzano: His Life and Work
5355:John Stillwell (16 October 2013).
5331:Mathematics: Its Power and Utility
5087:Aggarwal, M.L. (2021). "1. Sets".
4660:Topics in Contemporary Mathematics
4518:Sets, Logic and Axiomatic Theories
4442:. Courier Corporation. p. 2.
4436:Samuel Goldberg (1 January 1986).
3224:are the elements that are outside
3019:
2302:cardinality of the natural numbers
2208:; these are either finite sets or
2014:(or onto) if for every element of
1969:an "output" that is an element of
755:. The latter notation may be read
685:. Any such set can be written as {
642:
489:, this is written in shorthand as
262:{..., −3, −2, −1, 0, 1, 2, 3, ...}
27:Collection of mathematical objects
14:
8473:List of mathematical logic topics
5872:. Princeton, N.J.: Van Nostrand.
5803:"Gödel's Incompleteness Theorems"
5625:José Ferreirós (16 August 2007).
5334:. Cengage Learning. p. 401.
4993:Measure, Integral and Probability
4392:of definite and separate objects
2214:(sets of the same cardinality as
409:Classifying methods of definition
405:":" instead of the vertical bar.
8674:
8662:
8356:
6015:
5939:
5328:Karl J. Smith (7 January 2008).
5063:Elementary Set Theory, Part I/II
4811:. Cengage Learning. p. 36.
4727:Discrete Algorithmic Mathematics
4690:. Cengage Learning. p. 13.
4684:Susanna S. Epp (4 August 2010).
4663:. Cengage Learning. p. 47.
4603:Charles Roberts (24 June 2009).
4396:of our intuition or our thought.
4297:takes the concept of a set as a
2533:is a set of nonempty subsets of
2368:. For example, the power set of
1930:{\displaystyle \mathbf {Q} ^{+}}
1917:
1823:
1723:
1636:
1598:
1542:
1446:
1398:
1328:
1280:
1229:
229:Infinite sets in roster notation
184:This notation was introduced by
5585:Combinatorics of Set Partitions
5582:Toufik Mansour (27 July 2012).
5280:. World Scientific. p. 3.
5247:Yiannis N. Moschovakis (1994).
5163:Peter Comninos (6 April 2010).
4784:. Cengage Learning. p. 3.
4757:. Elsevier Science. p. 1.
4311:Gödel's incompleteness theorems
4305:(statements) about sets, using
2619:is the set of all elements (of
2119:{blue, white, red, blue, white}
1023:, then the region representing
8642:List of category theory topics
5652:Steve Russ (9 December 2004).
5385:Advanced Mathematical Thinking
4636:. W. H. Freeman. p. 220.
4193:; this property is called the
4177:are equal if every element of
3575:
3561:
3553:
3545:
3537:
3529:
3521:
3507:
3303:under one or more operations.
3059:
3047:
3041:
3029:
2958:as the absolute complement of
2094:, is the number of members of
349: is an integer, and
298:Such a definition is called a
1:
8317:History of mathematical logic
5944:The dictionary definition of
5761:The Principles of Mathematics
5274:Arthur Charles Fleck (2001).
5104:Chhaya Ganit (Ekadash Shreni)
4853:Frank Ruda (6 October 2011).
3486:Inclusion–exclusion principle
3398:. Another example is the set
3140:{1, 2, 3} − {3, 4, 5} = {1, 2
2340:is the set of all subsets of
1190:, which are contained in the
1165:, which are contained in the
1140:, which are contained in the
533:, which can also be read as "
499:, which can also be read as "
137:. A set may also be called a
67:of the set and are typically
8242:Primitive recursive function
5382:David Tall (11 April 2006).
4439:Probability: An Introduction
4224:represent different tuples.
2644:. In set-builder notation,
2491:unpaired. (There is never a
2480:will leave some elements of
2275:of that line, of the entire
2251:{\displaystyle \mathbb {N} }
2229:{\displaystyle \mathbb {N} }
2195:{\displaystyle \mathbb {N} }
2158:{\displaystyle \mathbb {N} }
1852:{\displaystyle \mathbb {C} }
1830:{\displaystyle \mathbf {C} }
1752:{\displaystyle \mathbb {R} }
1730:{\displaystyle \mathbf {R} }
1571:{\displaystyle \mathbb {Q} }
1549:{\displaystyle \mathbf {Q} }
1427:{\displaystyle \mathbb {Z} }
1405:{\displaystyle \mathbf {Z} }
1309:{\displaystyle \mathbb {N} }
1287:{\displaystyle \mathbf {N} }
1262:{\displaystyle \mathbb {Z} }
1236:{\displaystyle \mathbf {Z} }
1208:{\displaystyle \mathbb {C} }
1183:{\displaystyle \mathbb {R} }
1158:{\displaystyle \mathbb {Q} }
1133:{\displaystyle \mathbb {Z} }
1108:{\displaystyle \mathbb {N} }
1046:loops divide the plane into
291:be the set of colors of the
8637:Glossary of category theory
8511:Zermelo–Fraenkel set theory
8463:Mathematical constructivism
5712:. OUP Oxford. p. 430.
5685:. OUP Oxford. p. 249.
4947:. West Publishing Company.
4941:Ralph C. Steinlage (1987).
4469:. MIT Press. p. 1070.
2314:Zermelo–Fraenkel set theory
1788:{\displaystyle {\sqrt {2}}}
950:{1, 2, 3, 4} ⊆ {1, 2, 3, 4}
860:B is a proper superset of A
320:can be defined as follows:
103:Zermelo–Fraenkel set theory
8723:
8632:Mathematical structuralism
8569:Intuitionistic type theory
8405:Foundations of Mathematics
7306:Schröder–Bernstein theorem
7033:Monadic predicate calculus
6692:Foundations of mathematics
6475:von Neumann–Bernays–Gödel
5916:Cambridge University Press
4466:Introduction To Algorithms
4154:
4060:
3483:
3134:{1, 2, 3} ∩ {3, 4, 5} = {3
2564:
2537:, such that every element
2519:
2435:elements, as shown above.
2327:
2293:
2071:
1269:) typeface. These include
708:
670:
648:{\displaystyle \emptyset }
616:
470:
393:is the set of all numbers
309:
15:
8536:List of set theory topics
8352:
8339:Philosophy of mathematics
8288:Automated theorem proving
7459:
7413:Von Neumann–Bernays–Gödel
7054:
6276:One-to-one correspondence
6013:
5910:Velleman, Daniel (2006).
5887:Stoll, Robert R. (1979).
5733:Georg Cantor (Nov 1895).
4730:. CRC Press. p. 11.
4609:. CRC Press. p. 45.
4303:mathematical propositions
4240:do not contain themselves
4200:. As a consequence, e.g.
4078:Paradoxes of the Infinite
3343:= {rock, paper, scissors}
2376:. The power set of a set
2078:The cardinality of a set
2062:one-to-one correspondence
1582:(that is, the set of all
693:is the element. The set {
5844:Harvard University Press
5429:10.1515/crll.1878.84.242
5136:Felix Hausdorff (2005).
3444:, one and only one pair
3380:in the game if the pair
2940:, it is also called the
2290:The continuum hypothesis
1384:(often, authors exclude
727:is described as being a
715:If every element of set
441:listing all its elements
208:represent the same set.
113:Definition and notation
34:A set of polygons in an
8516:Constructive set theory
7989:Self-verifying theories
7810:Tarski's axiomatization
6761:Tarski's undefinability
6756:incompleteness theorems
4185:, and every element of
3008:but not both. One has
2427:elements. For example,
2382:is commonly written as
2334:The power set of a set
2211:countably infinite sets
2143:. For example, the set
1989:. A function is called
1767:numbers (which include
983:Euler and Venn diagrams
673:Singleton (mathematics)
571:is an integer, and 0 ≤
164:, separated by commas:
8617:Higher category theory
8521:Descriptive set theory
8426:Mathematical induction
8363:Mathematics portal
7974:Proof of impossibility
7622:propositional variable
6932:Propositional calculus
6234:Constructible universe
6061:Constructibility (V=L)
5483:10.1073/pnas.50.6.1143
5431:(inactive 2024-09-19).
5409:Cantor, Georg (1878).
4889:John F. Lucas (1990).
4580:"Introduction to Sets"
4514:Stoll, Robert (1974).
4329:Alternative set theory
4148:
4117:
4101:
4045:
3586:
3495:
3066:
2962:(in the universal set
2815:
2794:
2772:
2741:
2695:
2589:
2252:
2230:
2196:
2159:
2036:, and each element of
1931:
1853:
1831:
1797:transcendental numbers
1789:
1753:
1731:
1661:
1572:
1550:
1526:
1428:
1406:
1378:
1310:
1288:
1263:
1237:
1215:
1209:
1184:
1159:
1134:
1109:
1008:
929:is a proper subset of
870:, and is not equal to
838:. This can be written
649:
436:extensional definition
423:intensional definition
385:In this notation, the
377:
46:
38:
8579:Univalent foundations
8564:Dependent type theory
8554:Axiom of reducibility
8232:Kolmogorov complexity
8185:Computably enumerable
8085:Model complete theory
7877:Principia Mathematica
6937:Propositional formula
6766:Banach–Tarski paradox
6457:Principia Mathematica
6291:Transfinite induction
6150:(i.e. set difference)
5764:, chapter VI: Classes
5739:Mathematische Annalen
4359:Principia Mathematica
4271:defines a set as any
4087:
4046:
3587:
3493:
3067:
2800:
2778:
2747:
2712:
2696:
2631:. It may be denoted
2574:
2469:with the elements of
2444:is infinite (whether
2253:
2231:
2197:
2160:
1932:
1854:
1832:
1790:
1754:
1732:
1662:
1573:
1551:
1527:
1429:
1407:
1379:
1311:
1289:
1264:
1238:
1210:
1185:
1160:
1135:
1115:are contained in the
1110:
1089:
990:
944:{1, 3} ⊂ {1, 2, 3, 4}
650:
519:is not an element of
473:Element (mathematics)
378:
44:
33:
8702:Mathematical objects
8574:Homotopy type theory
8501:Axiomatic set theory
8180:Church–Turing thesis
8167:Computability theory
7376:continuum hypothesis
6894:Square of opposition
6752:Gödel's completeness
6531:Burali-Forti paradox
6286:Set-builder notation
6239:Continuum hypothesis
6179:Symmetric difference
5889:Set Theory and Logic
5631:. Birkhäuser Basel.
