20:
8245:
8315:
5927:
2336:
2119:
788:
Another way to rigorously deal with ordered pairs is to define them formally in the context of set theory. This can be done in several ways and has the advantage that existence and the characteristic property can be proven from the axioms that define the set theory. One of the most cited versions of
6193:
cf introduction to Wiener's paper in van
Heijenoort 1967:224. van Heijenoort observes that the resulting set that represents the ordered pair "has a type higher by 2 than the elements (when they are of the same type)"; he offers references that show how, under certain circumstances, the type can be
4377:
4123:
1564:
925:
5253:
Early in the development of the set theory, before paradoxes were discovered, Cantor followed Frege by defining the ordered pair of two sets as the class of all relations that hold between these sets, assuming that the notion of relation is primitive:
5907:
499:
2331:{\displaystyle \pi _{2}(p)=\bigcup \left\{\left.a\in \bigcup p\,\right|\,\bigcup p\neq \bigcap p\rightarrow a\notin \bigcap p\right\}=\bigcup \left\{\left.a\in \{x,y\}\,\right|\,\{x,y\}\neq \{x\}\rightarrow a\notin \{x\}\right\}=\bigcup \{y\}=y.}
1692:
4842:
2025:
1903:
5653:
5338:
defined the ordered pair so that its projections could be proper classes as well as sets. (The
Kuratowski definition does not allow this.) He first defined ordered pairs whose projections are sets in Kuratowski's manner. He then
772:
A more satisfactory approach is to observe that the characteristic property of ordered pairs given above is all that is required to understand the role of ordered pairs in mathematics. Hence the ordered pair can be taken as a
4238:
4017:
1408:
1337:
761:
must be different, but in an ordered pair they may be equal and that while the order of listing the elements of a set doesn't matter, in an ordered pair changing the order of distinct entries changes the ordered pair.
827:
2446:
6175:(pbk.). van Heijenoort states the simplification this way: "By giving a definition of the ordered pair of two elements in terms of class operations, the note reduced the theory of relations to that of classes".
4659:
5439:
2112:
5034:
5510:
4884:
5150:
2605:
2762:
2920:
1205:
4573:
1094:
2835:
5769:
367:
5208:
5225:, defined as a set of ordered pairs, to have a type only 1 higher than the type of its arguments. This definition works only if the set of natural numbers is infinite. This is the case in
4704:
1785:
4191:
1395:
4453:
1004:
813:
of some sort. Hence if the ordered pair is not taken as primitive, it must be defined as a set. Several set-theoretic definitions of the ordered pair are given below( see also ).
793:, published in 1970. Even those mathematical textbooks that give an informal definition of ordered pairs will often mention the formal definition of Kuratowski in an exercise.
5311:
612:
6624:
5521:
582:. In such situations, the context will usually make it clear which meaning is intended. For additional clarification, the ordered pair may be denoted by the variant notation
1572:
350:
304:
4233:
4012:
785:, published in 1954. However, this approach also has its drawbacks as both the existence of ordered pairs and their characteristic property must be axiomatically assumed.
6389:
6342:
4506:
2655:
8772:
7299:
6362:
4907:
4688:
4473:
2503:
2476:
4421:
4163:
4143:
2364:
6409:
5221:, the QuineâRosser pair has the same type as its projections and hence is termed a "type-level" ordered pair. Hence this definition has the advantage of enabling a
1909:
5684:
2681:
2521:
The above
Kuratowski definition of the ordered pair is "adequate" in that it satisfies the characteristic property that an ordered pair must satisfy, namely that
1793:
1210:
5764:
5744:
4401:
4211:
7382:
6523:
6163:
Wiener's paper "A Simplification of the logic of relations" is reprinted, together with a valuable commentary on pages 224ff in van
Heijenoort, Jean (1967),
6135:
has argued that the set-theoretical implementations of the concept of the ordered pair is a paradigm for the clarification of philosophical ideas (see "
5918:
The definition helps to avoid so called accidental theorems like (a,a) = {{a}}, and {a} â (a,b), if
Kuratowski's definition (a,b) = {{a}, {a,b}} was used.
4578:
769:. However, as is sometimes pointed out, no harm will come from relying on this description and almost everyone thinks of ordered pairs in this manner.
5346:
2045:
6139:", section 53). The general notion of such definitions or implementations are discussed in Thomas Forster "Reasoning about theoretical entities".
5444:
7696:
8922:
7854:
6313:
6094:
6067:
6043:
1119:
6642:
4519:
8461:
8281:
7709:
7032:
4372:{\displaystyle \varphi (x):=\sigma =\{\sigma (\alpha )\mid \alpha \in x\}=(x\setminus \mathbb {N} )\cup \{n+1:n\in (x\cap \mathbb {N} )\}.}
1021:
2369:
8789:
7294:
6004:
4118:{\displaystyle \sigma (x):={\begin{cases}x,&{\text{if }}x\notin \mathbb {N} ,\\x+1,&{\text{if }}x\in \mathbb {N} .\end{cases}}}
7714:
7704:
7441:
6647:
7192:
6638:
4145:
increments its argument if it is a natural number and leaves it as is otherwise; the number 0 does not appear as functional value of
1559:{\displaystyle (\exists Y\in p:x\in Y)\land (\forall Y_{1},Y_{2}\in p:Y_{1}\neq Y_{2}\rightarrow (x\notin Y_{1}\lor x\notin Y_{2})).}
7850:
6172:
6118:
4923:
6425:
7947:
7691:
6516:
5039:
2524:
149:
8767:
7252:
6945:
6686:
5723:
2942:
2689:
920:{\displaystyle \left(a,b\right):=\left\{\left\{\left\{a\right\},\,\emptyset \right\},\,\left\{\left\{b\right\}\right\}\right\}.}
2929:
definition is merely a trivial variant of the
Kuratowski definition, and as such is of no independent interest. The definition
19:
8208:
7910:
7673:
7668:
7493:
6914:
6598:
2841:
1356:
8541:
8420:
8203:
7986:
7903:
7616:
7547:
7424:
6666:
2768:
8784:
7274:
8128:
7954:
7640:
6873:
8777:
7279:
5327:
5315:
This definition is inadmissible in most modern formalized set theories and is methodologically similar to defining the
8415:
8378:
7611:
7350:
6608:
6509:
5257:
2984:
version keeps having cardinality 2, which is something one might expect of any "pair", including any "ordered pair".)
8006:
8001:
5241:
showed that the existence of such a type-level ordered pair (or even a "type-raising by 1" ordered pair) implies the
5155:
7935:
7525:
6919:
6887:
6578:
806:
717:
In some introductory mathematics textbooks an informal (or intuitive) definition of ordered pair is given, such as
6652:
8466:
8358:
8346:
8341:
8225:
8174:
8071:
7569:
7530:
7007:
5952:
5514:
This renders possible pairs whose projections are proper classes. The QuineâRosser definition above also admits
2035:
2031:
789:
this definition is due to
Kuratowski (see below) and his definition was used in the second edition of Bourbaki's
765:
This "definition" is unsatisfactory because it is only descriptive and is based on an intuitive understanding of
8066:
6681:
8274:
7996:
7535:
7387:
7370:
7093:
6573:
5714:. Ordered triples which are defined as ordered pairs do not have this property with respect to ordered pairs.
1716:
6433:. Academia-Bruylant. The publisher has graciously consented to permit diffusion of this monograph via the web.
4847:
5245:. For an extensive discussion of the ordered pair in the context of Quinian set theories, see Holmes (1998).
4168:
8886:
8804:
8679:
8631:
8445:
8368:
7898:
7875:
7836:
7722:
7663:
7309:
7229:
7073:
7017:
6630:
6213:
6132:
3985:
5902:{\displaystyle f(a_{1},b_{1})=f(a_{2},b_{2}){\text{ if and only if }}a_{1}=a_{2}{\text{ and }}b_{1}=b_{2}.}
494:{\displaystyle (a_{1},b_{1})=(a_{2},b_{2}){\text{ if and only if }}a_{1}=a_{2}{\text{ and }}b_{1}=b_{2}.}
8838:
8719:
8531:
8351:
8188:
7915:
7893:
7860:
7753:
7599:
7584:
7557:
7508:
7392:
7327:
7152:
7118:
7113:
6987:
6818:
6795:
5912:
5222:
3762:
is in the left hand side, and thus in the right hand side. Because equal sets have equal elements, one of
2506:
971:
942:
249:
8927:
8754:
8668:
8588:
8568:
8546:
8118:
7971:
7763:
7481:
7217:
7123:
6982:
6967:
6848:
6823:
933:
8244:
6241:
This differs from
Hausdorff's definition in not requiring the two elements 0 and 1 to be distinct from
1687:{\displaystyle (\forall Y_{1},Y_{2}\in p:Y_{1}\neq Y_{2}\rightarrow (x\notin Y_{1}\lor x\notin Y_{2}))}
6444:
4837:{\displaystyle (A,B):=\varphi \cup \psi =\{\varphi (a):a\in A\}\cup \{\varphi (b)\cup \{0\}:b\in B\}.}
8932:
8828:
8818:
8652:
8583:
8536:
8476:
8363:
8091:
8053:
7930:
7734:
7574:
7498:
7476:
7304:
7262:
7161:
7128:
6992:
6780:
6691:
6204:
5987:
2934:
1105:
585:
161:
145:
4041:
8823:
8734:
8647:
8642:
8637:
8451:
8393:
8331:
8267:
8220:
8111:
8096:
8076:
8033:
7920:
7870:
7796:
7741:
7678:
7471:
7466:
7414:
7182:
7171:
6843:
6743:
6671:
6662:
6658:
6593:
6588:
5930:
4424:
4380:
2949:
2945:
309:
263:
4216:
3995:
8746:
8741:
8526:
8481:
8388:
8249:
8018:
7981:
7966:
7959:
7942:
7728:
7594:
7520:
7503:
7456:
7269:
7178:
7012:
6997:
6957:
6909:
6894:
6882:
6838:
6813:
6583:
6532:
6367:
6320:
5975:
4482:
2952:, then 2 is defined as the set {0, 1} = {0, {0}}, which is indistinguishable from the pair (0, 0)
2610:
2039:
7746:
7202:
6151:"Set-Theoretical Representations of Ordered Pairs and Their Adequacy for the Logic of Relations"
4430:
8603:
8440:
8432:
8403:
8373:
8304:
8184:
7991:
7801:
7791:
7683:
7564:
7399:
7375:
7156:
7140:
7045:
7022:
6899:
6868:
6833:
6728:
6563:
6347:
6292:
6168:
6114:
6090:
6063:
6039:
5999:
5242:
5238:
4892:
4664:
4458:
3981:
2481:
2454:
2020:{\displaystyle \bigcup p=\bigcup {\bigg \{}\{x\},\{x,y\}{\bigg \}}=\{x\}\cup \{x,y\}=\{x,y\}.}
810:
753:
This is usually followed by a comparison to a set of two elements; pointing out that in a set
516:
504:
241:
23:
4406:
4148:
4128:
2343:
8891:
8881:
8866:
8861:
8729:
8383:
8198:
8193:
8086:
8043:
7865:
7826:
7821:
7806:
7632:
7589:
7486:
7284:
7234:
6808:
6770:
6222:
5963:
2683:. There are other definitions, of similar or lesser complexity, that are equally adequate:
1898:{\displaystyle \bigcap p=\bigcap {\bigg \{}\{x\},\{x,y\}{\bigg \}}=\{x\}\cap \{x,y\}=\{x\},}
946:
778:
777:, whose associated axiom is the characteristic property. This was the approach taken by the
774:
579:
6394:
8760:
8698:
8516:
8336:
8179:
8169:
8123:
8106:
8061:
8023:
7925:
7845:
7652:
7579:
7552:
7540:
7446:
7360:
7334:
7289:
7257:
7058:
6860:
6803:
6753:
6718:
6676:
6429:
6136:
5660:
5234:
5226:
5218:
1015:
536:
245:
27:
6010:
2660:
5648:{\displaystyle (x,y,z)=(\{0\}\times s(x))\cup (\{1\}\times s(y))\cup (\{2\}\times s(z))}
8896:
8693:
8674:
8578:
8563:
8520:
8456:
8398:
8164:
8143:
8101:
8081:
7976:
7831:
7429:
7419:
7409:
7404:
7338:
7212:
7088:
6977:
6972:
6950:
6551:
5749:
5729:
4386:
4196:
3989:
3009:
2510:
821:
113:
5926:
5686:
which has an inserted empty set allows tuples to have the uniqueness property that if
8916:
8901:
8871:
8703:
8617:
8612:
8138:
7816:
7323:
7108:
7098:
7068:
7053:
6723:
6422:
5335:
575:
8851:
8846:
8664:
8593:
8410:
8314:
8038:
7885:
7786:
7778:
7658:
7606:
7515:
7451:
7434:
7365:
7224:
7083:
6785:
6568:
6486:
5515:
5331:
157:
1207:
When the first and the second coordinates are identical, the definition obtains:
8876:
8511:
8148:
8028:
7207:
7197:
7144:
6828:
6748:
6733:
6613:
6558:
5316:
5230:
5214:
961:
928:
153:
72:
144:(sometimes, lists in a computer science context) of length 2. Ordered pairs of
87:), is a pair of objects in which their order is significant. The ordered pair (
8856:
8724:
8627:
8290:
7078:
6933:
6904:
6710:
5441:
where the component
Cartesian products are Kuratowski pairs of sets and where
802:
6280:
8659:
8622:
8573:
8471:
8230:
8133:
7186:
7103:
7063:
7027:
6963:
6775:
6765:
6738:
5974:. In this context the characteristic property above is a consequence of the
6227:
6208:
1332:{\displaystyle (x,\ x)_{K}=\{\{x\},\{x,\ x\}\}=\{\{x\},\ \{x\}\}=\{\{x\}\}}
824:
proposed the first set theoretical definition of the ordered pair in 1914:
160:.) The entries of an ordered pair can be other ordered pairs, enabling the
4889:
Extracting all the elements of the pair that do not contain 0 and undoing
8215:
8013:
7461:
7166:
6760:
6272:
5518:
as projections. Similarly the triple is defined as a 3-tuple as follows:
1567:
1566:
In the case that the left and right coordinates are identical, the right
141:
5986:. While different objects may have the universal property, they are all
5966:
represents the set of ordered pairs, with the first element coming from
226:
of the pair. Alternatively, the objects are called the first and second
7811:
6603:
6464:
165:
31:
2448:, but the previous formula also takes into account the case when x=y)
8684:
8506:
6501:
2030:
This is how we can extract the first coordinate of a pair (using the
548:
6308:
2441:{\displaystyle \{y\}=\{\left.a\in \{x,y\}\,\right|\,a\notin \{x\}\}}
6165:
From Frege to Gödel: A Source Book in
Mathematical Logic, 1979â1931
6060:
Sets, Functions and Logic / An Introduction to Abstract Mathematics
8556:
8323:
7355:
6701:
6546:
5925:
4917:
can be recovered from the elements of the pair that do contain 0.
4654:{\displaystyle \psi (x):=\sigma \cup \{0\}=\varphi (x)\cup \{0\}.}
574:
notation may be used for other purposes, most notably as denoting
137:
5319:
of a set as the class of all sets equipotent with the given set.
1096:"where 1 and 2 are two distinct objects different from a and b."
5911:
This definition is acceptable because this extension of ZF is a
3837:, and this combination contradicts the axiom of regularity, as {
2933:
is so-called because it requires two rather than three pairs of
927:
He observed that this definition made it possible to define the
8263:
6505:
5982:
can be identified with morphisms from 1 (a one element set) to
5434:{\displaystyle (x,y)=(\{0\}\times s(x))\cup (\{1\}\times s(y))}
2950:
von Neumann's set-theoretic construction of the natural numbers
964:
where all elements in a class must be of the same "type". With
5726:(ZF) axiomatically by just adding to ZF a new function symbol
2107:{\displaystyle \pi _{1}(p)=\bigcup \bigcap p=\bigcup \{x\}=x.}
8259:
5029:{\displaystyle (\{\{a,0\},\{b,c,1\}\},\{\{d,2\},\{e,f,3\}\})}
4479:
simply increments every natural number in it. In particular,
6184:
cf introduction to Wiener's paper in van Heijenoort 1967:224
6150:
5746:
of arity 2 (it is usually omitted) and a defining axiom for
5505:{\displaystyle s(x)=\{\emptyset \}\cup \{\{t\}\mid t\in x\}}
2988:
Proving that definitions satisfy the characteristic property
5217:
and in outgrowths thereof such as the axiomatic set theory
5145:{\displaystyle \{\{a,1\},\{b,c,2\},\{d,3,0\},\{e,f,4,0\}\}}
4111:
2600:{\displaystyle (a,b)=(x,y)\leftrightarrow (a=x)\land (b=y)}
2389:
2223:
2154:
6087:
Proof, Logic, and Conjecture / The Mathematician's Toolbox
6017:(definition Def5 of "ordered pairs" as { { x,y }, { x } })
2757:{\displaystyle (a,b)_{\text{reverse}}:=\{\{b\},\{a,b\}\};}
2607:. In particular, it adequately expresses 'order', in that
3841:} has no minimal element under the relation "element of."
1108:
offered the now-accepted definition of the ordered pair (
6209:"Sur la notion de l'ordre dans la Théorie des Ensembles"
3984:(1953) employed a definition of the ordered pair due to
2915:{\displaystyle (a,b)_{\text{01}}:=\{\{0,a\},\{1,b\}\}.}
1200:{\displaystyle (a,\ b)_{K}\;:=\ \{\{a\},\ \{a,\ b\}\}.}
6062:(3rd ed.), Chapman & Hall / CRC, p. 79,
4568:{\displaystyle \varphi (x)\neq \{0\}\cup \varphi (y).}
4508:
does never contain the number 0, so that for any sets
968:
nested within an additional set, its type is equal to
588:
507:
of all ordered pairs whose first entry is in some set
252:) are defined in terms of ordered pairs, cf. picture.
6397:
6370:
6350:
6323:
6038:(4th ed.), Pearson / Prentice Hall, p. 50,
5772:
5752:
5732:
5663:
5524:
5447:
5349:
5260:
5158:
5042:
4926:
4895:
4850:
4707:
4667:
4581:
4522:
4485:
4461:
4433:
4409:
4389:
4241:
4219:
4199:
4171:
4151:
4131:
4020:
3998:
2844:
2771:
2692:
2663:
2613:
2527:
2484:
2457:
2372:
2346:
2122:
2048:
1912:
1796:
1719:
1575:
1411:
1359:
1213:
1122:
1089:{\displaystyle (a,b):=\left\{\{a,1\},\{b,2\}\right\}}
1024:
974:
830:
370:
312:
266:
6260:
Lectures in Logic and Set Theory. Vol. 2: Set Theory
2830:{\displaystyle (a,b)_{\text{short}}:=\{a,\{a,b\}\};}
2509:, in the sense that their domains and codomains are
2116:
This is how the second coordinate can be extracted:
8837:
8800:
8712:
8602:
8490:
8431:
8322:
8297:
8157:
8052:
7884:
7777:
7629:
7322:
7245:
7139:
7043:
6932:
6859:
6794:
6709:
6700:
6622:
6539:
6317:, on: Boise State, March 29, 2009. The author uses
5978:of the product and the fact that elements of a set
2941:satisfies the characteristic property requires the
809:, then all mathematical objects must be defined as
6485:
6403:
6383:
6356:
6336:
5901:
5758:
5738:
5678:
5647:
5504:
5433:
5305:
5202:
5144:
5028:
4901:
4878:
4836:
4682:
4653:
4567:
4500:
4467:
4447:
4415:
4395:
4371:
4227:
4205:
4185:
4157:
4137:
4117:
4006:
2914:
2829:
2756:
2675:
2649:
2599:
2497:
2470:
2440:
2366:, then the set {y} could be obtained more simply:
2358:
2330:
2106:
2019:
1897:
1779:
1686:
1558:
1389:
1331:
1199:
1088:
998:
919:
606:
493:
344:
298:
156:since an ordered pair need not be an element of a
1961:
1927:
1845:
1811:
6283:Also see Tourlakis (2003), Proposition III.10.1.
4014:be the set of natural numbers and define first
6262:. Cambridge Univ. Press. Proposition III.10.1.
5203:{\displaystyle a,b,c,d,e,f\notin \mathbb {N} }
8275:
6517:
675:), respectively. In contexts where arbitrary
8:
5624:
5618:
5591:
5585:
5558:
5552:
5499:
5484:
5478:
5475:
5469:
5463:
5410:
5404:
5377:
5371:
5297:
5279:
5139:
5136:
5112:
5106:
5088:
5082:
5064:
5058:
5046:
5043:
5020:
5017:
4999:
4993:
4981:
4978:
4972:
4969:
4951:
4945:
4933:
4930:
4828:
4813:
4807:
4789:
4783:
4756:
4645:
4639:
4618:
4612:
4544:
4538:
4363:
4325:
4299:
4272:
2906:
2903:
2891:
2885:
2873:
2870:
2821:
2818:
2806:
2797:
2748:
2745:
2733:
2727:
2721:
2718:
2435:
2432:
2426:
2410:
2398:
2385:
2379:
2373:
2316:
2310:
2296:
2290:
2278:
2272:
2266:
2254:
2244:
2232:
2092:
2086:
2011:
1999:
1993:
1981:
1975:
1969:
1956:
1944:
1938:
1932:
1889:
1883:
1877:
1865:
1859:
1853:
1840:
1828:
1822:
1816:
1774:
1771:
1759:
1753:
1747:
1744:
1326:
1323:
1317:
1314:
1308:
1305:
1299:
1290:
1284:
1281:
1275:
1272:
1257:
1251:
1245:
1242:
1191:
1188:
1173:
1164:
1158:
1155:
1078:
1066:
1060:
1048:
993:
984:
978:
975:
601:
589:
175:objects). For example, the ordered triple (
8282:
8268:
8260:
7343:
6938:
6706:
6524:
6510:
6502:
6423:Elementary Set Theory with a Universal Set
6167:, Harvard University Press, Cambridge MA,
2968:are of the same type, the elements of the
1148:
797:Defining the ordered pair using set theory
6396:
6375:
6369:
6349:
6328:
6322:
6226:
6109:Fletcher, Peter; Patty, C. Wayne (1988),
5890:
5877:
5868:
5862:
5849:
5840:
5831:
5818:
5796:
5783:
5771:
5751:
5731:
5662:
5523:
5446:
5348:
5259:
5196:
5195:
5157:
5041:
4925:
4894:
4849:
4706:
4666:
4580:
4521:
4484:
4460:
4432:
4408:
4388:
4356:
4355:
4315:
4314:
4240:
4221:
4220:
4218:
4198:
4179:
4178:
4170:
4150:
4130:
4101:
4100:
4089:
4064:
4063:
4052:
4036:
4019:
4000:
3999:
3997:
3988:which requires a prior definition of the
2861:
2843:
2788:
2770:
2709:
2691:
2662:
2612:
2526:
2489:
2483:
2462:
2456:
2419:
2413:
2371:
2345:
2253:
2247:
2175:
2169:
2127:
2121:
2053:
2047:
1960:
1959:
1926:
1925:
1911:
1844:
1843:
1810:
1809:
1795:
1780:{\displaystyle p=(x,y)=\{\{x\},\{x,y\}\}}
1718:
1672:
1653:
1631:
1618:
1599:
1586:
1574:
1541:
1522:
1500:
1487:
1468:
1455:
1410:
1358:
1233:
1212:
1142:
1121:
1023:
973:
960:} to make the definition compatible with
889:
877:
829:
741:is a notation specifying the two objects
614:, but this notation also has other uses.
587:
482:
469:
460:
454:
441:
432:
423:
410:
391:
378:
369:
333:
320:
311:
287:
274:
265:
199:)), i.e., as one pair nested in another.
34:is associated with the set of all pairs (
6036:Analysis / With an Introduction to Proof
5722:Ordered pairs can also be introduced in
4879:{\displaystyle \varphi ''A\cup \psi ''B}
3786:}, then by similar reasoning as above, {
18:
6026:
4311:
4186:{\displaystyle x\setminus \mathbb {N} }
4175:
6089:, W. H. Freeman and Co., p. 164,
6080:
6078:
1390:{\displaystyle \forall Y\in p:x\in Y.}
1014:About the same time as Wiener (1914),
511:and whose second entry is in some set
95:) is different from the ordered pair (
7:
999:{\displaystyle \{\{a\},\emptyset \}}
152:. (Technically, this is an abuse of
148:are sometimes called 2-dimensional
5466:
4693:Finally, define the ordered pair (
4690:does always contain the number 0.
3872:}, again contradicting regularity.
2956:. Yet another disadvantage of the
1579:
1448:
1415:
1360:
990:
878:
14:
6111:Foundations of Higher Mathematics
6015:Journal of Formalized Mathematics
3790:} is in the right hand side, so {
8313:
8243:
5306:{\displaystyle (x,y)=\{R:xRy\}.}
607:{\textstyle \langle a,b\rangle }
26:associates to each point in the
6466:Set Theory From Cantor to Cohen
2960:pair is the fact that, even if
713:Informal and formal definitions
697:) is a common notation for the
6443:Frege, Gottlob (1893). "144".