4295:Axiomatic set theory
4285:Axiomatic set theory
3598:
3503:
3310:. A relation from a
3012:
2989:symmetric difference
2804:symmetric difference
2648:
2296:Continuum hypothesis
2279:, and indeed of any
2240:
2218:
2184:
2171:infinite cardinality
2147:
2105:= {blue, white, red}
2022:that maps to it, and
1912:
1841:
1819:
1775:
1741:
1719:
1594:
1560:
1538:
1442:
1416:
1394:
1324:
1298:
1276:
1251:
1225:
1197:
1172:
1147:
1122:
1097:
765:B is a superset of A
639:
556:= {blue, white, red}
455:ostensive definition
326:
312:Set-builder notation
306:Set-builder notation
300:semantic description
247:{0, 1, 2, 3, 4, ...}
240:nonnegative integers
223:{1, 2, 3, ..., 1000}
179:= {blue, white, red}
158:enumeration notation
101:, more specifically
69:mathematical objects
8334:Mathematical object
8225:P versus NP problem
8190:Computable function
7984:Reverse mathematics
7910:Logical consequence
7787:primitive recursive
7782:elementary function
7555:Free/bound variable
7408:Tarski–Grothendieck
6927:Logical connectives
6857:Logical equivalence
6707:Logical consequence
6492:Tarski–Grothendieck
5474:1963PNAS...50.1143C
5250:Notes on Set Theory
4916:Weisstein, Eric W.
4412:Functional Analysis
4096:is translated with
3238:The cardinality of
2942:relative complement
2818:Given any two sets
2703:absolute complement
439:describes a set by
269:Semantic definition
253:and the set of all
8559:Simple type theory
8506:Zermelo set theory
8453:Mathematical proof
8413:Mathematical logic
8132:Transfer principle
8095:Semantics of logic
8080:Categorical theory
8056:Non-standard model
7570:Logical connective
6697:Information theory
6646:Mathematical logic
6081:Limitation of size
5893:Dover Publications
5022:www.mathsisfun.com
4808:Elementary Algebra
4584:www.mathsisfun.com
4339:Class (set theory)
4102:
4041:
4039:
3582:
3496:
3160:} × {1, 2, 3} = {(
3088:is the set of all
3062:
2932:. Especially when
2816:
2795:
2773:
2742:
2691:
2590:
2528:partition of a set
2522:Partition of a set
2281:finite-dimensional
2248:
2226:
2192:
2155:
2100:. For example, if
1927:
1849:
1827:
1785:
1749:
1727:
1657:
1588:improper fractions
1568:
1546:
1522:
1424:
1402:
1374:
1306:
1284:
1259:
1233:
1216:
1205:
1180:
1155:
1130:
1105:
1042:sets in which the
1009:
697:} and the element
645:
515:". The statement "
373:
47:
39:
8697:Concepts in logic
8650:
8649:
8531:Russell's paradox
8446:Natural deduction
8370:
8369:
8302:Abstract category
8105:Theories of truth
7915:Rule of inference
7905:Natural deduction
7886:
7885:
7431:
7430:
7136:Cartesian product
7041:
7040:
6947:Many-valued logic
6922:Boolean functions
6805:Russell's paradox
6780:diagonal argument
6677:First-order logic
6612:
6611:
6521:Russell's paradox
6470:Zermelo–Fraenkel
6371:Dedekind-infinite
6244:Diagonal argument
6143:Cartesian product
6007:Set (mathematics)
5891:. Mineola, N.Y.:
5832:Dauben, Joseph W.
5787:978-3-7643-5749-8
5719:978-0-19-255683-7
5692:978-0-19-850535-8
5665:978-0-19-151370-1
5638:978-3-7643-8349-7
5595:978-1-4398-6333-6
5568:978-1-931914-41-3
5395:978-0-306-47203-9
5368:978-3-319-01577-4
5341:978-0-495-38913-2
5314:978-1-939512-07-9
5287:978-981-02-4500-9
5260:978-3-540-94180-4
5233:978-1-139-43943-5
5176:978-1-84628-292-8
5149:978-0-8218-3835-8
5073:978-962-209-026-2
5003:978-1-85233-781-0
4954:978-0-314-29531-6
4922:Wolfram MathWorld
4902:978-0-912675-73-2
4866:978-1-4411-7413-0
4818:978-0-618-95134-5
4791:978-1-285-60843-3
4764:978-1-4831-5039-0
4737:978-1-4398-6375-6
4697:978-0-495-39132-6
4670:978-1-133-10742-2
4643:978-0-7167-6297-3
4616:978-1-4200-6956-3
4560:978-0-07-136841-4
4476:978-0-262-03293-3
4449:978-0-486-65252-8
4422:978-81-224-0801-0
4307:first-order logic
4236:Russell's paradox
4189:is an element of
4181:is an element of
4128:Russell's paradox
3458:, it is called a
3345:of shapes in the
3115:is an element of
3107:is an element of
3077:cartesian product
2658:
2547:pairwise disjoint
1859:, the set of all
1783:
1769:algebraic numbers
1759:, the set of all
1618:
1578:, the set of all
1434:, the set of all
1316:, the set of all
1003:is a superset of
901:is any subset of
799:is equivalent to
485:is an element of
350:
8714:
8679:
8678:
8677:
8667:
8666:
8658:
8612:Category of sets
8584:Girard's paradox
8496:Naive set theory
8436:Axiomatic system
8403:Major topics in
8397:
8390:
8383:
8374:
8361:
8360:
8312:History of logic
8307:Category of sets
8200:Decision problem
7979:Ordinal analysis
7920:Sequent calculus
7818:Boolean algebras
7758:
7757:
7732:
7703:logical/constant
7457:
7443:
7366:Zermelo–Fraenkel
7117:Set operations:
7052:
6989:
6820:
6800:Löwenheim–Skolem
6687:Formal semantics
6639:
6632:
6625:
6616:
6594:Bertrand Russell
6584:John von Neumann
6569:Abraham Fraenkel
6564:Richard Dedekind
6526:Suslin's problem
6437:Cantor's theorem
6154:De Morgan's laws
6019:
5986:
5979:
5972:
5963:
5958:
5943:
5929:
5906:
5883:
5871:
5868:Naive Set Theory
5857:
5841:
5818:
5817:
5815:
5814:
5798:
5792:
5791:
5771:
5765:
5756:Bertrand Russell
5753:
5747:
5746:
5730:
5724:
5723:
5703:
5697:
5696:
5676:
5670:
5669:
5649:
5643:
5642:
5622:
5616:
5606:
5600:
5599:
5579:
5573:
5572:
5552:
5546:
5536:
5530:
5520:
5514:
5513:
5503:
5485:
5468:(6): 1143–1148.
5453:
5447:
5446:
5440:
5432:
5406:
5400:
5399:
5379:
5373:
5372:
5352:
5346:
5345:
5325:
5319:
5318:
5298:
5292:
5291:
5271:
5265:
5264:
5244:
5238:
5237:
5217:
5200:
5190:
5181:
5180:
5160:
5154:
5153:
5133:
5124:
5114:
5108:
5107:
5099:
5093:
5092:
5084:
5078:
5077:
5057:
5051:
5041:
5032:
5031:
5029:
5028:
5014:
5008:
5007:
4987:
4978:
4968:
4959:
4958:
4938:
4932:
4931:
4929:
4928:
4913:
4907:
4906:
4886:
4871:
4870:
4850:
4839:
4829:
4823:
4822:
4802:
4796:
4795:
4775:
4769:
4768:
4748:
4742:
4741:
4721:
4715:
4708:
4702:
4701:
4681:
4675:
4674:
4654:
4648:
4647:
4627:
4621:
4620:
4600:
4594:
4593:
4591:
4590:
4576:
4565:
4564:
4544:
4538:
4537:
4521:
4511:
4502:
4492:
4481:
4480:
4460:
4454:
4453:
4433:
4427:
4426:
4406:
4400:
4398:
4379:
4334:Category of sets
4299:primitive notion
4269:Naïve set theory
4262:Cantor's paradox
4258:}, cannot exist.