6011:TarskiâGrothendieck Set Theory
6005:TarskiâGrothendieck set theory
5837:
5811:
5802:
5776:
5673:
5667:
5642:
5639:
5633:
5615:
5609:
5606:
5600:
5582:
5576:
5573:
5567:
5549:
5543:
5525:
5457:
5451:
5428:
5425:
5419:
5401:
5395:
5392:
5386:
5368:
5362:
5350:
5273:
5261:
5023:
4927:
4801:
4795:
4768:
4762:
4750:
4744:
4735:
4729:
4720:
4708:
4677:
4671:
4633:
4627:
4606:
4600:
4591:
4585:
4559:
4553:
4532:
4526:
4495:
4489:
4360:
4346:
4319:
4305:
4284:
4278:
4266:
4260:
4251:
4245:
4193:is the set of the elements of
4030:
4024:
2858:
2845:
2785:
2772:
2706:
2693:
2644:
2632:
2626:
2614:
2594:
2582:
2576:
2564:
2561:
2558:
2546:
2540:
2528:
2281:
2191:
2139:
2133:
2065:
2059:
1738:
1726:
1681:
1678:
1640:
1637:
1576:
1550:
1547:
1509:
1506:
1445:
1439:
1412:
1230:
1214:
1139:
1123:
1037:
1025:
429:
403:
397:
371:
339:
313:
293:
267:
136:Ordered pairs are also called
125:}, equals the unordered pair {
1:
8204:History of mathematical logic
6452:. Jena: Verlag Hermann Pohle.
5657:The use of the singleton set
345:{\displaystyle (a_{2},b_{2})}
299:{\displaystyle (a_{1},b_{1})}
8923:Basic concepts in set theory
8129:Primitive recursive function
4228:{\displaystyle \mathbb {N} }
4007:{\displaystyle \mathbb {N} }
1401:is the second coordinate of
16:Pair of mathematical objects
6446:Grundgesetze der Arithmetik
6384:{\displaystyle \sigma _{2}}
6337:{\displaystyle \sigma _{1}}
5970:and the second coming from
5724:ZermeloâFraenkel set theory
4501:{\displaystyle \varphi (x)}
4455:as well. Applying function
3925:, contradicting regularity.
2972:pair are not. (However, if
2943:ZermeloâFraenkel set theory
2650:{\displaystyle (a,b)=(b,a)}
2032:iterated-operation notation
1349:is the first coordinate of
941:had taken types, and hence
352:be ordered pairs. Then the
8951:
8773:von NeumannâBernaysâGödel
7193:SchröderâBernstein theorem
6920:Monadic predicate calculus
6579:Foundations of mathematics
6484:Morse, Anthony P. (1965).
6463:Kanamori, Akihiro (2007).
6420:Holmes, M. Randall (1998)
6009:Trybulec, Andrzej, 1989, "
5842: if and only if
4448:{\displaystyle \sigma ''x}
1099:
1009:
816:
434: if and only if
8574:One-to-one correspondence
8311:
8239:
8226:Philosophy of mathematics
8175:Automated theorem proving
7346:
7300:Von NeumannâBernaysâGödel
6941:
6275:proof of the adequacy of
6258:Tourlakis, George (2003)
3528:}, a contradiction. Thus
3446:, which also contradicts
1694:is trivially true, since
1018:proposed his definition:
807:foundation of mathematics
30:an ordered pair. The red
6357:{\displaystyle \varphi }
6297:Logic for Mathematicians
6113:, PWS-Kent, p. 80,
6085:Wolf, Robert S. (1998),
4902:{\displaystyle \varphi }
4886:in alternate notation).
4701:) as the disjoint union
4683:{\displaystyle \psi (x)}
4468:{\displaystyle \varphi }
2948:. Moreover, if one uses
2498:{\displaystyle \pi _{2}}
2471:{\displaystyle \pi _{1}}
1405:" can be formulated as:
1353:" can be formulated as:
1341:Given some ordered pair
679:-tuples are considered,
364:of the ordered pair is:
234:, or the left and right
7876:Self-verifying theories
7697:Tarski's axiomatization
6648:Tarski's undefinability
6643:incompleteness theorems
6214:Fundamenta Mathematicae
6034:Lay, Steven R. (2005),
5328:MorseâKelley set theory
5249:CantorâFrege definition
4416:{\displaystyle \sigma }
4158:{\displaystyle \sigma }
4138:{\displaystyle \sigma }
3977:QuineâRosser definition
2359:{\displaystyle x\neq y}
1100:Kuratowski's definition
230:, the first and second
8532:Constructible universe
8359:Constructibility (V=L)
8250:Mathematics portal
7861:Proof of impossibility
7509:propositional variable
6819:Propositional calculus
6405:
6385:
6358:
6338:
6228:10.4064/fm-2-1-161-171
6058:Devlin, Keith (2004),
5948:
5913:conservative extension
5903:
5760:
5740:
5680:
5649:
5506:
5435:
5307:
5204:
5146:
5030:
4920:For example, the pair
4903:
4880:
4838:
4684:
4655:
4569:
4502:
4469:
4449:
4417:
4397:
4373:
4229:
4207:
4187:
4159:
4139:
4119:
4008:
3397:}} would also equal {{
2916:
2831:
2758:
2677:
2651:
2601:
2499:
2472:
2442:
2360:
2332:
2108:
2036:arbitrary intersection
2021:
1899:
1781:
1688:
1560:
1391:
1333:
1201:
1090:
1010:Hausdorff's definition
1000:
921:
751:
627:is usually denoted by
608:
495:
346:
300:
164:definition of ordered
68:
8755:Principia Mathematica
8589:Transfinite induction
8448:(i.e. set difference)
8119:Kolmogorov complexity
8072:Computably enumerable
7972:Model complete theory
7764:Principia Mathematica
6824:Propositional formula
6653:BanachâTarski paradox
6406:
6404:{\displaystyle \psi }
6386:
6359:
6339:
5951:A category-theoretic
5929:
5904:
5761:
5741:
5681:
5650:
5507:
5436:
5308:
5205:
5147:
5031:
4904:
4881:
4839:
4685:
4656:
4570:
4503:
4470:
4450:
4418:
4398:
4374:
4230:
4208:
4188:
4160:
4140:
4120:
4009:
3536:is the case, so that
2917:
2832:
2759:
2678:
2652:
2602:
2507:generalized functions
2500:
2473:
2443:
2361:
2333:
2109:
2022:
1900:
1782:
1689:
1561:
1392:
1334:
1202:
1091:
1001:
939:Principia Mathematica
934:Principia Mathematica
922:
719:
609:
496:
347:
301:
238:of the ordered pair.
202:In the ordered pair (
187:) can be defined as (
22:
8829:Burali-Forti paradox
8584:Set-builder notation
8537:Continuum hypothesis
8477:Symmetric difference
8067:ChurchâTuring thesis
8054:Computability theory
7263:continuum hypothesis
6781:Square of opposition
6639:Gödel's completeness
6395:
6368:
6348:
6321:
5988:naturally isomorphic
5933:for the set product
5770:
5750:
5730:
5718:Axiomatic definition
5679:{\displaystyle s(x)}
5661:
5522:
5445:
5347:
5258:
5156:
5040:
4924:
4893:
4848:
4705:
4665:
4579:
4520:
4483:
4459:
4431:
4407:
4387:
4239:
4217:
4197:
4169:
4149:
4129:
4018:
3996:
3885:Again, we see that {
3774:} must be the case.
2842:
2769:
2690:
2661:
2611:
2525:
2482:
2455:
2370:
2344:
2120:
2046:
1910:
1794:
1717:
1573:
1409:
1357:
1211:
1120:
1106:Kazimierz Kuratowski
1022:
972:
828:
721:For any two objects
701:-th component of an
586:
368:
310:
264:
111:. In contrast, the
8790:TarskiâGrothendieck
8221:Mathematical object
8112:P versus NP problem
8077:Computable function
7871:Reverse mathematics
7797:Logical consequence
7674:primitive recursive
7669:elementary function
7442:Free/bound variable
7295:TarskiâGrothendieck
6814:Logical connectives
6744:Logical equivalence
6594:Logical consequence
6205:Kuratowski, Casimir
5931:Commutative diagram
5694:-tuple and b is an
2946:axiom of regularity
2676:{\displaystyle b=a}
1710:is never the case.
945:of all arities, as
817:Wiener's definition
801:If one agrees that
729:, the ordered pair
617:The left and right
8379:Limitation of size
8019:Transfer principle
7982:Semantics of logic
7967:Categorical theory
7943:Non-standard model
7457:Logical connective
6584:Information theory
6533:Mathematical logic
6474:p. 22, footnote 59
6428:2011-04-11 at the
6401:
6381:
6354:
6334:
6309:Holmes, M. Randall
6194:reduced to 1 or 0.
5976:universal property
5949:
5899:
5756:
5736:
5676:
5645:
5502:
5431:
5330:makes free use of
5303:
5200:
5142:
5026:
4899:
4876:
4834:
4680:
4651:
4565:
4498:
4465:
4445:
4413:
4393:
4369:
4225:
4203:
4183:
4155:
4135:
4115:
4110:
4004:
3409:which contradicts
2912:
2827:
2754:
2673:
2647:
2597:
2495:
2468:
2438:
2356:
2328:
2104:
2017:
1895:
1777:
1684:
1556:
1387:
1329:
1197:
1086:
996:
917:
604:
491:
342:
296:
242:Cartesian products
171:(ordered lists of
69:
8910:
8909:
8819:Russell's paradox
8768:ZermeloâFraenkel
8669:Dedekind-infinite
8542:Diagonal argument
8441:Cartesian product
8305:Set (mathematics)
8257:
8256:
8189:Abstract category
7992:Theories of truth
7802:Rule of inference
7792:Natural deduction
7773:
7772:
7318:
7317:
7023:Cartesian product
6928:
6927:
6834:Many-valued logic
6809:Boolean functions
6692:Russell's paradox
6667:diagonal argument
6564:First-order logic
6492:. Academic Press.
6293:J. Barkley Rosser
6149:Dipert, Randall.
6096:978-0-7167-3050-7
6069:978-1-58488-449-1
6045:978-0-13-148101-5
6000:Cartesian product
5871:
5843:
5759:{\displaystyle f}
5739:{\displaystyle f}
5243:axiom of infinity
5239:J. Barkley Rosser
4425:sometimes denoted
4396:{\displaystyle x}
4206:{\displaystyle x}
4092:
4055:
3921:is an element of
3856:is an element of
3500:were true, then {
3249:. By hypothesis,
3233:}, which implies
2864:
2791:
2712:
1298:
1268:
1225:
1184:
1172:
1154:
1134:
749:, in that order.