4223:
4219:
4207:
4203:
4157:Naive set theory
4151:Naive set theory
4120:Bertrand Russell
4071:, was coined by
4050:
4048:
4047:
4042:
4040:
4033:
4029:
4025:
4024:
4023:
4005:
4004:
3992:
3991:
3979:
3978:
3960:
3959:
3948:
3944:
3928:
3923:
3913:
3908:
3904:
3900:
3899:
3895:
3894:
3893:
3881:
3880:
3854:
3850:
3849:
3848:
3836:
3835:
3818:
3814:
3813:
3812:
3800:
3799:
3777:
3772:
3768:
3764:
3763:
3759:
3758:
3739:
3735:
3734:
3718:
3714:
3713:
3697:
3693:
3692:
3669:
3665:
3664:
3663:
3645:
3644:
3632:
3631:
3619:
3618:
3591:
3589:
3588:
3583:
3578:
3564:
3556:
3548:
3540:
3532:
3524:
3510:
3475:
3457:
3451:
3443:
3437:
3431:
3421:
3415:
3403:
3397:
3391:
3379:
3373:
3367:
3360:
3354:
3344:
3337:
3327:
3318:
3285:abstract algebra
3263:
3259:
3255:
3251:
3247:
3234:
3227:
3223:
3219:
3214:
3192:De Morgan's laws
3185:
3147:
3141:
3135:
3129:
3118:
3114:
3110:
3106:
3102:
3087:
3071:
3069:
3068:
3063:
3007:
3003:
2999:
2983:
2965:
2957:
2951:
2947:
2939:
2935:
2931:
2927:
2923:
2913:
2893:
2889:
2885:
2866:
2842:
2825:
2821:
2792:
2770:
2760:
2756:
2739:
2729:
2723:
2700:
2698:
2697:
2692:
2660:
2659:
2656:
2643:
2636:
2630:
2622:
2618:
2606:
2602:
2598:
2561:Basic operations
2511:
2500:
2490:
2479:
2468:
2462:
2443:
2434:
2430:
2426:
2422:
2411:
2405:
2396:
2392:
2381:
2375:
2371:
2367:
2361:
2355:
2345:
2339:
2261:uncountable sets
2257:
2255:
2254:
2249:
2247:
2235:
2233:
2232:
2227:
2225:
2201:
2199:
2198:
2193:
2191:
2164:
2162:
2161:
2156:
2154:
2122:
2120:
2114:
2112:
2106:
2099:
2093:
2091:
2083:
2043:
2039:
2035:
2031:
2021:
2017:
2007:
1999:
1988:
1980:
1972:
1968:
1964:
1960:
1936:
1934:
1933:
1928:
1926:
1925:
1920:
1900:
1889:
1858:
1856:
1855:
1850:
1848:
1836:
1834:
1833:
1828:
1826:
1811:
1803:
1794:
1792:
1791:
1786:
1784:
1779:
1758:
1756:
1755:
1750:
1748:
1736:
1734:
1733:
1728:
1726:
1712:
1707:
1705:
1704:
1701:
1698:
1689:
1684:
1682:
1681:
1678:
1675:
1666:
1664:
1663:
1658:
1656:
1652:
1639:
1619:
1611:
1601:
1580:rational numbers
1577:
1575:
1574:
1569:
1567:
1555:
1553:
1552:
1547:
1545:
1531:
1529:
1528:
1523:
1449:
1433:
1431:
1430:
1425:
1423:
1411:
1409:
1408:
1403:
1401:
1387:
1383:
1381:
1380:
1375:
1331:
1315:
1313:
1312:
1307:
1305:
1293:
1291:
1290:
1285:
1283:
1268:
1266:
1265:
1260:
1258:
1242:
1240:
1239:
1234:
1232:
1214:
1212:
1211:
1206:
1204:
1189:
1187:
1186:
1181:
1179:
1164:
1162:
1161:
1156:
1154:
1142:rational numbers
1139:
1137:
1136:
1131:
1129:
1114:
1112:
1111:
1106:
1104:
1077:
1073:
1069:
1065:
1061:
1057:
1053:
1049:
1045:
1041:
1030:
1026:
1022:
1018:
977:
965:
951:
945:
925:for cases where
924:
914:
896:
886:
857:
847:
822:is not equal to
798:
788:
754:
744:
662:
658:
654:
652:
651:
646:
634:
608:
601:
592:
585:
576:
557:
550:
532:
523:" is written as
498:
488:
484:
480:
400:
396:
392:
382:
380:
379:
374:
351:
348:
319:
290:
279:
265:
263:
250:
248:
224:
218:
207:
203:
180:
172:
136:
132:
128:
18:Naive set theory
8722:
8721:
8717:
8716:
8715:
8713:
8712:
8711:
8687:
8686:
8685:
8675:
8673:
8661:
8653:
8651:
8646:
8594:Category theory
8588:
8540:
8477:
8407:
8401:
8371:
8366:
8355:
8348:
8293:Category theory
8283:Algebraic logic
8266:
8237:Lambda calculus
8175:Church encoding
8161:
8137:Truth predicate
7993:
7959:Complete theory
7882:
7751:
7747:
7743:
7738:
7730:
7450: and
7446:
7441:
7427:
7403:New Foundations
7371:axiom of choice
7354:
7316:Gödel numbering
7256: and
7248:
7152:
7037:
6987:
6968:
6917:Boolean algebra
6903:
6867:Equiconsistency
6832:Classical logic
6809:
6790:Halting problem
6778: and
6754: and
6742: and
6741:
6736:Theorems (
6731:
6648:
6643:
6613:
6608:
6535:
6514:
6498:
6463:New Foundations
6410:
6300:
6219:Cardinal number
6202:
6188:
6129:
6020:
6011:
5995:
5990:
5956:
5936:
5926:
5909:
5903:
5886:
5880:
5862:Halmos, Paul R.
5860:
5854:
5830:
5827:
5822:
5821:
5812:
5810:
5800:
5799:
5795:
5788:
5773:
5772:
5768:
5754:
5750:
5732:
5731:
5727:
5720:
5705:
5704:
5700:
5693:
5678:
5677:
5673:
5666:
5651:
5650:
5646:
5639:
5624:
5623:
5619:
5607:
5603:
5596:
5581:
5580:
5576:
5569:
5554:
5553:
5549:
5537:
5533:
5521:
5517:
5455:
5454:
5450:
5433:
5423:(84): 242–258.
5408:
5407:
5403:
5396:
5381:
5380:
5376:
5369:
5354:
5353:
5349:
5342:
5327:
5326:
5322:
5315:
5300:
5299:
5295:
5288:
5273:
5272:
5268:
5261:
5246:
5245:
5241:
5234:
5219:
5218:
5203:
5191:
5184:
5177:
5162:
5161:
5157:
5150:
5135:
5134:
5127:
5115:
5111:
5101:
5100:
5096:
5086:
5085:
5081:
5074:
5059:
5058:
5054:
5042:
5035:
5026:
5024:
5016:
5015:
5011:
5004:
4989:
4988:
4981:
4969:
4962:
4955:
4944:College Algebra
4940:
4939:
4935:
4926:
4924:
4915:
4914:
4910:
4903:
4888:
4887:
4874:
4867:
4852:
4851:
4842:
4830:
4826:
4819:
4804:
4803:
4799:
4792:
4777:
4776:
4772:
4765:
4750:
4749:
4745:
4738:
4723:
4722:
4718:
4709:
4705:
4698:
4683:
4682:
4678:
4671:
4656:
4655:
4651:
4644:
4629:
4628:
4624:
4617:
4602:
4601:
4597:
4588:
4586:
4578:
4577:
4568:
4561:
4546:
4545:
4541:
4534:
4513:
4512:
4505:
4493:
4484:
4477:
4462:
4461:
4457:
4450:
4435:
4434:
4430:
4423:
4408:
4407:
4403:
4381:
4380:
4373:
4368:
4363:
4324:Algebra of sets
4319:
4309:. According to
4287:
4221:
4217:
4205:
4201:
4165:. Two sets are
4159:
4153:
4073:Bernard Bolzano
4065:
4059:
4053:
4038:
4037:
4015:
3996:
3983:
3970:
3969:
3965:
3961:
3937:
3933:
3932:
3921:
3920:
3906:
3905:
3885:
3866:
3865:
3861:
3840:
3827:
3826:
3822:
3804:
3791:
3790:
3786:
3785:
3781:
3770:
3769:
3750:
3746:
3726:
3722:
3705:
3701:
3684:
3680:
3679:
3675:
3673:
3655:
3636:
3623:
3610:
3609:
3605:
3596:
3595:
3501:
3500:
3488:
3482:
3463:
3453:
3445:
3439:
3433:
3423:
3417:
3405:
3399:
3393:
3392:is a member of
3381:
3375:
3369:
3362:
3356:
3350:
3339:
3329:
3323:
3314:
3277:
3261:
3257:
3253:
3249:
3239:
3232:
3225:
3221:
3217:
3196:
3151:
3145:
3139:
3133:
3127:
3116:
3112:
3108:
3104:
3092:
3079:
3010:
3009:
3005:
3001:
2991:
2967:
2963:
2953:
2949:
2945:
2937:
2936:is a subset of
2933:
2929:
2925:
2915:
2905:
2894:are said to be
2891:
2887:
2876:
2858:
2834:
2823:
2819:
2814:
2793:
2784:
2771:
2762:
2758:
2754:
2740:
2731:
2725:
2719:
2651:
2646:
2645:
2638:
2632:
2628:
2620:
2616:
2604:
2603:is a subset of
2600:
2596:
2592:Suppose that a
2588:
2569:
2567:Algebra of sets
2563:
2524:
2518:
2502:
2496:
2481:
2470:
2464:
2453:
2439:
2432:
2428:
2424:
2413:
2412:elements, then
2407:
2401:
2394:
2383:
2377:
2373:
2369:
2363:
2357:
2351:
2341:
2335:
2332:
2326:
2318:axiom of choice
2298:
2292:
2284:Euclidean space
2238:
2237:
2216:
2215:
2182:
2181:
2167:natural numbers
2145:
2144:
2135:
2118:
2116:
2110:
2108:
2101:
2095:
2087:
2085:
2079:
2076:
2070:
2041:
2037:
2033:
2029:
2019:
2015:
2005:
1997:
1986:
1978:
1970:
1966:
1962:
1958:
1943:
1915:
1910:
1909:
1891:
1890:, for example,
1864:
1861:complex numbers
1839:
1838:
1817:
1816:
1807:
1801:
1773:
1772:
1739:
1738:
1717:
1716:
1702:
1699:
1696:
1695:
1693:
1691:
1679:
1676:
1673:
1672:
1670:
1668:
1667:. For example,
1609:
1605:
1592:
1591:
1558:
1557:
1536:
1535:
1440:
1439:
1414:
1413:
1392:
1391:
1385:
1322:
1321:
1318:natural numbers
1296:
1295:
1274:
1273:
1249:
1248:
1245:blackboard bold
1223:
1222:
1195:
1194:
1192:complex numbers
1170:
1169:
1145:
1144:
1120:
1119:
1095:
1094:
1092:natural numbers
1084:
1075:
1071:
1067:
1063:
1059:
1055:
1051:
1047:
1043:
1039:
1028:
1024:
1020:
1019:is a subset of
1016:
999:
994:is a subset of
985:
969:
960:
949:
943:
916:
906:
888:
878:
849:
839:
814:is a subset of
790:
780:
746:
736:
713:
707:
675:
669:
660:
656:
637:
636:
632:
621:
615:
610:
603:
596:
594:
587:
580:
559:
552:
545:
524:
490:
486:
482:
478:
475:
469:
411:
398:
394:
390:
324:
323:
317:
314:
308:
296:
288:
285:
277:
271:
266:
261:
260:
251:
246:
245:
231:
226:
222:
216:
205:
201:
182:
175:
173:
167:
151:
149:Roster notation
134:
130:
126:
119:capital letters
115:
28:
25:
12:
11:
5:
8720:
8718:
8710:
8709:
8704:
8699:
8689:
8688:
8684:
8683:
8671:
8648:
8647:
8645:
8644:
8639:
8634:
8629:
8627:∞-topos theory
8624:
8619:
8614:
8609:
8604:
8598:
8596:
8590:
8589:
8587:
8586:
8581:
8576:
8571:
8566:
8561:
8556:
8550:
8548:
8542:
8541:
8539:
8538:
8533:
8528:
8523:
8518:
8513:
8508:
8503:
8498:
8493:
8487:
8485:
8479:
8478:
8476:
8475:
8470:
8465:
8460:
8455:
8450:
8449:
8448:
8443:
8441:Hilbert system
8438:
8428:
8423:
8417:
8415:
8409:
8408:
8402:
8400:
8399:
8392:
8385:
8377:
8368:
8367:
8353:
8350:
8349:
8347:
8346:
8341:
8336:
8331:
8326:
8325:
8324:
8314:
8309:
8304:
8295:
8290:
8285:
8280:
8278:Abstract logic
8274:
8272:
8268:
8267:
8265:
8264:
8259:
8257:Turing machine
8254:
8249:
8244:
8239:
8234:
8229:
8228:
8227:
8222:
8217:
8212:
8207:
8197:
8195:Computable set
8192:
8187:
8182:
8177:
8171:
8169:
8163:
8162:
8160:
8159:
8154:
8149:
8144:
8139:
8134:
8129:
8124:
8123:
8122:
8117:
8112:
8102:
8097:
8092:
8090:Satisfiability
8087:
8082:
8077:
8076:
8075:
8065:
8064:
8063:
8053:
8052:
8051:
8046:
8041:
8036:
8031:
8021:
8020:
8019:
8014:
8007:Interpretation
8003:
8001:
7995:
7994:
7992:
7991:
7986:
7981:
7976:
7971:
7961:
7956:
7955:
7954:
7953:
7952:
7942:
7937:
7927:
7922:
7917:
7912:
7907:
7902:
7896:
7894:
7888:
7887:
7884:
7883:
7881:
7880:
7872:
7871:
7870:
7869:
7864:
7863:
7862:
7857:
7852:
7832:
7831:
7830:
7828:minimal axioms
7825:
7814:
7813:
7812:
7801:
7800:
7799:
7794:
7789:
7784:
7779:
7774:
7761:
7759:
7740:
7739:
7737:
7736:
7735:
7734:
7722:
7717:
7716:
7715:
7710:
7705:
7700:
7690:
7685:
7680:
7675:
7674:
7673:
7668:
7658:
7657:
7656:
7651:
7646:
7641:
7631:
7626:
7625:
7624:
7619:
7614:
7604:
7603:
7602:
7597:
7592:
7587:
7582:
7577:
7567:
7562:
7557:
7552:
7551:
7550:
7545:
7540:
7535:
7525:
7520:
7518:Formation rule
7515:
7510:
7509:
7508:
7503:
7493:
7492:
7491:
7481:
7476:
7471:
7466:
7460:
7454:
7437:Formal systems
7433:
7432:
7429:
7428:
7426:
7425:
7420:
7415:
7410:
7405:
7400:
7395:
7390:
7385:
7380:
7379:
7378:
7373:
7362:
7360:
7356:
7355:
7353:
7352:
7351:
7350:
7340:
7335:
7334:
7333:
7326:Large cardinal
7323:
7318:
7313:
7308:
7303:
7289:
7288:
7287:
7282:
7277:
7262:
7260:
7250:
7249:
7247:
7246:
7245:
7244:
7239:
7234:
7224:
7219:
7214:
7209:
7204:
7199:
7194:
7189:
7184:
7179:
7174:
7169:
7163:
7161:
7154:
7153:
7151:
7150:
7149:
7148:
7143:
7138:
7133:
7128:
7123:
7115:
7114:
7113:
7108:
7098:
7093:
7091:Extensionality
7088:
7086:Ordinal number
7083:
7073:
7068:
7067:
7066:
7055:
7049:
7043:
7042:
7039:
7038:
7036:
7035:
7030:
7025:
7020:
7015:
7010:
7005:
7004:
7003:
6993:
6992:
6991:
6978:
6976:
6970:
6969:
6967:
6966:
6965:
6964:
6959:
6954:
6944:
6939:
6934:
6929:
6924:
6919:
6913:
6911:
6905:
6904:
6902:
6901:
6896:
6891:
6886:
6881:
6876:
6871:
6870:
6869:
6859:
6854:
6849:
6844:
6839:
6834:
6828:
6826:
6817:
6811:
6810:
6808:
6807:
6802:
6797:
6792:
6787:
6782:
6770:Cantor's
6768:
6763:
6758:
6748:
6746:
6733:
6732:
6730:
6729:
6724:
6719:
6714:
6709:
6704:
6699:
6694:
6689:
6684:
6679:
6674:
6669:
6668:
6667:
6656:
6654:
6650:
6649:
6644:
6642:
6641:
6634:
6627:
6619:
6610:
6609:
6607:
6606:
6601:
6599:Thoralf Skolem
6596:
6591:
6586:
6581:
6576:
6571:
6566:
6561:
6556:
6551:
6545:
6543:
6537:
6536:
6534:
6533:
6528:
6523:
6517:
6515:
6513:
6512:
6509:
6503:
6500:
6499:
6497:
6496:
6495:
6494:
6489:
6484:
6483:
6482:
6467:
6466:
6465:
6453:
6452:
6451:
6440:
6439:
6434:
6429:
6424:
6418:
6416:
6412:
6411:
6409:
6408:
6403:
6398:
6393:
6384:
6379:
6374:
6364:
6359:
6358:
6357:
6352:
6347:
6337:
6327:
6322:
6317:
6311:
6309:
6302:
6301:
6299:
6298:
6293:
6288:
6283:
6281:Ordinal number
6278:
6273:
6268:
6263:
6262:
6261:
6256:
6246:
6241:
6236:
6231:
6226:
6216:
6211:
6205:
6203:
6201:
6200:
6197:
6193:
6190:
6189:
6187:
6186:
6181:
6176:
6171:
6166:
6161:
6159:Disjoint union
6156:
6151:
6145:
6139:
6137:
6131:
6130:
6128:
6127:
6126:
6125:
6120:
6109:
6108:
6106:Martin's axiom
6103:
6098:
6093:
6088:
6083:
6078:
6073:
6071:Extensionality
6068:
6063:
6058:
6057:
6056:
6051:
6046:
6036:
6030:
6028:
6022:
6021:
6014:
6012:
6010:
6009:
6003:
6001:
5997:
5996:
5991:
5989:
5988:
5981:
5974:
5966:
5960:
5959:
5951:
5935:
5934:External links
5932:
5931:
5930:
5924:
5907:
5901:
5884:
5878:
5858:
5852:
5826:
5823:
5820:
5819:
5793:
5786:
5766:
5748:
5725:
5718:
5698:
5691:
5671:
5664:
5658:. OUP Oxford.
5644:
5637:
5617:
5601:
5594:
5574:
5567:
5547:
5531:
5515:
5448:
5401:
5394:
5374:
5367:
5347:
5340:
5320:
5313:
5293:
5286:
5266:
5259:
5239:
5232:
5201:
5182:
5175:
5155:
5148:
5125:
5109:
5094:
5079:
5072:
5052:
5033:
5009:
5002:
4979:
4960:
4953:
4933:
4908:
4901:
4872:
4865:
4840:
4824:
4817:
4797:
4790:
4770:
4763:
4743:
4736:
4716:
4710:A. Kanamori, "
4703:
4696:
4676:
4669:
4649:
4642:
4622:
4615:
4595:
4566:
4559:
4539:
4532:
4503:
4482:
4475:
4455:
4448:
4428:
4421:
4401:
4370:
4369:
4367:
4364:
4362:
4361:
4356:
4351:
4346:
4344:Family of sets
4341:
4336:
4331:
4326:
4320:
4318:
4315:
4286:
4283:
4266:
4265:
4259:
4196:extensionality
4155:Main article:
4152:
4149:
4061:Main article:
4058:
4055:
4036:
4032:
4028:
4022:
4018:
4014:
4011:
4008:
4003:
3999:
3995:
3990:
3986:
3982:
3977:
3973:
3968:
3964:
3958:
3955:
3952:
3947:
3943:
3940:
3936:
3931:
3926:
3924:
3922:
3919:
3916:
3911:
3909:
3907:
3903:
3898:
3892:
3888:
3884:
3879:
3876:
3873:
3869:
3864:
3860:
3857:
3853:
3847:
3843:
3839:
3834:
3830:
3825:
3821:
3817:
3811:
3807:
3803:
3798:
3794:
3789:
3784:
3780:
3775:
3773:
3771:
3767:
3762:
3757:
3753:
3749:
3745:
3742:
3738:
3733:
3729:
3725:
3721:
3717:
3712:
3708:
3704:
3700:
3696:
3691:
3687:
3683:
3678:
3674:
3672:
3668:
3662:
3658:
3654:
3651:
3648:
3643:
3639:
3635:
3630:
3626:
3622:
3617:
3613:
3608:
3604:
3603:
3581:
3577:
3573:
3570:
3567:
3563:
3559:
3555:
3551:
3547:
3543:
3539:
3535:
3531:
3527:
3523:
3519:
3516:
3513:
3509:
3484:Main article:
3481:
3478:
3276:
3273:
3188:
3187:
3149:
3143:
3137:
3131:
3121:
3120:
3073:
3061:
3058:
3055:
3052:
3049:
3046:
3043:
3040:
3037:
3034:
3031:
3028:
3025:
3021:
3017:
2985:
2914:(also written
2903:set difference
2899:
2852:
2801:
2782:set difference
2779:
2748:
2713:
2707:
2706:
2690:
2687:
2684:
2681:
2678:
2675:
2672:
2669:
2666:
2663:
2654:
2575:
2565:Main article:
2562:
2559:
2520:Main article:
2517:
2514:
2328:Main article:
2325:
2322:
2294:Main article:
2291:
2288:
2246:
2224:
2205:countable sets
2190:
2153:
2134:
2131:
2072:Main article:
2069:
2066:
2046:
2045:
2023:
2009:
1942:
1939:
1924:
1919:
1903:
1902:
1847:
1825:
1814:
1782:
1747:
1725:
1714:
1655:
1651:
1648:
1645:
1642:
1638:
1634:
1631:
1628:
1625:
1622:
1617:
1614:
1608:
1604:
1600:
1566:
1544:
1533:
1521:
1518:
1515:
1512:
1509:
1506:
1503:
1500:
1497:
1494:
1491:
1488:
1485:
1482:
1479:
1476:
1473:
1470:
1467:
1464:
1461:
1458:
1455:
1452:
1448:
1422:
1400:
1389:
1373:
1370:
1367:
1364:
1361:
1358:
1355:
1352:
1349:
1346:
1343:
1340:
1337:
1334:
1330:
1304:
1282:
1257:
1231:
1203:
1178:
1153:
1128:
1103:
1083:
1080:
984:
981:
980:
979:
967:
954:
953:
947:
941:
733:contained in B
709:Main article:
706:
703:
671:Main article:
668:
667:Singleton sets
665:
644:
617:Main article:
614:
611:
595:
579:
549:= {1, 2, 3, 4}
471:Main article:
468:
465:
464:
463:
450:
431:
410:
407:
372:
369:
366:
363:
360:
357:
354:
346:
343:
340:
337:
334:
331:
310:Main article:
307:
304:
286:
275:
270:
267:
259:
244:
230:
227:
221:
174:
171:= {4, 2, 1, 3}
166:
162:curly brackets
150:
147:
114:
111:
92:extensionality
26:
13:
10:
9:
6:
4:
3:
2:
8719:
8708:
8705:
8703:
8700:
8698:
8695:
8694:
8692:
8682:
8672:
8670:
8665:
8660:
8656:
8643:
8640:
8638:
8635:
8633:
8630:
8628:
8625:
8623:
8620:
8618:
8615:
8613:
8610:
8608:
8605:
8603:
8600:
8599:
8597:
8595:
8591:
8585:
8582:
8580:
8577:
8575:
8572:
8570:
8567:
8565:
8562:
8560:
8557:
8555:
8552:
8551:
8549:
8547:
8543:
8537:
8534:
8532:
8529:
8527:
8524:
8522:
8519:
8517:
8514:
8512:
8509:
8507:
8504:
8502:
8499:
8497:
8494:
8492:
8489:
8488:
8486:
8484:
8480:
8474:
8471:
8469:
8466:
8464:
8461:
8459:
8456:
8454:
8451:
8447:
8444:
8442:
8439:
8437:
8434:
8433:
8432:
8431:Formal system
8429:
8427:
8424:
8422:
8419:
8418:
8416:
8414:
8410:
8406:
8398:
8393:
8391:
8386:
8384:
8379:
8378:
8375:
8365:
8364:
8359:
8351:
8345:
8342:
8340:
8337:
8335:
8332:
8330:
8327:
8323:
8320:
8319:
8318:
8315:
8313:
8310:
8308:
8305:
8303:
8299:
8296:
8294:
8291:
8289:
8286:
8284:
8281:
8279:
8276:
8275:
8273:
8269:
8263:
8260:
8258:
8255:
8253:
8252:Recursive set
8250:
8248:
8245:
8243:
8240:
8238:
8235:
8233:
8230:
8226:
8223:
8221:
8218:
8216:
8213:
8211:
8208:
8206:
8203:
8202:
8201:
8198:
8196:
8193:
8191:
8188:
8186:
8183:
8181:
8178:
8176:
8173:
8172:
8170:
8168:
8164:
8158:
8155:
8153:
8150:
8148:
8145:
8143:
8140:
8138:
8135:
8133:
8130:
8128:
8125:
8121:
8118:
8116:
8113:
8111:
8108:
8107:
8106:
8103:
8101:
8098:
8096:
8093:
8091:
8088:
8086:
8083:
8081:
8078:
8074:
8071:
8070:
8069:
8066:
8062:
8061:of arithmetic
8059:
8058:
8057:
8054:
8050:
8047:
8045:
8042:
8040:
8037:
8035:
8032:
8030:
8027:
8026:
8025:
8022:
8018:
8015:
8013:
8010:
8009:
8008:
8005:
8004:
8002:
8000:
7996:
7990:
7987:
7985:
7982:
7980:
7977:
7975:
7972:
7969:
7968:from ZFC
7965:
7962:
7960:
7957:
7951:
7948:
7947:
7946:
7943:
7941:
7938:
7936:
7933:
7932:
7931:
7928:
7926:
7923:
7921:
7918:
7916:
7913:
7911:
7908:
7906:
7903:
7901:
7898:
7897:
7895:
7893:
7889:
7879:
7878:
7874:
7873:
7868:
7867:non-Euclidean
7865:
7861:
7858:
7856:
7853:
7851:
7850:
7846:
7845:
7843:
7840:
7839:
7837:
7833:
7829:
7826:
7824:
7821:
7820:
7819:
7815:
7811:
7808:
7807:
7806:
7802:
7798:
7795:
7793:
7790:
7788:
7785:
7783:
7780:
7778:
7775:
7773:
7770:
7769:
7767:
7763:
7762:
7760:
7755:
7749:
7744:Example
7741:
7733:
7728:
7727:
7726:
7723:
7721:
7718:
7714:
7711:
7709:
7706:
7704:
7701:
7699:
7696:
7695:
7694:
7691:
7689:
7686:
7684:
7681:
7679:
7676:
7672:
7669:
7667:
7664:
7663:
7662:
7659:
7655:
7652:
7650:
7647:
7645:
7642:
7640:
7637:
7636:
7635:
7632:
7630:
7627:
7623:
7620:
7618:
7615:
7613:
7610:
7609:
7608:
7605:
7601:
7598:
7596:
7593:
7591:
7588:
7586:
7583:
7581:
7578:
7576:
7573:
7572:
7571:
7568:
7566:
7563:
7561:
7558:
7556:
7553:
7549:
7546:
7544:
7541:
7539:
7536:
7534:
7531:
7530:
7529:
7526:
7524:
7521:
7519:
7516:
7514:
7511:
7507:
7504:
7502:
7501:by definition
7499:
7498:
7497:
7494:
7490:
7487:
7486:
7485:
7482:
7480:
7477:
7475:
7472:
7470:
7467:
7465:
7462:
7461:
7458:
7455:
7453:
7449:
7444:
7438:
7434:
7424:
7421:
7419:
7416:
7414:
7411:
7409:
7406:
7404:
7401:
7399:
7396:
7394:
7391:
7389:
7388:Kripke–Platek
7386:
7384:
7381:
7377:
7374:
7372:
7369:
7368:
7367:
7364:
7363:
7361:
7357:
7349:
7346:
7345:
7344:
7341:
7339:
7336:
7332:
7329:
7328:
7327:
7324:
7322:
7319:
7317:
7314:
7312:
7309:
7307:
7304:
7301:
7297:
7293:
7290:
7286:
7283:
7281:
7278:
7276:
7273:
7272:
7271:
7267:
7264:
7263:
7261:
7259:
7255:
7251:
7243:
7240:
7238:
7235:
7233:
7232:constructible
7230:
7229:
7228:
7225:
7223:
7220:
7218:
7215:
7213:
7210:
7208:
7205:
7203:
7200:
7198:
7195:
7193:
7190:
7188:
7185:
7183:
7180:
7178:
7175:
7173:
7170:
7168:
7165:
7164:
7162:
7160:
7155:
7147:
7144:
7142:
7139:
7137:
7134:
7132:
7129:
7127:
7124:
7122:
7119:
7118:
7116:
7112:
7109:
7107:
7104:
7103:
7102:
7099:
7097:
7094:
7092:
7089:
7087:
7084:
7082:
7078:
7074:
7072:
7069:
7065:
7062:
7061:
7060:
7057:
7056:
7053:
7050:
7048:
7044:
7034:
7031:
7029:
7026:
7024:
7021:
7019:
7016:
7014:
7011:
7009:
7006:
7002:
6999:
6998:
6997:
6994:
6990:
6985:
6984:
6983:
6980:
6979:
6977:
6975:
6971:
6963:
6960:
6958:
6955:
6953:
6950:
6949:
6948:
6945:
6943:
6940:
6938:
6935:
6933:
6930:
6928:
6925:
6923:
6920:
6918:
6915:
6914:
6912:
6910:
6909:Propositional
6906:
6900:
6897:
6895:
6892:
6890:
6887:
6885:
6882:
6880:
6877:
6875:
6872:
6868:
6865:
6864:
6863:
6860:
6858:
6855:
6853:
6850:
6848:
6845:
6843:
6840:
6838:
6837:Logical truth
6835:
6833:
6830:
6829:
6827:
6825:
6821:
6818:
6816:
6812:
6806:
6803:
6801:
6798:
6796:
6793:
6791:
6788:
6786:
6783:
6781:
6777:
6773:
6769:
6767:
6764:
6762:
6759:
6757:
6753:
6750:
6749:
6747:
6745:
6739:
6734:
6728:
6725:
6723:
6720:
6718:
6715:
6713:
6710:
6708:
6705:
6703:
6700:
6698:
6695:
6693:
6690:
6688:
6685:
6683:
6680:
6678:
6675:
6673:
6670:
6666:
6663:
6662:
6661:
6658:
6657:
6655:
6651:
6647:
6640:
6635:
6633:
6628:
6626:
6621:
6620:
6617:
6605:
6604:Ernst Zermelo
6602:
6600:
6597:
6595:
6592:
6590:
6589:Willard Quine
6587:
6585:
6582:
6580:
6577:
6575:
6572:
6570:
6567:
6565:
6562:
6560:
6557:
6555:
6552:
6550:
6547:
6546:
6544:
6542:
6541:Set theorists
6538:
6532:
6529:
6527:
6524:
6522:
6519:
6518:
6516:
6510:
6508:
6505:
6504:
6501:
6493:
6490:
6488:
6487:Kripke–Platek
6485:
6481:
6478:
6477:
6476:
6473:
6472:
6471:
6468:
6464:
6461:
6460:
6459:
6458:
6454:
6450:
6447:
6446:
6445:
6442:
6441:
6438:
6435:
6433:
6430:
6428:
6425:
6423:
6420:
6419:
6417:
6413:
6407:
6404:
6402:
6399:
6397:
6394:
6392:
6390:
6385:
6383:
6380:
6378:
6375:
6372:
6368:
6365:
6363:
6360:
6356:
6353:
6351:
6348:
6346:
6343:
6342:
6341:
6338:
6335:
6331:
6328:
6326:
6323:
6321:
6318:
6316:
6313:
6312:
6310:
6307:
6303:
6297:
6294:
6292:
6289:
6287:
6284:
6282:
6279:
6277:
6274:
6272:
6269:
6267:
6264:
6260:
6257:
6255:
6252:
6251:
6250:
6247:
6245:
6242:
6240:
6237:
6235:
6232:
6230:
6227:
6224:
6220:
6217:
6215:
6212:
6210:
6207:
6206:
6204:
6198:
6195:
6194:
6191:
6185:
6182:
6180:
6177:
6175:
6172:
6170:
6167:
6165:
6162:
6160:
6157:
6155:
6152:
6149:
6146:
6144:
6141:
6140:
6138:
6136:
6132:
6124:
6123:specification
6121:
6119:
6116:
6115:
6114:
6111:
6110:
6107:
6104:
6102:
6099:
6097:
6094:
6092:
6089:
6087:
6084:
6082:
6079:
6077:
6074:
6072:
6069:
6067:
6064:
6062:
6059:
6055:
6052:
6050:
6047:
6045:
6042:
6041:
6040:
6037:
6035:
6032:
6031:
6029:
6027:
6023:
6018:
6008:
6005:
6004:
6002:
5998:
5994:
5987:
5982:
5980:
5975:
5973:
5968:
5967:
5964:
5955:
5952:
5950:at Wiktionary
5949:
5948:
5942:
5938:
5937:
5933:
5927:
5925:0-521-67599-5
5921:
5917:
5913:
5908:
5904:
5902:0-486-63829-4
5898:
5894:
5890:
5885:
5881:
5879:0-387-90092-6
5875:
5870:
5869:
5863:
5859:
5855:
5853:0-691-02447-2
5849:
5845:
5840:
5839:
5833:
5829:
5828:
5824:
5808:
5804:
5797:
5794:
5789:
5783:
5779:
5778:
5770:
5767:
5763:
5762:
5757:
5752:
5749:
5745:(4): 481–512.