517:Cartesian product
463:
435:
218:, and the object
24:Analytic geometry
8940:
8892:Bertrand Russell
8882:John von Neumann
8867:Abraham Fraenkel
8862:Richard Dedekind
8824:Suslin's problem
8735:Cantor's theorem
8452:De Morgan's laws
8317:
8284:
8277:
8270:
8261:
8248:
8247:
8199:History of logic
8194:Category of sets
8087:Decision problem
7866:Ordinal analysis
7807:Sequent calculus
7705:Boolean algebras
7645:
7644:
7619:
7590:logical/constant
7344:
7330:
7253:ZermeloâFraenkel
7004:Set operations:
6939:
6876:
6707:
6687:LöwenheimâSkolem
6574:Formal semantics
6526:
6519:
6512:
6503:
6494:
6493:
6491:
6488:A Theory of Sets
6481:
6475:
6473:
6471:
6460:
6454:
6453:
6451:
6440:
6434:
6418:
6412:
6410:
6408:
6407:
6402:
6390:
6388:
6387:
6382:
6380:
6379:
6363:
6361:
6360:
6355:
6343:
6341:
6340:
6335:
6333:
6332:
6314:On Ordered Pairs
6306:
6300:
6290:
6284:
6269:
6263:
6256:
6250:
6239:
6233:
6232:
6230:
6201:
6195:
6191:
6185:
6182:
6176:
6161:
6155:
6154:
6146:
6140:
6130:
6124:
6123:
6106:
6100:
6099:
6082:
6073:
6072:
6055:
6049:
6048:
6031:
5964:category of sets
5908:
5906:
5905:
5900:
5895:
5894:
5882:
5881:
5872:
5869:
5867:
5866:
5854:
5853:
5844:
5841:
5836:
5835:
5823:
5822:
5801:
5800:
5788:
5787:
5765:
5763:
5762:
5757:
5745:
5743:
5742:
5737:
5685:
5683:
5682:
5677:
5654:
5652:
5651:
5646:
5511:
5509:
5508:
5503:
5440:
5438:
5437:
5432:
5323:Morse definition
5312:
5310:
5309:
5304:
5209:
5207:
5206:
5201:
5199:
5151:
5149:
5148:
5143:
5035:
5033:
5032:
5027:
4908:
4906:
4905:
4900:
4885:
4883:
4882:
4877:
4872:
4858:
4843:
4841:
4840:
4835:
4689:
4687:
4686:
4681:
4660:
4658:
4657:
4652:
4575:Further, define
4574:
4572:
4571:
4566:
4507:
4505:
4504:
4499:
4474:
4472:
4471:
4466:
4454:
4452:
4451:
4446:
4441:
4422:
4420:
4419:
4414:
4402:
4400:
4399:
4394:
4378:
4376:
4375:
4370:
4359:
4318:
4234:
4232:
4231:
4226:
4224:
4212:
4210:
4209:
4204:
4192:
4190:
4189:
4184:
4182:
4164:
4162:
4161:
4156:
4144:
4142:
4141:
4136:
4124:
4122:
4121:
4116:
4114:
4113:
4104:
4093:
4090:
4067:
4056:
4053:
4013:
4011:
4010:
4005:
4003:
2921:
2919:
2918:
2913:
2866:
2865:
2862:
2836:
2834:
2833:
2828:
2793:
2792:
2789:
2763:
2761:
2760:
2755:
2714:
2713:
2710:
2682:
2680:
2679:
2674:
2657:is false unless
2656:
2654:
2653:
2648:
2606:
2604:
2603:
2598:
2504:
2502:
2501:
2496:
2494:
2493:
2477:
2475:
2474:
2469:
2467:
2466:
2447:
2445:
2444:
2439:
2418:
2414:
2365:
2363:
2362:
2357:
2337:
2335:
2334:
2329:
2303:
2299:
2252:
2248:
2210:
2206:
2174:
2170:
2132:
2131:
2113:
2111:
2110:
2105:
2058:
2057:
2026:
2024:
2023:
2018:
1965:
1964:
1931:
1930:
1904:
1902:
1901:
1896:
1849:
1848:
1815:
1814:
1786:
1784:
1783:
1778:
1709:
1693:
1691:
1690:
1685:
1677:
1676:
1658:
1657:
1636:
1635:
1623:
1622:
1604:
1603:
1591:
1590:
1565:
1563:
1562:
1557:
1546:
1545:
1527:
1526:
1505:
1504:
1492:
1491:
1473:
1472:
1460:
1459:
1396:
1394:
1393:
1388:
1345:, the property "
1338:
1336:
1335:
1330:
1296:
1266:
1238:
1237:
1223:
1206:
1204:
1203:
1198:
1182:
1170:
1152:
1147:
1146:
1132:
1095:
1093:
1092:
1087:
1085:
1081:
1005:
1003:
1002:
997:
926:
924:
923:
918:
913:
909:
908:
904:
885:
881:
873:
849:
845:
805:is an appealing
775:primitive notion
760:
756:
748:
744:
740:
728:
724:
692:
691:
682:
665:
652:
641:
630:
622:
621:
613:
611:
610:
605:
580:real number line
573:
500:
498:
497:
492:
487:
486:
474:
473:
464:
461:
459:
458:
446:
445:
436:
433:
428:
427:
415:
414:
396:
395:
383:
382:
351:
349:
348:
343:
338:
337:
325:
324:
305:
303:
302:
297:
292:
291:
279:
278:
246:binary relations
66:
60:
58:
57:
54:
51:
8950:
8949:
8943:
8942:
8941:
8939:
8938:
8937:
8913:
8912:
8911:
8906:
8833:
8812:
8796:
8761:New Foundations
8708:
8598:
8517:Cardinal number
8500:
8486:
8427:
8318:
8309:
8293:
8288:
8258:
8253:
8242:
8235:
8180:Category theory
8170:Algebraic logic
8153:
8124:Lambda calculus
8062:Church encoding
8048:
8024:Truth predicate
7880:
7846:Complete theory
7769:
7638:
7634:
7630:
7625:
7617:
7337: and
7333:
7328:
7314:
7290:New Foundations
7258:axiom of choice
7241:
7203:Gödel numbering
7143: and
7135:
7039:
6924:
6874:
6855:
6804:Boolean algebra
6790:
6754:Equiconsistency
6719:Classical logic
6696:
6677:Halting problem
6665: and
6641: and
6629: and
6628:
6623:Theorems (
6618:
6535:
6530:
6499:
6497:
6483:
6482:
6478:
6469:
6462:
6461:
6457:
6449:
6442:
6441:
6437:
6430:Wayback Machine
6419:
6415:
6393:
6392:
6371:
6366:
6365:
6346:
6345:
6324:
6319:
6318:
6307:
6303:
6291:
6287:
6281:here (opthreg).
6270:
6266:
6257:
6253:
6240:
6236:
6203:
6202:
6198:
6192:
6188:
6183:
6179:
6162:
6158:
6148:
6147:
6143:
6137:Word and Object
6131:
6127:
6121:
6108:
6107:
6103:
6097:
6084:
6083:
6076:
6070:
6057:
6056:
6052:
6046:
6033:
6032:
6028:
6024:
5996:
5946:
5939:
5924:
5922:Category theory
5886:
5873:
5870: and
5858:
5845:
5827:
5814:
5792:
5779:
5768:
5767:
5748:
5747:
5728:
5727:
5720:
5659:
5658:
5520:
5519:
5443:
5442:
5345:
5344:
5325:
5256:
5255:
5251:
5154:
5153:
5038:
5037:
4922:
4921:
4891:
4890:
4865:
4851:
4846:
4845:
4703:
4702:
4663:
4662:
4577:
4576:
4518:
4517:
4481:
4480:
4457:
4456:
4434:
4429:
4428:
4405:
4404:
4385:
4384:
4237:
4236:
4215:
4214:
4195:
4194:
4167:
4166:
4147:
4146:
4127:
4126:
4109:
4108:
4087:
4072:
4071:
4050:
4037:
4016:
4015:
3994:
3993:
3990:natural numbers
3979:
3735:
3727:
3684:
3676:
3630:
3622:
3614:
3606:
3592:
3568:
3560:
3389:}}. But then {{
3300:
3288:
3209:
3197:
3142:
3097:
3085:
3034:
2990:
2955:
2937:. Proving that
2857:
2840:
2839:
2784:
2767:
2766:
2705:
2688:
2687:
2659:
2658:
2609:
2608:
2523:
2522:
2519:
2485:
2480:
2479:
2458:
2453:
2452:
2391:
2388:
2368:
2367:
2342:
2341:
2225:
2222:
2221:
2217:
2156:
2153:
2152:
2148:
2123:
2118:
2117:
2049:
2044:
2043:
2040:arbitrary union
1908:
1907:
1792:
1791:
1715:
1714:
1708:
1701:
1695:
1668:
1649:
1627:
1614:
1595:
1582:
1571:
1570:
1537:
1518:
1496:
1483:
1464:
1451:
1407:
1406:
1355:
1354:
1229:
1209:
1208:
1138:
1118:
1117:
1102:
1047:
1043:
1020:
1019:
1016:Felix Hausdorff
1012:
970:
969:
956:}} instead of {
894:
890:
863:
862:
858:
857:
853:
835:
831:
826:
825:
819:
799:
758:
754:
746:
742:
730:
726:
722:
715:
690:
685:
684:
683:
680:
670:
663:
657:
650:
644:
639:
633:
628:
619:
618:
584:
583:
563:
537:binary relation
478:
465:
462: and
450:
437:
419:
406:
387:
374:
366:
365:
329:
316:
308:
307:
283:
270:
262:
261:
258:
55:
52:
47:
46:
44:
43:
28:Euclidean plane
17:
12:
11:
5:
8948:
8947:
8944:
8936:
8935:
8930:
8925:
8915:
8914:
8908:
8907:
8905:
8904:
8899:
8897:Thoralf Skolem
8894:
8889:
8884:
8879:
8874:
8869:
8864:
8859:
8854:
8849:
8843:
8841:
8835:
8834:
8832:
8831:
8826:
8821:
8815:
8813:
8811:
8810:
8807:
8801:
8798:
8797:
8795:
8794:
8793:
8792:
8787:
8782:
8781:
8780:
8765:
8764:
8763:
8751:
8750:
8749:
8738:
8737:
8732:
8727:
8722:
8716:
8714:
8710:
8709:
8707:
8706:
8701:
8696:
8691:
8682:
8677:
8672:
8662:
8657:
8656:
8655:
8650:
8645:
8635:
8625:
8620:
8615:
8609:
8607:
8600:
8599:
8597:
8596:
8591:
8586:
8581:
8579:Ordinal number
8576:
8571:
8566:
8561:
8560:
8559:
8554:
8544:
8539:
8534:
8529:
8524:
8514:
8509:
8503:
8501:
8499:
8498:
8495:
8491:
8488:
8487:
8485:
8484:
8479:
8474:
8469:
8464:
8459:
8457:Disjoint union
8454:
8449:
8443:
8437:
8435:
8429:
8428:
8426:
8425:
8424:
8423:
8418:
8407:
8406:
8404:Martin's axiom
8401:
8396:
8391:
8386:
8381:
8376:
8371:
8369:Extensionality
8366:
8361:
8356:
8355:
8354:
8349:
8344:
8334:
8328:
8326:
8320:
8319:
8312:
8310:
8308:
8307:
8301:
8299:
8295:
8294:
8289:
8287:
8286:
8279:
8272:
8264:
8255:
8254:
8240:
8237:
8236:
8234:
8233:
8228:
8223:
8218:
8213:
8212:
8211:
8201:
8196:
8191:
8182:
8177:
8172:
8167:
8165:Abstract logic
8161:
8159:
8155:
8154:
8152:
8151:
8146:
8144:Turing machine
8141:
8136:
8131:
8126:
8121:
8116:
8115:
8114:
8109:
8104:
8099:
8094:
8084:
8082:Computable set
8079:
8074:
8069:
8064:
8058:
8056:
8050:
8049:
8047:
8046:
8041:
8036:
8031:
8026:
8021:
8016:
8011:
8010:
8009:
8004:
7999:
7989:
7984:
7979:
7977:Satisfiability
7974:
7969:
7964:
7963:
7962:
7952:
7951:
7950:
7940:
7939:
7938:
7933:
7928:
7923:
7918:
7908:
7907:
7906:
7901:
7894:Interpretation
7890:
7888:
7882:
7881:
7879:
7878:
7873:
7868:
7863:
7858:
7848:
7843:
7842:
7841:
7840:
7839:
7829:
7824:
7814:
7809:
7804:
7799:
7794:
7789:
7783:
7781:
7775:
7774:
7771:
7770:
7768:
7767:
7759:
7758:
7757:
7756:
7751:
7750:
7749:
7744:
7739:
7719:
7718:
7717:
7715:minimal axioms
7712:
7701:
7700:
7699:
7688:
7687:
7686:
7681:
7676:
7671:
7666:
7661:
7648:
7646:
7627:
7626:
7624:
7623:
7622:
7621:
7609:
7604:
7603:
7602:
7597:
7592:
7587:
7577:
7572:
7567:
7562:
7561:
7560:
7555:
7545:
7544:
7543:
7538:
7533:
7528:
7518:
7513:
7512:
7511:
7506:
7501:
7491:
7490:
7489:
7484:
7479:
7474:
7469:
7464:
7454:
7449:
7444:
7439:
7438:
7437:
7432:
7427:
7422:
7412:
7407:
7405:Formation rule
7402:
7397:
7396:
7395:
7390:
7380:
7379:
7378:
7368:
7363:
7358:
7353:
7347:
7341:
7324:Formal systems
7320:
7319:
7316:
7315:
7313:
7312:
7307:
7302:
7297:
7292:
7287:
7282:
7277:
7272:
7267:
7266:
7265:
7260:
7249:
7247:
7243:
7242:
7240:
7239:
7238:
7237:
7227:
7222:
7221:
7220:
7213:Large cardinal
7210:
7205:
7200:
7195:
7190:
7176:
7175:
7174:
7169:
7164:
7149:
7147:
7137:
7136:
7134:
7133:
7132:
7131:
7126:
7121:
7111:
7106:
7101:
7096:
7091:
7086:
7081:
7076:
7071:
7066:
7061:
7056:
7050:
7048:
7041:
7040:
7038:
7037:
7036:
7035:
7030:
7025:
7020:
7015:
7010:
7002:
7001:
7000:
6995:
6985:
6980:
6978:Extensionality
6975:
6973:Ordinal number
6970:
6960:
6955:
6954:
6953:
6942:
6936:
6930:
6929:
6926:
6925:
6923:
6922:
6917:
6912:
6907:
6902:
6897:
6892:
6891:
6890:
6880:
6879:
6878:
6865:
6863:
6857:
6856:
6854:
6853:
6852:
6851:
6846:
6841:
6831:
6826:
6821:
6816:
6811:
6806:
6800:
6798:
6792:
6791:
6789:
6788:
6783:
6778:
6773:
6768:
6763:
6758:
6757:
6756:
6746:
6741:
6736:
6731:
6726:
6721:
6715:
6713:
6704:
6698:
6697:
6695:
6694:
6689:
6684:
6679:
6674:
6669:
6657:Cantor's
6655:
6650:
6645:
6635:
6633:
6620:
6619:
6617:
6616:
6611:
6606:
6601:
6596:
6591:
6586:
6581:
6576:
6571:
6566:
6561:
6556:
6555:
6554:
6543:
6541:
6537:
6536:
6531:
6529:
6528:
6521:
6514:
6506:
6496:
6495:
6476:
6472:. Elsevier BV.
6455:
6435:
6413:
6400:
6378:
6374:
6353:
6331:
6327:
6301:
6299:. McGrawâHill.
6285:
6264:
6251:
6234:
6221:(1): 161â171.