5744:
5741:(in German).
5740:
5736:
5729:
5726:
5721:
5715:
5711:
5710:
5702:
5699:
5694:
5688:
5684:
5683:
5675:
5672:
5667:
5661:
5657:
5656:
5648:
5645:
5640:
5634:
5630:
5629:
5621:
5618:
5614:
5610:
5605:
5602:
5597:
5591:
5588:. CRC Press.
5587:
5586:
5578:
5575:
5570:
5564:
5560:
5559:
5551:
5548:
5544:
5540:
5535:
5532:
5528:
5524:
5519:
5516:
5511:
5507:
5502:
5497:
5493:
5489:
5484:
5479:
5475:
5471:
5467:
5463:
5459:
5452:
5449:
5444:
5438:
5430:
5426:
5422:
5418:
5417:
5412:
5405:
5402:
5397:
5391:
5387:
5386:
5378:
5375:
5370:
5364:
5360:
5359:
5351:
5348:
5343:
5337:
5333:
5332:
5324:
5321:
5316:
5310:
5306:
5305:
5297:
5294:
5289:
5283:
5279:
5278:
5270:
5267:
5262:
5256:
5252:
5251:
5243:
5240:
5235:
5229:
5225:
5224:
5216:
5214:
5212:
5210:
5208:
5206:
5202:
5198:
5194:
5189:
5187:
5183:
5178:
5172:
5168:
5167:
5159:
5156:
5151:
5145:
5141:
5140:
5132:
5130:
5126:
5122:
5118:
5113:
5110:
5105:
5098:
5095:
5090:
5083:
5080:
5075:
5069:
5065:
5064:
5056:
5053:
5049:
5045:
5040:
5038:
5034:
5023:
5019:
5018:"Set Symbols"
5013:
5010:
5005:
4999:
4995:
4994:
4986:
4984:
4980:
4976:
4972:
4967:
4965:
4961:
4956:
4950:
4946:
4945:
4937:
4934:
4923:
4919:
4912:
4909:
4904:
4898:
4894:
4893:
4885:
4883:
4881:
4879:
4877:
4873:
4868:
4862:
4858:
4857:
4849:
4847:
4845:
4841:
4837:
4833:
4828:
4825:
4820:
4814:
4810:
4809:
4801:
4798:
4793:
4787:
4783:
4782:
4774:
4771:
4766:
4760:
4756:
4755:
4747:
4744:
4739:
4733:
4729:
4728:
4720:
4717:
4713:
4707:
4704:
4699:
4693:
4689:
4688:
4680:
4677:
4672:
4666:
4662:
4661:
4653:
4650:
4645:
4639:
4635:
4634:
4626:
4623:
4618:
4612:
4608:
4607:
4599:
4596:
4585:
4581:
4575:
4573:
4571:
4567:
4562:
4556:
4552:
4551:
4543:
4540:
4535:
4533:9780716704577
4529:
4525:
4520:
4519:
4510:
4508:
4504:
4500:
4496:
4491:
4489:
4487:
4483:
4478:
4472:
4468:
4467:
4459:
4456:
4451:
4445:
4441:
4440:
4432:
4429:
4424:
4418:
4414:
4413:
4405:
4402:
4397:
4395:
4391:
4385:
4378:
4376:
4372:
4365:
4360:
4357:
4355:
4352:
4350:
4347:
4345:
4342:
4340:
4337:
4335:
4332:
4330:
4327:
4325:
4322:
4321:
4316:
4314:
4312:
4308:
4304:
4300:
4296:
4292:
4284:
4282:
4280:
4276:
4275:
4270:
4263:
4260:
4257:
4253:
4250:is a set and
4249:
4245:
4241:
4237:
4234:
4233:
4232:
4230:
4225:
4215:
4211:
4199:
4197:
4192:
4188:
4184:
4180:
4176:
4172:
4168:
4164:
4158:
4150:
4147:
4145:
4141:
4137:
4131:
4129:
4125:
4121:
4116:
4112:
4110:
4106:
4099:
4095:
4091:
4086:
4082:
4080:
4079:
4074:
4070:
4064:
4056:
4054:
4051:
4034:
4030:
4026:
4020:
4016:
4012:
4009:
4006:
4001:
3997:
3993:
3988:
3984:
3980:
3975:
3971:
3966:
3962:
3956:
3953:
3950:
3945:
3941:
3938:
3934:
3929:
3925:
3917:
3914:
3910:
3901:
3896:
3890:
3886:
3882:
3877:
3874:
3871:
3867:
3862:
3858:
3855:
3851:
3845:
3841:
3837:
3832:
3828:
3823:
3819:
3815:
3809:
3805:
3801:
3796:
3792:
3787:
3782:
3778:
3774:
3765:
3760:
3755:
3751:
3747:
3743:
3740:
3736:
3731:
3727:
3723:
3719:
3715:
3710:
3706:
3702:
3698:
3694:
3689:
3685:
3681:
3676:
3670:
3666:
3660:
3656:
3652:
3649:
3646:
3641:
3637:
3633:
3628:
3624:
3620:
3615:
3611:
3606:
3592:
3579:
3571:
3568:
3565:
3557:
3549:
3541:
3533:
3525:
3517:
3514:
3511:
3492:
3487:
3479:
3477:
3474:
3470:
3466:
3461:
3456:
3449:
3442:
3436:
3430:
3426:
3420:
3413:
3409:
3404:of all pairs
3402:
3396:
3389:
3385:
3378:
3372:
3365:
3359:
3353:
3348:
3342:
3336:
3332:
3326:
3322:
3317:
3313:
3309:
3304:
3302:
3298:
3294:
3290:
3286:
3282:
3274:
3272:
3270:
3265:
3246:
3242:
3236:
3230:
3215:
3212:
3208:
3204:
3200:
3193:
3183:
3179:
3175:
3171:
3167:
3163:
3159:
3155:
3150:
3144:
3138:
3132:
3126:
3125:
3124:
3100:
3096:
3091:
3090:ordered pairs
3086:
3082:
3078:
3074:
3056:
3050:
3044:
3038:
3032:
3026:
3023:
3015:
2998:
2994:
2990:
2986:
2982:
2978:
2974:
2970:
2961:
2956:
2943:
2922:
2918:
2912:
2908:
2904:
2900:
2897:
2883:
2879:
2874:
2870:
2865:
2861:
2857:
2853:
2850:
2846:
2841:
2837:
2833:
2829:
2828:
2827:
2813:
2809:
2805:
2799:
2791:
2787:
2783:
2777:
2769:
2765:
2752:
2746:
2738:
2734:
2728:
2722:
2717:
2711:
2704:
2685:
2682:
2679:
2676:
2673:
2670:
2667:
2661:
2652:
2641:
2635:
2626:
2614:
2610:
2609:
2608:
2595:
2594:universal set
2587:
2583:
2579:
2573:
2568:
2560:
2558:
2556:
2552:
2548:
2544:
2540:
2536:
2532:
2529:
2523:
2515:
2513:
2509:
2505:
2499:
2494:
2488:
2484:
2477:
2473:
2467:
2460:
2456:
2451:
2447:
2442:
2436:
2420:
2416:
2410:
2404:
2398:
2390:
2386:
2380:
2366:
2360:
2354:
2349:
2344:
2338:
2331:
2323:
2321:
2319:
2315:
2311:
2307:
2303:
2297:
2289:
2287:
2285:
2282:
2278:
2274:
2270:
2269:straight line
2265:
2263:
2262:
2213:
2212:
2207:
2206:
2179:
2174:
2172:
2168:
2142:
2141:
2132:
2130:
2127:
2124:
2104:
2098:
2090:
2082:
2075:
2067:
2065:
2063:
2059:
2055:
2051:
2027:
2024:
2013:
2010:
2003:
1995:
1992:
1991:
1990:
1984:
1976:
1957:) from a set
1956:
1955:
1950:
1949:
1940:
1938:
1922:
1906:
1899:
1895:
1887:
1883:
1879:
1875:
1871:
1867:
1862:
1815:
1812:
1810:
1804:
1798:
1780:
1770:
1766:
1762:
1715:
1711:
1688:
1653:
1649:
1646:
1643:
1640:
1632:
1629:
1626:
1623:
1620:
1615:
1612:
1606:
1602:
1589:
1585:
1581:
1534:
1516:
1513:
1510:
1507:
1504:
1501:
1498:
1495:
1492:
1489:
1486:
1483:
1480:
1477:
1474:
1471:
1468:
1465:
1462:
1459:
1456:
1450:
1437:
1390:
1368:
1365:
1362:
1359:
1356:
1353:
1350:
1347:
1344:
1341:
1338:
1332:
1319:
1272:
1271:
1270:
1246:
1219:
1193:
1168:
1143:
1118:
1093:
1088:
1081:
1079:
1037:
1032:
1014:
1013:Euler diagram
1006:
1002:
997:
993:
989:
982:
976:
972:
968:
964:
959:
958:
957:
948:
942:
939:
938:
937:
934:
932:
928:
923:
919:
913:
909:
904:
900:
895:
891:
885:
881:
875:
873:
869:
865:
861:
856:
852:
846:
842:
837:
833:
832:proper subset
829:
825:
821:
817:
813:
808:
806:
802:
797:
793:
787:
783:
778:
774:
770:
766:
762:
758:
753:
749:
743:
739:
734:
730:
726:
722:
718:
712:
704:
702:
700:
696:
692:
688:
684:
680:
679:singleton set
674:
666:
664:
630:
626:
620:
613:The empty set
612:
607:
600:
591:
584:
578:
574:
570:
566:
562:
555:
548:
542:
540:
536:
531:
527:
522:
518:
514:
510:
506:
502:
497:
493:
481:is a set and
474:
466:
461:
457:
456:
451:
448:
447:
442:
438:
437:
432:
429:
425:
424:
419:
418:
417:
415:
408:
406:
404:
388:
383:
370:
364:
361:
358:
355:
352:
344:
341:
338:
332:
329:
321:
313:
305:
303:
301:
294:
283:
274:
268:
258:
256:
243:
241:
236:
228:
220:
214:
209:
199:
195:
191:
187:
186:Ernst Zermelo
178:
170:
165:
163:
159:
155:
148:
146:
144:
140:
124:
120:
112:
110:
108:
104:
100:
95:
93:
89:
84:
82:
78:
74:
70:
66:
62:
61:
56:
52:
43:
37:
36:Euler diagram
32:
23:
19:
8607:Topos theory
8490:
8458:Model theory
8421:Peano axioms
8354:
8152:Ultraproduct
7999:Model theory
7964:Independence
7900:Formal proof
7892:Proof theory
7875:
7848:
7805:real numbers
7777:second-order
7688:Substitution
7565:Metalanguage
7506:conservative
7479:Axiom schema
7423:Constructive
7393:Morse–Kelley
7359:Set theories
7338:Aleph number
7331:inaccessible
7237:Grothendieck
7158:
7121:intersection
7008:Higher-order
6996:Second-order
6942:Truth tables
6899:Venn diagram
6682:Formal proof
6554:Georg Cantor
6549:Paul Bernays
6480:Morse–Kelley
6455:
6388:
6387:Subset
6334:hereditarily
6305:
6296:Venn diagram
6254:ordered pair