6196:
6186:
6177:
6156:
6141:
6125:
6119:
6101:
6095:
6074:
6068:
6050:
6044:
6025:
6023:
6020:
6019:
6018:
6007:
6002:
5995:
5992:
5944:
5937:
5923:
5920:
5898:
5893:
5889:
5885:
5880:
5876:
5865:
5861:
5857:
5852:
5848:
5839:
5834:
5830:
5826:
5821:
5817:
5813:
5810:
5807:
5804:
5799:
5795:
5791:
5786:
5782:
5778:
5775:
5755:
5735:
5719:
5716:
5675:
5672:
5669:
5666:
5644:
5641:
5638:
5635:
5632:
5629:
5626:
5623:
5620:
5617:
5614:
5611:
5608:
5605:
5602:
5599:
5596:
5593:
5590:
5587:
5584:
5581:
5578:
5575:
5572:
5569:
5566:
5563:
5560:
5557:
5554:
5551:
5548:
5545:
5542:
5539:
5536:
5533:
5530:
5527:
5516:proper classes
5501:
5498:
5495:
5492:
5489:
5486:
5483:
5480:
5477:
5474:
5471:
5468:
5465:
5462:
5459:
5456:
5453:
5450:
5430:
5427:
5424:
5421:
5418:
5415:
5412:
5409:
5406:
5403:
5400:
5397:
5394:
5391:
5388:
5385:
5382:
5379:
5376:
5373:
5370:
5367:
5364:
5361:
5358:
5355:
5352:
5332:proper classes
5324:
5321:
5302:
5299:
5296:
5293:
5290:
5287:
5284:
5281:
5278:
5275:
5272:
5269:
5266:
5263:
5250:
5247:
5198:
5194:
5191:
5188:
5185:
5182:
5179:
5176:
5173:
5170:
5167:
5164:
5161:
5141:
5138:
5135:
5132:
5129:
5126:
5123:
5120:
5117:
5114:
5111:
5108:
5105:
5102:
5099:
5096:
5093:
5090:
5087:
5084:
5081:
5078:
5075:
5072:
5069:
5066:
5063:
5060:
5057:
5054:
5051:
5048:
5045:
5036:is encoded as
5025:
5022:
5019:
5016:
5013:
5010:
5007:
5004:
5001:
4998:
4995:
4992:
4989:
4986:
4983:
4980:
4977:
4974:
4971:
4968:
4965:
4962:
4959:
4956:
4953:
4950:
4947:
4944:
4941:
4938:
4935:
4932:
4929:
4898:
4875:
4871:
4868:
4864:
4861:
4857:
4854:
4833:
4830:
4827:
4824:
4821:
4818:
4815:
4812:
4809:
4806:
4803:
4800:
4797:
4794:
4791:
4788:
4785:
4782:
4779:
4776:
4773:
4770:
4767:
4764:
4761:
4758:
4755:
4752:
4749:
4746:
4743:
4740:
4737:
4734:
4731:
4728:
4725:
4722:
4719:
4716:
4713:
4710:
4679:
4676:
4673:
4670:
4650:
4647:
4644:
4641:
4638:
4635:
4632:
4629:
4626:
4623:
4620:
4617:
4614:
4611:
4608:
4605:
4602:
4599:
4596:
4593:
4590:
4587:
4584:
4564:
4561:
4558:
4555:
4552:
4549:
4546:
4543:
4540:
4537:
4534:
4531:
4528:
4525:
4497:
4494:
4491:
4488:
4464:
4444:
4440:
4437:
4412:
4392:
4368:
4365:
4362:
4358:
4354:
4351:
4348:
4345:
4342:
4339:
4336:
4333:
4330:
4327:
4324:
4321:
4317:
4313:
4310:
4307:
4304:
4301:
4298:
4295:
4292:
4289:
4286:
4283:
4280:
4277:
4274:
4271:
4268:
4265:
4262:
4259:
4256:
4253:
4250:
4247:
4244:
4223:
4202:
4181:
4177:
4174:
4154:
4134:
4112:
4107:
4103:
4099:
4096:
4088:
4086:
4083:
4080:
4077:
4074:
4073:
4070:
4066:
4062:
4059:
4051:
4049:
4046:
4043:
4042:
4040:
4035:
4032:
4029:
4026:
4023:
4002:
3978:
3975:
3974:
3973:
3926:
3883:
3882:
3875:
3874:
3873:
3842:
3733:
3725:
3682:
3674:
3628:
3620:
3612:
3604:
3590:
3566:
3554:
3553:
3489:
3488:
3456:
3455:
3419:
3418:
3298:
3286:
3267:
3266:
3215:
3207:
3195:
3172:
3140:
3095:
3083:
3010:if and only if
2989:
2986:
2953:
2923:
2922:
2911:
2908:
2905:
2902:
2899:
2896:
2893:
2890:
2887:
2884:
2881:
2878:
2875:
2872:
2869:
2860:
2856:
2853:
2850:
2847:
2837:
2826:
2823:
2820:
2817:
2814:
2811:
2808:
2805:
2802:
2799:
2796:
2787:
2783:
2780:
2777:
2774:
2764:
2753:
2750:
2747:
2744:
2741:
2738:
2735:
2732:
2729:
2726:
2723:
2720:
2717:
2708:
2704:
2701:
2698:
2695:
2672:
2669:
2666:
2646:
2643:
2640:
2637:
2634:
2631:
2628:
2625:
2622:
2619:
2616:
2596:
2593:
2590:
2587:
2584:
2581:
2578:
2575:
2572:
2569:
2566:
2563:
2560:
2557:
2554:
2551:
2548:
2545:
2542:
2539:
2536:
2533:
2530:
2518:
2515:
2511:proper classes
2492:
2488:
2465:
2461:
2437:
2434:
2431:
2428:
2425:
2422:
2417:
2412:
2409:
2406:
2403:
2400:
2397:
2394:
2390:
2387:
2384:
2381:
2378:
2375:
2355:
2352:
2349:
2327:
2324:
2321:
2318:
2315:
2312:
2309:
2306:
2302:
2298:
2295:
2292:
2289:
2286:
2283:
2280:
2277:
2274:
2271:
2268:
2265:
2262:
2259:
2256:
2251:
2246:
2243:
2240:
2237:
2234:
2231:
2228:
2224:
2220:
2216:
2213:
2209:
2205:
2202:
2199:
2196:
2193:
2190:
2187:
2184:
2181:
2178:
2173:
2168:
2165:
2162:
2159:
2155:
2151:
2147:
2144:
2141:
2138:
2135:
2130:
2126:
2103:
2100:
2097:
2094:
2091:
2088:
2085:
2082:
2079:
2076:
2073:
2070:
2067:
2064:
2061:
2056:
2052:
2028:
2027:
2016:
2013:
2010:
2007:
2004:
2001:
1998:
1995:
1992:
1989:
1986:
1983:
1980:
1977:
1974:
1971:
1968:
1963:
1958:
1955:
1952:
1949:
1946:
1943:
1940:
1937:
1934:
1929:
1924:
1921:
1918:
1915:
1905:
1894:
1891:
1888:
1885:
1882:
1879:
1876:
1873:
1870:
1867:
1864:
1861:
1858:
1855:
1852:
1847:
1842:
1839:
1836:
1833:
1830:
1827:
1824:
1821:
1818:
1813:
1808:
1805:
1802:
1799:
1776:
1773:
1770:
1767:
1764:
1761:
1758:
1755:
1752:
1749:
1746:
1743:
1740:
1737:
1734:
1731:
1728:
1725:
1722:
1706:
1699:
1683:
1680:
1675:
1671:
1667:
1664:
1661:
1656:
1652:
1648:
1645:
1642:
1639:
1634:
1630:
1626:
1621:
1617:
1613:
1610:
1607:
1602:
1598:
1594:
1589:
1585:
1581:
1578:
1555:
1552:
1549:
1544:
1540:
1536:
1533:
1530:
1525:
1521:
1517:
1514:
1511:
1508:
1503:
1499:
1495:
1490:
1486:
1482:
1479:
1476:
1471:
1467:
1463:
1458:
1454:
1450:
1447:
1444:
1441:
1438:
1435:
1432:
1429:
1426:
1423:
1420:
1417:
1414:
1397:The property "
1386:
1383:
1380:
1377:
1374:
1371:
1368:
1365:
1362:
1328:
1325:
1322:
1319:
1316:
1313:
1310:
1307:
1304:
1301:
1295:
1292:
1289:
1286:
1283:
1280:
1277:
1274:
1271:
1265:
1262:
1259:
1256:
1253:
1250:
1247:
1244:
1241:
1236:
1232:
1228:
1222:
1219:
1216:
1196:
1193:
1190:
1187:
1181:
1178:
1175:
1169:
1166:
1163:
1160:
1157:
1151:
1145:
1141:
1137:
1131:
1128:
1125:
1101:
1098:
1084:
1080:
1077:
1074:
1071:
1068:
1065:
1062:
1059:
1056:
1053:
1050:
1046:
1042:
1039:
1036:
1033:
1030:
1027:
1011:
1008:
995:
992:
989:
986:
983:
980:
977:
952:Wiener used {{
916:
912:
907:
903:
900:
897:
893:
888:
884:
880:
876:
872:
869:
866:
861:
856:
852:
848:
844:
841:
838:
834:
822:Norbert Wiener
818:
815:
798:
795:
791:Theory of Sets
783:Theory of Sets
714:
711:
686:
666:
653:
642:
631:
603:
600:
597:
594:
591:
576:open intervals
527:, and written
515:is called the
490:
485:
481:
477:
472:
468:
457:
453:
449:
444:
440:
431:
426:
422:
418:
413:
409:
405:
402:
399:
394:
390:
386:
381:
377:
373:
354:characteristic
341:
336:
332:
328:
323:
319:
315:
295:
290:
286:
282:
277:
273:
269:
257:
254:
214:is called the
210:), the object
114:unordered pair
15:
13:
10:
9:
6:
4:
3:
2:
8946:
8945:
8934:
8931:
8929:
8926:
8924:
8921:
8920:
8918:
8903:
8902:Ernst Zermelo
8900:
8898:
8895:
8893:
8890:
8888:
8887:Willard Quine
8885:
8883:
8880:
8878:
8875:
8873:
8870:
8868:
8865:
8863:
8860:
8858:
8855:
8853:
8850:
8848:
8845:
8844:
8842:
8840:
8839:Set theorists
8836:
8830:
8827:
8825:
8822:
8820:
8817:
8816:
8814:
8808:
8806:
8803:
8802:
8799:
8791:
8788:
8786:
8785:KripkeâPlatek
8783:
8779:
8776:
8775:
8774:
8771:
8770:
8769:
8766:
8762:
8759:
8758:
8757:
8756:
8752:
8748:
8745:
8744:
8743:
8740:
8739:
8736:
8733:
8731:
8728:
8726:
8723:
8721:
8718:
8717:
8715:
8711:
8705:
8702:
8700:
8697:
8695:
8692:
8690:
8688:
8683:
8681:
8678:
8676:
8673:
8670:
8666:
8663:
8661:
8658:
8654:
8651:
8649:
8646:
8644:
8641:
8640:
8639:
8636:
8633:
8629:
8626:
8624:
8621:
8619:
8616:
8614:
8611:
8610:
8608:
8605:
8601:
8595:
8592:
8590:
8587:
8585:
8582:
8580:
8577:
8575:
8572:
8570:
8567:
8565:
8562:
8558:
8555:
8553:
8550:
8549:
8548:
8545:
8543:
8540:
8538:
8535:
8533:
8530:
8528:
8525:
8522:
8518:
8515:
8513:
8510:
8508:
8505:
8504:
8502:
8496:
8493:
8492:
8489:
8483:
8480:
8478:
8475:
8473:
8470:
8468:
8465:
8463:
8460:
8458:
8455:
8453:
8450:
8447:
8444:
8442:
8439:
8438:
8436:
8434:
8430:
8422:
8421:specification
8419:
8417:
8414:
8413:
8412:
8409:
8408:
8405:
8402:
8400:
8397:
8395:
8392:
8390:
8387:
8385:
8382:
8380:
8377:
8375:
8372:
8370:
8367:
8365:
8362:
8360:
8357:
8353:
8350:
8348:
8345:
8343:
8340:
8339:
8338:
8335:
8333:
8330:
8329:
8327:
8325:
8321:
8316:
8306:
8303:
8302:
8300:
8296:
8292:
8285:
8280:
8278:
8273:
8271:
8266:
8265:
8262:
8252:
8251:
8246:
8238:
8232:
8229:
8227:
8224:
8222:
8219:
8217:
8214:
8210:
8207:
8206:
8205:
8202:
8200:
8197:
8195:
8192:
8190:
8186:
8183:
8181:
8178:
8176:
8173:
8171:
8168:
8166:
8163:
8162:
8160:
8156:
8150:
8147:
8145:
8142:
8140:
8139:Recursive set
8137:
8135:
8132:
8130:
8127:
8125:
8122:
8120:
8117:
8113:
8110:
8108:
8105:
8103:
8100:
8098:
8095:
8093:
8090:
8089:
8088:
8085:
8083:
8080:
8078:
8075:
8073:
8070:
8068:
8065:
8063:
8060:
8059:
8057:
8055:
8051:
8045:
8042:
8040:
8037:
8035:
8032:
8030:
8027:
8025:
8022:
8020:
8017:
8015:
8012:
8008:
8005:
8003:
8000:
7998:
7995:
7994:
7993:
7990:
7988:
7985:
7983:
7980:
7978:
7975:
7973:
7970:
7968:
7965:
7961:
7958:
7957:
7956:
7953:
7949:
7948:of arithmetic
7946:
7945:
7944:
7941:
7937:
7934:
7932:
7929:
7927:
7924:
7922:
7919:
7917:
7914:
7913:
7912:
7909:
7905:
7902:
7900:
7897:
7896:
7895:
7892:
7891:
7889:
7887:
7883:
7877:
7874:
7872:
7869:
7867:
7864:
7862:
7859:
7856:
7855:from ZFC
7852:
7849:
7847:
7844:
7838:
7835:
7834:
7833:
7830:
7828:
7825:
7823:
7820:
7819:
7818:
7815:
7813:
7810:
7808:
7805:
7803:
7800:
7798:
7795:
7793:
7790:
7788:
7785:
7784:
7782:
7780:
7776:
7766:
7765:
7761:
7760:
7755:
7754:non-Euclidean
7752:
7748:
7745:
7743:
7740:
7738:
7737:
7733:
7732:
7730:
7727:
7726:
7724:
7720:
7716:
7713:
7711:
7708:
7707:
7706:
7702:
7698:
7695:
7694:
7693:
7689:
7685:
7682:
7680:
7677:
7675:
7672:
7670:
7667:
7665:
7662:
7660:
7657:
7656:
7654:
7650:
7649:
7647:
7642:
7636:
7631:Example
7628:
7620:
7615:
7614:
7613:
7610:
7608:
7605:
7601:
7598:
7596:
7593:
7591:
7588:
7586:
7583:
7582:
7581:
7578:
7576:
7573:
7571:
7568:
7566:
7563:
7559:
7556:
7554:
7551:
7550:
7549:
7546:
7542:
7539:
7537:
7534:
7532:
7529:
7527:
7524:
7523:
7522:
7519:
7517:
7514:
7510:
7507:
7505:
7502:
7500:
7497:
7496:
7495:
7492:
7488:
7485:
7483:
7480:
7478:
7475:
7473:
7470:
7468:
7465:
7463:
7460:
7459:
7458:
7455:
7453:
7450:
7448:
7445:
7443:
7440:
7436:
7433:
7431:
7428:
7426:
7423:
7421:
7418:
7417:
7416:
7413:
7411:
7408:
7406:
7403:
7401:
7398:
7394:
7391:
7389:
7388:by definition
7386:
7385:
7384:
7381:
7377:
7374:
7373:
7372:
7369:
7367:
7364:
7362:
7359:
7357:
7354:
7352:
7349:
7348:
7345:
7342:
7340:
7336:
7331:
7325:
7321:
7311:
7308:
7306:
7303:
7301:
7298:
7296:
7293:
7291:
7288:
7286:
7283:
7281:
7278:
7276:
7275:KripkeâPlatek
7273:
7271:
7268:
7264:
7261:
7259:
7256:
7255:
7254:
7251:
7250:
7248:
7244:
7236:
7233:
7232:
7231:
7228:
7226:
7223:
7219:
7216:
7215:
7214:
7211:
7209:
7206:
7204:
7201:
7199:
7196:
7194:
7191:
7188:
7184:
7180:
7177:
7173:
7170:
7168:
7165:
7163:
7160:
7159:
7158:
7154:
7151:
7150:
7148:
7146:
7142:
7138:
7130:
7127:
7125:
7122:
7120:
7119:constructible
7117:
7116:
7115:
7112:
7110:
7107:
7105:
7102:
7100:
7097:
7095:
7092:
7090:
7087:
7085:
7082:
7080:
7077:
7075:
7072:
7070:
7067:
7065:
7062:
7060:
7057:
7055:
7052:
7051:
7049:
7047:
7042:
7034:
7031:
7029:
7026:
7024:
7021:
7019:
7016:
7014:
7011:
7009:
7006:
7005:
7003:
6999:
6996:
6994:
6991:
6990:
6989:
6986:
6984:
6981:
6979:
6976:
6974:
6971:
6969:
6965:
6961:
6959:
6956:
6952:
6949:
6948:
6947:
6944:
6943:
6940:
6937:
6935:
6931:
6921:
6918:
6916:
6913:
6911:
6908:
6906:
6903:
6901:
6898:
6896:
6893:
6889:
6886:
6885:
6884:
6881:
6877:
6872:
6871:
6870:
6867:
6866:
6864:
6862:
6858:
6850:
6847:
6845:
6842:
6840:
6837:
6836:
6835:
6832:
6830:
6827:
6825:
6822:
6820:
6817:
6815:
6812:
6810:
6807:
6805:
6802:
6801:
6799:
6797:
6796:Propositional
6793:
6787:
6784:
6782:
6779:
6777:
6774:
6772:
6769:
6767:
6764:
6762:
6759:
6755:
6752:
6751:
6750:
6747:
6745:
6742:
6740:
6737:
6735:
6732:
6730:
6727:
6725:
6724:Logical truth
6722:
6720:
6717:
6716:
6714:
6712:
6708:
6705:
6703:
6699:
6693:
6690:
6688:
6685:
6683:
6680:
6678:
6675:
6673:
6670:
6668:
6664:
6660:
6656:
6654:
6651:
6649:
6646:
6644:
6640:
6637:
6636:
6634:
6632:
6626:
6621:
6615:
6612:
6610:
6607:
6605:
6602:
6600:
6597:
6595:
6592:
6590:
6587:
6585:
6582:
6580:
6577:
6575:
6572:
6570:
6567:
6565:
6562:
6560:
6557:
6553:
6550:
6549:
6548:
6545:
6544:
6542:
6538:
6534:
6527:
6522:
6520:
6515:
6513:
6508:
6507:
6504:
6500:
6490:
6489:
6480:
6477:
6468:
6467:
6459:
6456:
6448:
6447:
6439:
6436:
6432:
6431:
6427:
6424:
6417:
6414:
6398:
6376:
6372:
6351:
6329:
6325:
6316:
6315:
6310:
6305:
6302:
6298:
6294:
6289:
6286:
6282:
6278:
6274:
6271:For a formal
6268:
6265:
6261:
6255:
6252:
6248:
6244:
6238:
6235:
6229:
6224:
6220:
6216:
6215:
6210:
6206:
6200:
6197:
6190:
6187:
6181:
6178:
6174:
6173:0-674-32449-8
6170:
6166:
6160:
6157:
6152:
6145:
6142:
6138:
6134:
6129:
6126:
6122:
6120:0-87150-164-3
6116:
6112:
6105:
6102:
6098:
6092:
6088:
6081:
6079:
6075:
6071:
6065:
6061:
6054:
6051:
6047:
6041:
6037:
6030:
6027:
6021:
6016:
6012:
6008:
6006:
6003:
6001:
5998:
5997:
5993:
5991:
5989:
5985:
5981:
5977:
5973:
5969:
5965:
5961:
5957:
5954:
5943:
5936:
5932:
5928:
5921:
5919:
5916:
5914:
5909:
5896:
5891:
5887:
5883:
5878:
5874:
5863:
5859:
5855:
5850:
5846:
5832:
5828:
5824:
5819:
5815:
5808:
5805:
5797:
5793:
5789:
5784:
5780:
5773:
5753:
5733:
5725:
5717:
5715:
5713:
5709:
5705:
5701:
5697:
5693:
5689:
5670:
5664:
5655:
5636:
5630:
5627:
5621:
5612:
5603:
5597:
5594:
5588:
5579:
5570:
5564:
5561:
5555:
5546:
5540:
5537:
5534:
5531:
5528:
5517:
5512:
5496:
5493:
5490:
5487:
5481:
5472:
5460:
5454:
5448:
5422:
5416:
5413:
5407:
5398:
5389:
5383:
5380:
5374:
5365:
5359:
5356:
5353:
5342:
5337:
5333:
5329:
5322:
5320:
5318:
5313:
5300:
5294:
5291:
5288:
5285:
5282:
5276:
5270:
5267:
5264:
5248:
5246:
5244:
5240:
5236:
5232:
5229:, but not in
5228:
5224:
5220:
5216:
5211:
5192:
5189:
5186:
5183:
5180:
5177:
5174:
5171:
5168:
5165:
5162:
5159:
5133:
5130:
5127:
5124:
5121:
5118:
5115:
5109:
5103:
5100:
5097:
5094:
5091:
5085:
5079:
5076:
5073:
5070:
5067:
5061:
5055:
5052:
5049:
5014:
5011:
5008:
5005:
5002:
4996:
4990:
4987:
4984:
4975:
4966:
4963:
4960:
4957:
4954:
4948:
4942:
4939:
4936:
4918:
4916:
4912:
4896:
4887:
4873:
4869:
4866:
4862:
4859:
4855:
4852:
4831:
4825:
4822:
4819:
4816:
4810:
4804:
4798:
4792:
4786:
4780:
4777:
4774:
4771:
4765:
4759:
4753:
4747:
4741:
4738:
4732:
4726:
4723:
4717:
4714:
4711:
4700:
4696:
4691:
4674:
4668:
4648:
4642:
4636:
4630:
4624:
4621:
4615:
4609:
4603:
4597:
4594:
4588:
4582:
4562:
4556:
4550:
4547:
4541:
4535:
4529:
4523:
4515:
4511:
4492:
4486:
4478:
4462:
4442:
4438:
4435:
4426:
4410:
4390:
4382:
4366:
4352:
4349:
4343:
4340:
4337:
4334:
4331:
4328:
4322:
4308:
4302:
4296:
4293:
4290:
4287:
4281:
4275:
4269:
4263:
4257:
4254:
4248:
4242:
4200:
4172:
4152:
4132:
4125:The function
4105:
4097:
4094:
4084:
4081:
4078:
4075:
4068:
4060:
4057:
4047:
4044:
4038:
4033:
4027:
4021:
3991:
3987:
3983:
3976:
3971:
3967:
3963:
3959:
3955:
3951:
3947:
3943:
3939:
3935:
3931:
3927:
3924:
3920:
3917:implies that
3916:
3912:
3908:
3904:
3903:
3902:
3900:
3896:
3892:
3888:
3880:
3876:
3871:
3867:
3863:
3859:
3855:
3851:
3847:
3843:
3840:
3836:
3832:
3828:
3824:
3820:
3816:
3812:
3808:
3807:
3805:
3801:
3797:
3793:
3789:
3785:
3781:
3777:
3776:
3775:
3773:
3769:
3765:
3761:
3757:
3753:
3749:
3745:
3741:
3737:
3731:
3723:
3719:
3715:
3711:
3707:
3703:
3699:
3695:
3691:
3690:
3686:
3680:
3672:
3668:
3664:
3660:
3656:
3652:
3648:
3644:
3640:
3638:
3634:
3631:. Therefore,
3626:
3618:
3610:
3602:
3598:
3594:
3588:
3584:
3580:
3576:
3572:
3564:
3558:
3551:
3547:
3543:
3539:
3535:
3531:
3527:
3523:
3519:
3515:
3511:
3507:
3503:
3499:
3495:
3491:
3490:
3486:
3482:
3478:
3474:
3470:
3466:
3462:
3458:
3457:
3453:
3449:
3445:
3441:
3437:
3433:
3429:
3425:
3421:
3420:
3416:
3412:
3408:
3404:
3400:
3396:
3392:
3388:
3384:
3380:
3376:
3372:
3368:
3364:
3360:
3356:
3352:
3348:
3344:
3340:
3336:
3332:
3328:
3327:
3326:
3324:
3320:
3316:
3312:
3308:
3304:
3296:
3292:
3284:
3280:
3276:
3272:
3264:
3260:
3256:
3252:
3248:
3244:
3240:
3236:
3232:
3228:
3224:
3220:
3216:
3213:
3205:
3201:
3193:
3189:
3185:
3181:
3177:
3173:
3170:
3166:
3162:
3158:
3154:
3150:
3146:
3138:
3134:
3133:
3132:
3130:
3126:
3121:
3119:
3115:
3111:
3107:
3104:. Two cases:
3103:
3099:
3093:
3089:
3081:
3077:
3073:
3069:
3065:
3061:
3057:
3053:
3049:
3045:
3041:
3037:
3032:
3028:
3026:
3022:
3018:
3014:
3011:
3007:
3003:
2999:
2995:
2987:
2985:
2983:
2979:
2976: =
2975:
2971:
2967:
2963:
2959:
2951:
2947:
2944:
2940:
2936:
2932:
2928:
2909:
2900:
2897:
2894:
2888:
2882:
2879:
2876:
2867:
2854:
2851:
2848:
2838:
2824:
2815:
2812:
2809:
2803:
2800:
2794:
2781:
2778:
2775:
2765:
2751:
2742:
2739:
2736:
2730:
2724:
2715:
2702:
2699:
2696:
2686:
2685:
2684:
2670:
2667:
2664:
2641:
2638:
2635:
2629:
2623:
2620:
2617:
2591:
2588:
2585:
2579:
2573:
2570:
2567:
2555:
2552:
2549:
2543:
2537:
2534:
2531:
2516:
2514:
2512:
2508:
2490:
2486:
2463:
2459:
2449:
2429:
2423:
2420:
2415:
2407:
2404:
2401:
2395:
2392:
2382:
2376:
2353:
2350:
2347:
2338:
2325:
2322:
2319:
2313:
2307:
2304:
2300:
2293:
2287:
2284:
2275:
2269:
2263:
2260:
2257:
2249:
2241:
2238:
2235:
2229:
2226:
2218:
2214:
2211:
2207:
2203:
2200:
2197:
2194:
2188:
2185:
2182:
2179:
2176:
2171:
2166:
2163:
2160:
2157:
2149:
2145:
2142:
2136:
2128:
2124:
2114:
2101:
2098:
2095:
2089:
2083:
2080:
2077:
2074:
2071:
2068:
2062:
2054:
2050:
2041:
2037:
2033:
2014:
2008:
2005:
2002:
1996:
1990:
1987:
1984:
1978:
1972:
1966:
1953:
1950:
1947:
1941:
1935:
1922:
1919:
1916:
1913:
1906:
1892:
1886:
1880:
1874:
1871:
1868:
1862:
1856:
1850:
1837:
1834:
1831:
1825:
1819:
1806:
1803:
1800:
1797:
1790:
1789:
1788:
1768:
1765:
1762:
1756:
1750:
1741:
1735:
1732:
1729:
1723:
1720:
1711:
1705:
1698:
1673:
1669:
1665:
1662:
1659:
1654:
1650:
1646:
1643:
1632:
1628:
1624:
1619:
1615:
1611:
1608:
1605:
1600:
1596:
1592:
1587:
1583:
1569:
1553:
1542:
1538:
1534:
1531:
1528:
1523:
1519:
1515:
1512:
1501:
1497:
1493:
1488:
1484:
1480:
1477:
1474:
1469:
1465:
1461:
1456:
1452:
1442:
1436:
1433:
1430:
1427:
1424:
1421:
1418:
1404:
1400:
1384:
1381:
1378:
1375:
1372:
1369:
1366:
1363:
1352:
1348:
1344:
1339:
1320:
1311:
1302:
1293:
1287:
1278:
1269:
1263:
1260:
1254:
1248:
1239:
1234:
1226:
1220:
1217:
1194:
1185:
1179:
1176:
1167:
1161:
1149:
1143:
1135:
1129:
1126:
1115:
1111:
1107:
1097:
1082:
1075:
1072:
1069:
1063:
1057:
1054:
1051:
1044:
1040:
1034:
1031:
1028:
1017:
1007:
987:
981:
967:
963:
959:
955:
950:
948:
944:
940:
936:
935:
930:
914:
910:
905:
901:
898:
895:
891:
886:
882:
874:
870:
867:
864:
859:
854:
850:
846:
842:
839:
836:
832:
823:
814:
812:
808:
804:
796:
794:
792:
786:
784:
781:group in its
780:
776:
770:
768:
763:
750:
738:
734:
718:
712:
710:
708:
704:
700:
696:
689:
678:
674:
669:
661:
656:
648:
637:
626:
615:
598:
595:
592:
581:
577:
571:
567:
560:
558:
554:
550:
546:
542:
539:between sets
538:
534:
530:
526:
522:
518:
514:
510:
506:
501:
488:
483:
479:
475:
470:
466:
455:
451:
447:
442:
438:
424:
420:
416:
411:
407:
400:
392:
388:
384:
379:
375:
363:
359:
355:
334:
330:
326:
321:
317:
288:
284:
280:
275:
271:
255:
253:
251:
247:
243:
239:
237:
233:
229:
225:
221:
217:
213:
209:
205:
200:
198:
194:
190:
186:
182:
178:
174:
170:
168:
163:
159:
155:
151:
147:
143:
139:
134:
132:
128:
124:
120:
116:
115:
110:
106:
102:
98:
94:
90:
86:
82:
78:
74:
64:
50:
41:
37:
33:
29:
25:
21:
8928:Order theory
8852:Georg Cantor
8847:Paul Bernays
8778:MorseâKelley
8753:
8686:
8685:Subset
8632:hereditarily
8594:Venn diagram
8552:ordered pair
8551:
8467:Intersection
8411:Axiom schema
8241:
8039:Ultraproduct
7886:Model theory
7851:Independence
7787:Formal proof
7779:Proof theory
7762:
7735:
7692:real numbers
7664:second-order
7575:Substitution
7452:Metalanguage
7393:conservative
7366:Axiom schema
7310:Constructive
7280:MorseâKelley
7246:Set theories
7225:Aleph number
7218:inaccessible
7124:Grothendieck
7008:intersection
6895:Higher-order
6883:Second-order
6829:Truth tables
6786:Venn diagram
6569:Formal proof
6498:
6487:
6479:
6465:
6458:
6445:
6438:
6421:
6416:
6312:
6304:
6296:
6288:
6276:
6267:
6259:
6254:
6246:
6242:
6237:
6218:
6212:
6199:
6189:
6180:
6164:
6159:
6144:
6128:
6110:
6104:
6086:
6059:
6053:
6035:
6029:
6014:
5983:
5979:
5971:
5967:
5959:
5955:
5950:
5941:
5934:
5917:
5910:
5721:
5711:
5707:
5703:
5699:
5695:
5691:
5687:
5656:
5513:
5340:
5326:
5314:
5252:
5212:
4919:
4914:
4913:. Likewise,
4910:
4888:
4698:
4694:
4692:
4513:
4509:
4476:
4379:This is the
3980:
3969:
3965:
3961:
3957:
3953:
3949:
3945:
3941:
3940:}, and so: {
3937:
3933:
3929:
3922:
3918:
3914:
3910:
3906:
3905:The option {
3898:
3894:
3890:
3886:
3884:
3878:
3869:
3865:
3861:
3857:
3853:
3849:
3845:
3838:
3834:
3830:
3826:
3822:
3818:
3814:
3810:
3803:
3799:
3795:
3791:
3787:
3783:
3779:
3771:
3767:
3763:
3759:
3755:
3751:
3747:
3743:
3739:
3738:
3729:
3721:
3717:
3713:
3709:
3705:
3701:
3697:
3693:
3692:
3688:
3687:
3678:
3670:
3666:
3662:
3658:
3654:
3650:
3646:
3642:
3641:
3636:
3632:
3624:
3616:
3608:
3600:
3596:
3595:
3586:
3582:
3578:
3574:
3570:
3562:
3556:
3555:
3549:
3545:
3541:
3537:
3533:
3529:
3525:
3521:
3517:
3513:
3509:
3505:
3501:
3497:
3493:
3484:
3480:
3476:
3472:
3468:
3464:
3460:
3451:
3447:
3443:
3439:
3435:
3431:
3427:
3423:
3414:
3410:
3406:
3402:
3401:}}, so that
3398:
3394:
3390:
3386:
3382:
3378:
3374:
3370:
3366:
3362:
3358:
3354:
3350:
3346:
3342:
3338:
3334:
3330:
3322:
3318:
3314:
3310:
3306:
3302:
3294:
3290:
3282:
3278:
3274:
3270:
3268:
3262:
3258:
3254:
3250:
3246:
3242:
3238:
3234:
3230:
3226:
3222:
3218:
3211:
3203:
3199:
3191:
3187:
3183:
3179:
3175:
3168:
3164:
3160:
3156:
3152:
3148:
3144:
3136:
3128:
3124:
3122:
3117:
3113:
3109:
3105:
3101:
3100:
3091:
3087:
3079:
3075:
3071:
3067:
3063:
3059:
3055:
3051:
3047:
3043:
3039:
3035:
3030:
3029:
3024:
3020:
3016:
3012:
3005:
3001:
2997:
2993:
2991:
2981:
2977:
2973:
2969:
2965:
2961:
2957:
2938:
2930:
2926:
2924:
2520:
2450:
2339:
2115:
2029:
1712:
1703:
1696:
1402:
1398:
1350:
1346:
1342:
1340:
1113:
1109:
1103:
1013:
965:
957:
953:
951:
938:
932:
820:
800:
790:
787:
782:
771:
766:
764:
752:
736:
732:
720:
716:
706:
702:
698:
694:
687:
676:
672:
667:
659:
654:
646:
635:
624:
616:
569:
565:
561:
556:
552:
544:
540:
532:
528:
524:
520:
512:
508:
502:
361:
357:
353:
259:
256:Generalities
240:
235:
231:
227:
224:second entry
223:
219:
215:
211:
207:
203:
201:
196:
192:
188:
184:
180:
176:
172:
166:
158:vector space
135:
130:
126:
122:
118:
112:
108:
104:
100:
96:
92:
88:
84:
80:
77:ordered pair
76:
70:
62:
48:
42:) such that
39:
35:
8933:Type theory
8877:Thomas Jech
8720:Alternative
8699:Uncountable
8653:Ultrafilter
8512:Cardinality
8416:replacement
8364:Determinacy
8149:Type theory
8097:undecidable
8029:Truth value
7916:equivalence
7595:non-logical
7208:Enumeration
7198:Isomorphism
7145:cardinality
7129:Von Neumann
7094:Ultrafilter
7059:Uncountable
6993:equivalence
6910:Quantifiers
6900:Fixed-point
6869:First-order
6749:Consistency
6734:Proposition
6711:Traditional
6682:Lindström's
6672:Compactness
6614:Type theory
6559:Cardinality
5698:-tuple and
5231:type theory
5215:type theory
4235:go on with
3928:So we have
3742:: Suppose {
3467:}, so that
3459:Therefore {
3353:, and so {{
962:type theory
779:N. Bourbaki
248:(and hence
236:projections
232:coordinates
216:first entry
154:terminology
117:, denoted {
79:, denoted (
73:mathematics
8917:Categories
8872:Kurt Gödel
8857:Paul Cohen
8694:Transitive
8462:Identities
8446:Complement
8433:Operations
8394:Regularity
8332:Adjunction
8291:Set theory
7960:elementary
7653:arithmetic
7521:Quantifier
7499:functional
7371:Expression
7089:Transitive
7033:identities
7018:complement
6951:hereditary
6934:Set theory
6022:References
4844:(which is
3881:must hold.
3720:}}. Thus (
3669:}}. Thus (
3301:implies {{
3078:}}. Thus (
3031:Kuratowski
2451:Note that
803:set theory
623:of a pair
620:projection
228:components
103:), unless
8805:Paradoxes
8725:Axiomatic
8704:Universal
8680:Singleton
8675:Recursive
8618:Countable
8613:Amorphous
8472:Power set
8389:Power set
8347:dependent
8342:countable
8231:Supertask
8134:Recursion
8092:decidable
7926:saturated
7904:of models
7827:deductive
7822:axiomatic
7742:Hilbert's
7729:Euclidean
7710:canonical
7633:axiomatic
7565:Signature
7494:Predicate
7383:Extension
7305:Ackermann
7230:Operation
7109:Universal
7099:Recursive
7074:Singleton
7069:Inhabited
7054:Countable
7044:Types of
7028:power set
6998:partition
6915:Predicate
6861:Predicate
6776:Syllogism
6766:Soundness
6739:Inference
6729:Tautology
6631:paradoxes
6399:ψ
6373:σ
6352:φ
6326:σ
5628:×
5613:∪
5595:×
5580:∪
5562:×
5494:∈
5488:∣
5473:∪
5467:∅
5414:×
5399:∪
5381:×
5343:the pair
5341:redefined
5193:∉
5152:provided
4897:φ
4867:ψ
4863:∪
4853:φ
4823:∈
4805:∪
4793:φ
4787:∪
4778:∈
4760:φ
4742:ψ
4739:∪
4727:φ
4669:ψ
4661:By this,
4637:∪
4625:φ
4610:∪
4598:σ
4583:ψ
4551:φ
4548:∪
4536:≠
4524:φ
4487:φ
4475:to a set
4463:φ
4436:σ
4411:σ
4383:of a set
4381:set image
4353:∩
4344:∈
4323:∪
4312:∖
4294:∈
4291:α
4288:∣
4282:α
4276:σ
4258:σ
4243:φ
4176:∖
4153:σ
4133:σ
4098:∈
4061:∉
4022:σ
3758:}}. Then
3653:, then {{
3422:Suppose {
3329:Suppose {
3054:, then {{
2980:then the
2580:∧
2562:↔
2487:π
2460:π
2424:∉
2396:∈
2351:≠
2308:⋃
2288:∉
2282:→
2270:≠
2230:∈
2215:⋃
2201:⋂
2198:∉
2192:→
2186:⋂
2183:≠
2177:⋃
2164:⋃
2161:∈
2146:⋃
2125:π
2084:⋃
2075:⋂
2072:⋃
2051:π
1979:∪
1923:⋃
1914:⋃
1863:∩
1807:⋂
1798:⋂
1666:∉
1660:∨
1647:∉
1638:→
1625:≠
1606:∈
1580:∀
1535:∉
1529:∨
1516:∉
1507:→
1494:≠
1475:∈
1449:∀
1443:∧
1434:∈
1422:∈
1416:∃
1379:∈
1367:∈
1361:∀
991:∅
947:primitive
943:relations
937:as sets.