6169:Intersection
6113:Axiom schema
6006:
5946:
5911:
5888:
5867:
5837:
5811:. Retrieved
5806:
5796:
5776:
5769:
5759:
5751:
5742:
5738:
5728:
5708:
5701:
5681:
5674:
5654:
5647:
5627:
5620:
5604:
5584:
5577:
5557:
5550:
5534:
5518:
5465:
5461:
5451:
5437:cite journal
5420:
5414:
5404:
5384:
5377:
5357:
5350:
5330:
5323:
5303:
5296:
5276:
5269:
5249:
5242:
5222:
5165:
5158:
5138:
5112:
5103:
5097:
5088:
5082:
5062:
5055:
5025:. Retrieved
5021:
5012:
4992:
4943:
4936:
4925:. Retrieved
4921:
4911:
4891:
4855:
4827:
4807:
4800:
4780:
4773:
4753:
4746:
4726:
4719:
4706:
4686:
4679:
4659:
4652:
4632:
4625:
4605:
4598:
4587:. Retrieved
4583:
4549:
4542:
4517:
4465:
4458:
4438:
4431:
4411:
4404:
4393:
4389:
4387:
4383:
4288:
4279:well-defined
4278:
4274:well-defined
4272:
4267:
4255:
4251:
4247:
4243:
4239:
4226:
4206:{4, 6, 4, 2}
4194:
4190:
4186:
4182:
4178:
4174:
4170:
4162:
4160:
4143:
4139:
4135:
4133:
4118:
4114:
4108:
4105:Georg Cantor
4103:
4097:
4093:
4089:
4076:
4075:in his work
4068:
4066:
4052:
3593:
3497:
3472:
3468:
3464:
3454:
3452:is found in
3447:
3440:
3434:
3428:
3424:
3418:
3411:
3407:
3400:
3394:
3387:
3383:
3376:
3370:
3363:
3357:
3351:
3340:
3334:
3330:
3324:
3315:
3305:
3278:
3275:Applications
3269:Boolean ring
3266:
3244:
3240:
3237:
3228:
3210:
3206:
3202:
3198:
3195:
3194:states that
3189:
3181:
3177:
3173:
3169:
3165:
3161:
3157:
3153:
3122:
3098:
3094:
3084:
3080:
2996:
2992:
2980:
2976:
2972:
2968:
2959:
2954:
2920:
2916:
2910:
2906:
2895:
2881:
2877:
2872:
2868:
2863:
2859:
2856:intersection
2848:
2844:
2839:
2835:
2817:
2811:
2807:
2803:
2789:
2785:
2781:
2767:
2763:
2751:intersection
2750:
2736:
2732:
2726:
2720:
2715:
2702:
2639:
2633:
2624:
2591:
2585:
2581:
2577:
2554:
2542:
2538:
2534:
2530:
2525:
2507:
2503:
2497:
2486:
2482:
2475:
2471:
2465:
2458:
2454:
2440:
2437:
2418:
2414:
2408:
2402:
2399:
2388:
2384:
2378:
2364:
2358:
2352:
2342:
2336:
2333:
2299:
2266:
2259:
2209:
2203:
2178:real numbers
2175:
2170:
2138:
2136:
2128:
2125:
2102:
2096:
2088:
2080:
2077:
2061:
2057:
2053:
2049:
2047:
2004:elements of
2001:
1982:
1952:
1946:
1944:
1907:
1904:
1897:
1893:
1885:
1881:
1877:
1873:
1869:
1865:
1808:
1761:real numbers
1709:
1686:
1220:
1217:
1167:real numbers
1074:and outside
1036:Venn diagram
1033:
1010:
1004:
1000:
995:
991:
974:
970:
962:
955:
935:
930:
926:
921:
917:
911:
907:
902:
898:
893:
889:
883:
879:
876:
871:
867:
863:
859:
854:
850:
848:. Likewise,
844:
840:
835:
831:
830:is called a
827:
823:
819:
815:
811:
809:
804:
800:
795:
791:
785:
781:
776:
772:
769:relationship
764:
761:B includes A
760:
757:B contains A
756:
751:
747:
741:
737:
732:
728:
724:
720:
716:
714:
698:
694:
690:
686:
682:
678:
676:
655:, { },
628:
624:
622:
605:
598:
589:
582:
572:
568:
564:
560:
553:
546:
543:
538:
534:
529:
525:
520:
516:
512:
508:
504:
500:
495:
491:
476:
459:
453:
444:
440:
434:
427:
421:
412:
387:vertical bar
384:
322:
315:
299:
297:
272:
252:
235:infinite set
232:
210:
206:{4, 6, 4, 2}
183:
176:
168:
157:
153:
152:
142:
138:
116:
96:
85:
73:infinite set
64:
58:
54:
48:
8669:Mathematics
8546:Type theory
8526:Determinacy
8468:Modal logic
8262:Type theory
8210:undecidable
8142:Truth value
8029:equivalence
7708:non-logical
7321:Enumeration
7311:Isomorphism
7258:cardinality
7242:Von Neumann
7207:Ultrafilter
7172:Uncountable
7106:equivalence
7023:Quantifiers
7013:Fixed-point
6982:First-order
6862:Consistency
6847:Proposition
6824:Traditional
6795:Lindström's
6785:Compactness
6727:Type theory
6672:Cardinality
6579:Thomas Jech
6422:Alternative
6401:Uncountable
6355:Ultrafilter
6214:Cardinality
6118:replacement
6066:Determinacy
5957:(in German)
5609:Halmos 1960
5539:Halmos 1960
5523:Halmos 1960
5193:Halmos 1960
5117:Halmos 1960
5044:Halmos 1960
4971:Halmos 1960
4832:Halmos 1960
4495:Halmos 1960
3361:is the set
3299:, are sets
2450:uncountable
2310:independent
2258:are called
2202:are called
2074:Cardinality
2068:Cardinality
1985:element of
1983:exactly one
777:containment
729:subset of B
719:is also in
503:belongs to
446:enumerative
293:French flag
198:permutation
107:foundations
51:mathematics
8707:Set theory
8691:Categories
8681:Arithmetic
8622:∞-groupoid
8483:Set theory
8073:elementary
7766:arithmetic
7634:Quantifier
7612:functional
7484:Expression
7202:Transitive
7146:identities
7131:complement
7064:hereditary
7047:Set theory
6574:Kurt Gödel
6559:Paul Cohen
6396:Transitive
6164:Identities
6148:Complement
6135:Operations
6096:Regularity
6034:Adjunction
5993:Set theory
5842:. Boston:
5825:References
5813:2024-06-03
5611:, p.
5541:, p.
5525:, p.
5195:, p.
5139:Set Theory
5046:, p.
5027:2020-08-19
4973:, p.
4927:2020-08-19
4834:, p.
4589:2020-08-19
4497:, p.
4399:Here: p.85
4242:", i.e., {
4146:the class.
4063:Set theory
3287:, such as
3281:structures
3123:Examples:
3103:such that
2761:, denoted
2730:, denoted
2627:belong to
2623:) that do
2613:complement
2578:complement
2516:Partitions
2324:Power sets
2306:Paul Cohen
2121:| = 3
2113:| = 3
2084:, denoted
2054:surjection
2012:surjective
1765:irrational
936:Examples:
735:, written
537:is not in
467:Membership
414:Philosophy
397:such that
139:collection
125:, such as
99:set theory
22:Set theory
8344:Supertask
8247:Recursion
8205:decidable
8039:saturated
8017:of models
7940:deductive
7935:axiomatic
7855:Hilbert's
7842:Euclidean
7823:canonical
7746:axiomatic
7678:Signature
7607:Predicate
7496:Extension
7418:Ackermann
7343:Operation
7222:Universal
7212:Recursive
7187:Singleton
7182:Inhabited
7167:Countable
7157:Types of
7141:power set
7111:partition
7028:Predicate
6974:Predicate
6889:Syllogism
6879:Soundness
6852:Inference
6842:Tautology
6744:paradoxes
6507:Paradoxes
6427:Axiomatic
6406:Universal
6382:Singleton
6377:Recursive
6320:Countable
6315:Amorphous
6174:Power set
6091:Power set
6049:dependent
6044:countable
4354:Mereology
4349:Fuzzy set
4229:paradoxes
4222:(6, 4, 2)
4218:(2, 4, 6)
4210:multisets
4202:{2, 4, 6}
4098:aggregate
4013:∩
4010:…
4007:∩
3994:∩
3981:∩
3954:−
3939:−
3918:…
3883:∩
3875:−
3859:…
3838:∩
3802:∩
3779:−
3744:…
3653:∪
3650:…
3647:∪
3634:∪
3621:∪
3569:∩
3558:−
3515:∪
3308:relations
3054:∖
3045:∪
3036:∖
3020:Δ
2683:∉
2671:∈
2493:bijection
2446:countable
2429:{1, 2, 3}
2370:{1, 2, 3}
2348:empty set
2330:Power set
2316:with the
2058:bijection
2050:injection
2026:bijective
2002:different
1994:injective
1961:to a set
1941:Functions
1647:≠
1633:∈
1621:∣
1478:−
1469:−
866:contains
773:inclusion
689:}, where
643:∅
625:empty set
619:Empty set
362:≤
356:≤
342:∣
202:{2, 4, 6}
81:singleton
77:empty set
8602:Category
8329:Logicism
8322:timeline
8298:Concrete
8157:Validity
8127:T-schema
8120:Kripke's
8115:Tarski's
8110:semantic
8100:Strength
8049:submodel
8044:spectrum
8012:function
7860:Tarski's
7849:Elements
7836:geometry
7792:Robinson
7713:variable
7698:function
7671:spectrum
7661:Sentence
7617:variable
7560:Language
7513:Relation
7474:Automata
7464:Alphabet
7448:language
7302:-jection
7280:codomain
7266:Function
7227:Universe
7197:Infinite
7101:Relation
6884:Validity
6874:Argument
6772:theorem,
6511:Problems
6415:Theories
6391:Superset
6367:Infinite
6196:Concepts
6076:Infinity
6000:Overview
5864:(1960).
5834:(1979).
5510:16578557
4317:See also
4140:ensemble
3460:function
3416:, where
3321:codomain
3231:outside
2928:but not
2896:disjoint
2851:or both.
2452:), then
2140:infinite
1975:relation
1948:function
1799:such as
1771:such as
1436:integers
1117:integers
897:to mean
683:unit set
629:null set
604:green ∉
460:examples
282:integers
255:integers
213:ellipsis
190:sequence
60:elements
8655:Portals
8271:Related
8068:Diagram
7966: (
7945:Hilbert
7930:Systems
7925:Theorem
7803:of the
7748:systems
7528:Formula
7523:Grammar
7439: (
7383:General
7096:Forcing
7081:Element
7001:Monadic
6776:paradox
6717:Theorem
6653:General
6449:General
6444:Zermelo
6350:subbase
6332: (
6271:Forcing
6249:Element
6221: (
6199:Methods
6086:Pairing
5758:(1903)
5470:Bibcode
4198:of sets
4163:members
4057:History
3368:; thus
2952:. With
2886:, then
2273:segment
2123:, too.
2107:, then
1954:mapping
1706:
1694:
1683:
1671:
1669:−
862:, i.e.
826:, then
723:, then
705:Subsets
507:", or "
426:uses a
196:, or a
65:members
8034:finite
7797:Skolem
7750:
7725:Theory
7693:Symbol
7683:String
7666:atomic
7543:ground
7538:closed
7533:atomic
7489:ground
7452:syntax
7348:binary
7275:domain
7192:Finite
6957:finite
6815:Logics
6774:
6722:Theory
6340:Filter
6330:Finite
6266:Family
6209:Almost
6054:global
6039:Choice
6026:Axioms
5922:
5899:
5876:
5850:
5784:
5716:
5689:
5662:
5635:
5592:
5565:
5508:
5501:221287
5498:
5490:
5392:
5365:
5338:
5311:
5284:
5257:
5230:
5173:
5146:
5121:Sect.2
5070:
5000:
4951:
4899:
4863:
4815:
4788:
4761:
4734:
4694:
4667:
4640:
4613:
4557:
4530:
4473:
4446:
4419:
4291:axioms
4214:Tuples
3374:beats
3312:domain
3301:closed
3293:fields
3289:groups
3180:,2), (
3176:,1), (
3172:,3), (
3168:,2), (
3164:,1), (
3075:their
2987:their
2875:. If
2854:their
2830:their
2346:. The
2117:|
2109:|
2092:|
2086:|
1584:proper
1247:(e.g.
1062:, and
858:means
818:, but
767:. The
711:Subset
558:, and
511:is in
154:Roster
143:family
123:italic
8024:Model
7772:Peano
7629:Proof
7469:Arity
7398:Naive
7285:image
7217:Fuzzy
7177:Empty
7126:union
7071:Class
6712:Model
6702:Lemma
6660:Axiom
6432:Naive
6362:Fuzzy
6325:Empty
6308:types
6259:tuple
6229:Class
6223:large
6184:Union
6101:Union
5492:71858
5488:JSTOR
4918:"Set"
4366:Notes
4167:equal
4136:Menge
4124:class
4100:here.
4090:Menge
4069:Menge
3450:,...)
3319:to a
3297:rings
3205:)′ =
2832:union
2716:union
2551:union
2501:onto
2495:from
2433:2 = 8
2277:plane
1892:1 + 2
1243:) or
763:, or
745:, or
731:, or
659:, or
597:20 ∉
593:; and
588:12 ∈
575:≤ 19}
403:colon
194:tuple
88:equal
8147:Type
7950:list
7754:list
7731:list
7720:Term
7654:rank
7548:open
7442:list
7254:Maps
7159:sets
7018:Free
6988:list
6738:list
6665:list
6345:base
5920:ISBN
5897:ISBN
5874:ISBN
5848:ISBN
5782:ISBN
5714:ISBN
5687:ISBN
5660:ISBN
5633:ISBN
5590:ISBN
5563:ISBN
5506:PMID
5443:link
5421:1878
5390:ISBN
5363:ISBN
5336:ISBN
5309:ISBN
5282:ISBN
5255:ISBN
5228:ISBN
5171:ISBN
5144:ISBN
5068:ISBN
4998:ISBN
4949:ISBN
4897:ISBN
4861:ISBN
4813:ISBN
4786:ISBN
4759:ISBN
4732:ISBN
4692:ISBN
4665:ISBN
4638:ISBN
4611:ISBN
4555:ISBN
4528:ISBN
4471:ISBN
4444:ISBN
4417:ISBN
4220:and
4204:and
4173:and
4130:):
4092:for
3471:) =
3347:game
3295:and
3260:and
3252:and
3220:and
3209:′ ∩
3111:and
2901:the
2890:and
2871:and
2822:and
2810:and
2802:The
2780:The
2757:and
2749:The
2724:and
2714:The
2611:The
2576:The
2423:has
2406:has
2350:and
1951:(or
1805:and
1692:5 =
1690:and
1586:and
1090:The
1070:and
961:∅ ⊆
915:and
887:and
789:and
627:(or
623:The
602:and
586:and
581:4 ∈
428:rule
287:Let
276:Let
204:and
192:, a
83:.
53:, a
8491:Set
7834:of
7816:of
7764:of
7296:Sur
7270:Map
7077:Ur-
7059:Set
6306:Set
5947:set
5496:PMC
5478:doi
5425:doi
4094:set
3438:in
3355:to
3283:in
3235:).
3229:and
3184:,3)
3004:or
2966:),
2948:in
2944:of
2884:= ∅
2847:or
2826:,
2806:of
2753:of
2718:of
2637:or
2625:not
2615:of
2584:in
2580:of
2541:in
2512:.)
2448:or
2438:If
2400:If
2393:or
2372:is
2173:.
2165:of
2060:or
2000:to
1981:to
1868:= {
1837:or
1737:or
1590:):
1556:or
1412:or
1294:or
1011:An
834:of
810:If
775:or
563:= {
541:".
477:If
452:An
433:An
420:An
257:is
242:is
233:An
217:...
156:or
141:or
121:in
63:or
55:set
49:In
8693::
8220:NP
7844::
7838::
7768::
7445:),
7300:Bi
7292:In
5918:.
5914:.
5895:.
5846:.
5805:.
5743:46
5737:.
5613:28
5543:20
5527:19
5504:.
5494:.
5486:.
5476:.
5466:50
5464:.
5460:.
5439:}}
5435:{{
5419:.
5413:.
5204:^
5185:^
5128:^
5119:,
5036:^
5020:.
4982:^
4963:^
4920:.
4875:^
4843:^
4582:.
4569:^
4526:.
4506:^
4485:^
4374:^
4293:.
4281:.
4254:∉
4246:|
4144:is
4138:,
4111::
3476:.
3427:×
3410:,
3333:×
3291:,
3243:×
3201:∪
3186:}.
3156:,
3148:}.
3142:}.
3136:}.
3130:}.
3083:×
2995:Δ
2979:∩
2975:=
2971:\
2919:−
2909:\
2880:∩
2862:∩
2838:∪
2788:\
2766:∩
2735:∪
2607:.
2557:.
2526:A
2397:.
2286:.
2264:.
2064:.
1945:A
1896:∈
1884:∈
1880:,
1876:|
1874:bi
1872:+
1863::
1813:);
1708:∈
1685:∈
1388:);
1320::
1058:,
1034:A
973:⊆
933:.
920:⊃
910:⊂
892:⊃
882:⊂
874:.
853:⊋
843:⊊
807:.
803:=
794:⊆
784:⊆
759:,
750:⊇
740:⊆
677:A
663:.
635:,
577:,
567:|
551:,
528:∉
494:∈
365:19
302:.
133:,
129:,
8657::
8396:e
8389:t
8382:v
8300:/
8215:P
7970:)
7756:)
7752:(
7649:∀
7644:!
7639:∃
7600:=
7595:↔
7590:→
7585:∧
7580:∨
7575:¬
7298:/
7294:/
7268:/
7079:)
7075:(
6962:∞
6952:3
6740:)
6638:e
6631:t
6624:v
6389:·
6373:)
6369:(
6336:)
6225:)
5985:e
5978:t
5971:v
5928:.
5905:.
5882:.
5856:.
5816:.
5790:.
5722:.
5695:.
5668:.
5641:.
5615:.
5598:.
5571:.
5545:.
5529:.
5512:.
5480::
5472::
5445:)
5427::
5398:.
5371:.
5344:.
5317:.
5290:.
5263:.
5236:.
5199:.
5197:3
5179:.
5152:.
5123:.
5076:.
5050:.
5048:8
5030:.
5006:.
4977:.
4975:2
4957:.
4930:.
4905:.
4869:.
4838:.
4836:4
4821:.
4794:.
4767:.
4740:.
4700:.
4673:.
4646:.
4619:.
4592:.
4563:.
4536:.
4524:5
4501:.
4499:1
4479:.
4452:.
4425:.
4394:m
4390:M
4256:x
4252:x
4248:x
4244:x
4191:A
4187:B
4183:B
4179:A
4175:B
4171:A
4081:.
4035:.
4031:)
4027:|
4021:n
4017:A
4002:3
3998:A
3989:2
3985:A
3976:1
3972:A
3967:|
3963:(
3957:1
3951:n
3946:)
3942:1
3935:(
3930:+
3915:+
3902:)
3897:|
3891:n
3887:A
3878:1
3872:n
3868:A
3863:|
3856:+
3852:|
3846:3
3842:A
3833:1
3829:A
3824:|
3820:+
3816:|
3810:2
3806:A
3797:1
3793:A
3788:|
3783:(
3766:)
3761:|
3756:n
3752:A
3748:|
3741:+
3737:|
3732:3
3728:A
3724:|
3720:+
3716:|
3711:2
3707:A
3703:|
3699:+
3695:|
3690:1
3686:A
3682:|
3677:(
3671:=
3667:|
3661:n
3657:A
3642:3
3638:A
3629:2
3625:A
3616:1
3612:A
3607:|
3580:.
3576:|
3572:B
3566:A
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