879:∅
649:), or by
602:⟩
590:⟨
250:functions
162:recursive
142:sequences
8809:Problems
8713:Theories
8689:Superset
8665:Infinite
8494:Concepts
8374:Infinity
8298:Overview
8216:Logicism
8209:timeline
8185:Concrete
8044:Validity
8014:T-schema
8007:Kripke's
8002:Tarski's
7997:semantic
7987:Strength
7936:submodel
7931:spectrum
7899:function
7747:Tarski's
7736:Elements
7723:geometry
7679:Robinson
7600:variable
7585:function
7558:spectrum
7548:Sentence
7504:variable
7447:Language
7400:Relation
7361:Automata
7351:Alphabet
7335:language
7189:-jection
7167:codomain
7153:Function
7114:Universe
7084:Infinite
6988:Relation
6771:Validity
6761:Argument
6659:theorem,
6426:Archived
6295:, 1953.
6273:Metamath
6207:(1921).
5994:See also
5317:cardinal
5223:function
4870:″
4856:″
4439:″
4091:if
4054:if
3852:}, then
3704:, then {
3434:}. Then
3341:}. Then
3277:, then (
3257:. Hence
3155:}} = {{
2992:Prove: (
2517:Variants
1568:conjunct
1104:In 1921
362:property
358:defining
138:2-tuples
8747:General
8742:Zermelo
8648:subbase
8630: (
8569:Forcing
8547:Element
8519: (
8497:Methods
8384:Pairing
8158:Related
7955:Diagram
7853: (
7832:Hilbert
7817:Systems
7812:Theorem
7690:of the
7635:systems
7415:Formula
7410:Grammar
7326: (
7270:General
6983:Forcing
6968:Element
6888:Monadic
6663:paradox
6604:Theorem
6540:General
5953:product
4909:yields
4213:not in
3860:, from
3821:is in {
3740:Only if
3683:reverse
3675:reverse
3661:}} = {{
3643:Only if
3613:reverse
3605:reverse
3577:}} = {{
3567:reverse
3557:Reverse
3385:}} = {{
3377:}} = {{
3365:}} = {{
3313:}} = {{
3167:}} = {{
3102:Only if
3066:}} = {{
2927:reverse
2711:reverse
705:-tuple
578:on the
169:-tuples
150:vectors
146:scalars
59:
45:
32:ellipse
8638:Filter
8628:Finite
8564:Family
8507:Almost
8352:global
8337:Choice
8324:Axioms
7921:finite
7684:Skolem
7637:
7612:Theory
7580:Symbol
7570:String
7553:atomic
7430:ground
7425:closed
7420:atomic
7376:ground
7339:syntax
7235:binary
7162:domain
7079:Finite
6844:finite
6702:Logics
6661:
6609:Theory
6279:, see
6171:
6117:
6093:
6066:
6042:
5690:is an
5233:or in
4403:under
3992:. Let
3982:Rosser
3964:}, so
3877:Hence
3833:is in
3750:}} = {
3712:}} = {
3689:Short:
3599:. If (
3585:}} = (
3217:Thus {
3186:}} = (
3112:, and
2935:braces
1787:then:
1297:
1267:
1224:
1183:
1171:
1153:
1133:
662:) and
638:) and
549:subset
8730:Naive
8660:Fuzzy
8623:Empty
8606:types
8557:tuple
8527:Class
8521:large
8482:Union
8399:Union
7911:Model
7659:Peano
7516:Proof
7356:Arity
7285:Naive
7172:image
7104:Fuzzy
7064:Empty
7013:union
6958:Class
6599:Model
6589:Lemma
6547:Axiom
6470:(PDF)
6450:(PDF)
6277:short
6133:Quine
5962:in a
5706:then
5336:Morse
4165:. As
3986:Quine
3960:} = {
3956:} \ {
3952:} = {
3948:} \ {
3944:} = {
3936:} = {
3932:and {
3930:a = c
3915:a = c
3897:} = {
3879:a = c
3868:} = {
3848:} = {
3817:then
3802:} = {
3764:a = c
3734:short
3726:short
3702:b = d
3698:a = c
3696:: If
3651:b = d
3647:a = c
3645:. If
3637:a = c
3633:b = d
3520:} â {
3516:} = {
3508:} = {
3479:} = {
3471:and {
3469:c = a
3463:} = {
3426:} = {
3337:} = {
3229:} = {
3221:} = {
3038:. If
3000:) = (
2982:short
2970:short
2958:short
2954:short
2939:short
2931:short
2790:short
929:types
767:order
547:is a
140:, or
75:, an
8643:base
8034:Type
7837:list
7641:list
7618:list
7607:Term
7541:rank
7435:open
7329:list
7141:Maps
7046:sets
6905:Free
6875:list
6625:list
6552:list
6391:for
6364:and
6344:for
6245:and
6169:ISBN
6115:ISBN
6091:ISBN
6064:ISBN
6040:ISBN
4512:and
3954:c, d
3946:a, b
3938:c, d
3934:a, b
3913:and
3909:} =
3907:a, b
3899:c, d
3895:a, b
3893:or {
3889:} =
3887:a, b
3870:a, b
3866:c, d
3850:c, d
3846:a, b
3844:If {
3839:a, c
3829:and
3825:} =
3823:c, d
3813:} =
3811:a, b
3809:If {
3804:c, d
3800:a, b
3798:or {
3794:} =
3792:a, b
3788:a, b
3784:c, d
3772:c, d
3756:c, d
3748:a, b
3730:c, d
3722:a, b
3718:c, d
3710:a, b
3700:and
3679:c, d
3671:a, b
3667:c, d
3665:}, {
3659:a, b
3657:}, {
3649:and
3635:and
3625:d, c
3617:b, a
3609:c, d
3601:a, b
3587:b, a
3583:b, a
3581:}, {
3575:a, b
3573:}, {
3569:= {{
3563:a, b
3544:and
3395:a, b
3393:}, {
3381:}, {
3369:}, {
3357:}, {
3325:}}.
3317:}, {
3305:}, {
3241:and
3210:= {{
3178:}, {
3159:}, {
3147:}, {
3143:= {{
3137:a, b
3080:a, b
3070:}, {
3058:}, {
3046:and
3019:and
2964:and
2925:The
2505:are
2478:and
2340:(if
2038:and
2034:for
1006:'s.
811:sets
757:and
745:and
725:and
562:The
543:and
535:. A
523:and
503:The
356:(or
306:and
260:Let
244:and
222:the
8604:Set
7721:of
7703:of
7651:of
7183:Sur
7157:Map
6964:Ur-
6946:Set
6223:doi
6013:",
5766::
5235:NFU
5213:In
4427:by
3901:}.
3864:= {
3806:}.
3782:= {
3778:If
3770:= {
3766:or
3754:, {
3746:, {
3728:= (
3716:, {
3708:, {
3677:= (
3623:= (
3615:, (
3607:= (
3492:If
3289:= (
3269:If
3214:}}.
3198:= (
3171:}}.
3123:If
3086:= (
2042:):
1713:If
1116:):
931:of
551:of
519:of
505:set
191:, (
133:}.
71:In
65:= 1
8919::
8107:NP
7731::
7725::
7655::
7332:),
7187:Bi
7179:In
6311::
6217:.
6211:.
6077:^
5990:.
5958:Ă
5915:.
5710:=
5702:=
5334:.
5237:.
5227:NF
5219:NF
5210:.
4724::=
4697:,
4595::=
4516:,
4423:,
4255::=
4034::=
3968:=
3736:.
3694:If
3685:.
3639:.
3597:If
3593:.
3548:=
3540:=
3532:=
3524:,
3512:,
3504:,
3496:=
3487:}.
3483:,
3475:,
3450:â
3442:=
3438:=
3430:,
3413:â
3405:=
3373:,
3361:,
3349:=
3345:=
3333:,
3321:,
3309:,
3293:,
3281:,
3273:â
3261:=
3253:=
3245:=
3237:=
3225:,
3202:,
3190:,
3182:,
3174:{{
3163:,
3151:,
3131::
3127:=
3120:.
3116:â
3108:=
3098:.
3090:,
3074:,
3062:,
3050:=
3042:=
3036:If
3027:.
3023:=
3015:=
3008:)
3004:,
2996:,
2868::=
2863:01
2795::=
2716::=
2513:.
1702:â
1150::=
1112:,
1041::=
949:.
851::=
735:,
709:.
568:,
559:.
555:Ă
531:Ă
360:)
206:,
129:,
121:,
107:=
99:,
91:,
83:,
61:+
8687:·
8671:)
8667:(
8634:)
8523:)
8283:e
8276:t
8269:v
8187:/
8102:P
7857:)
7643:)
7639:(
7536:â
7531:!
7526:â
7487:=
7482:â
7477:â
7472:â§
7467:âš
7462:ÂŹ
7185:/
7181:/
7155:/
6966:)
6962:(
6849:â
6839:3
6627:)
6525:e
6518:t
6511:v
6411:.
6377:2
6330:1
6249:.
6247:b
6243:a
6231:.
6225::
6219:2
6153:.
5984:X
5980:X
5972:B
5968:A
5960:B
5956:A
5947:.
5945:2
5942:X
5940:Ă
5938:1
5935:X
5897:.
5892:2
5888:b
5884:=
5879:1
5875:b
5864:2
5860:a
5856:=
5851:1
5847:a
5838:)
5833:2
5829:b
5825:,
5820:2
5816:a
5812:(
5809:f
5806:=
5803:)
5798:1
5794:b
5790:,
5785:1
5781:a
5777:(
5774:f
5754:f
5734:f
5712:m
5708:n
5704:b
5700:a
5696:m
5692:n
5688:a
5674:)
5671:x
5668:(
5665:s
5643:)
5640:)
5637:z
5634:(
5631:s
5625:}
5622:2
5619:{
5616:(
5610:)
5607:)
5604:y
5601:(
5598:s
5592:}
5589:1
5586:{
5583:(
5577:)
5574:)
5571:x
5568:(
5565:s
5559:}
5556:0
5553:{
5550:(
5547:=
5544:)
5541:z
5538:,
5535:y
5532:,
5529:x
5526:(
5500:}
5497:x
5491:t
5485:}
5482:t
5479:{
5476:{
5470:}
5464:{
5461:=
5458:)
5455:x
5452:(
5449:s
5429:)
5426:)
5423:y
5420:(
5417:s
5411:}
5408:1
5405:{
5402:(
5396:)
5393:)
5390:x
5387:(
5384:s
5378:}
5375:0
5372:{
5369:(
5366:=
5363:)
5360:y
5357:,
5354:x
5351:(
5301:.
5298:}
5295:y
5292:R
5289:x
5286::
5283:R
5280:{
5277:=
5274:)
5271:y
5268:,
5265:x
5262:(
5197:N
5190:f
5187:,
5184:e
5181:,
5178:d
5175:,
5172:c
5169:,
5166:b
5163:,
5160:a
5140:}
5137:}
5134:0
5131:,
5128:4
5125:,
5122:f
5119:,
5116:e
5113:{
5110:,
5107:}
5104:0
5101:,
5098:3
5095:,
5092:d
5089:{
5086:,
5083:}
5080:2
5077:,
5074:c
5071:,
5068:b
5065:{
5062:,
5059:}
5056:1
5053:,
5050:a
5047:{
5044:{
5024:)
5021:}
5018:}
5015:3
5012:,
5009:f
5006:,
5003:e
5000:{
4997:,
4994:}
4991:2
4988:,
4985:d
4982:{
4979:{
4976:,
4973:}
4970:}
4967:1
4964:,
4961:c
4958:,
4955:b
4952:{
4949:,
4946:}
4943:0
4940:,
4937:a
4934:{
4931:{
4928:(
4915:B
4911:A
4874:B
4860:A
4832:.
4829:}
4826:B
4820:b
4817::
4814:}
4811:0
4808:{
4802:)
4799:b
4796:(
4790:{
4784:}
4781:A
4775:a
4772::
4769:)
4766:a
4763:(
4757:{
4754:=
4751:]
4748:B
4745:[
4736:]
4733:A
4730:[
4721:)
4718:B
4715:,
4712:A
4709:(
4699:B
4695:A
4678:)
4675:x
4672:(
4649:.
4646:}
4643:0
4640:{
4634:)
4631:x
4628:(
4622:=
4619:}
4616:0
4613:{
4607:]
4604:x
4601:[
4592:)
4589:x
4586:(
4563:.
4560:)
4557:y
4554:(
4545:}
4542:0
4539:{
4533:)
4530:x
4527:(
4514:y
4510:x
4496:)
4493:x
4490:(
4477:x
4443:x
4391:x
4367:.
4364:}
4361:)
4357:N
4350:x
4347:(
4341:n
4338::
4335:1
4332:+
4329:n
4326:{
4320:)
4316:N
4309:x
4306:(
4303:=
4300:}
4297:x
4285:)
4279:(
4273:{
4270:=
4267:]
4264:x
4261:[
4252:)
4249:x
4246:(
4222:N
4201:x
4180:N
4173:x
4106:.
4102:N
4095:x
4085:,
4082:1
4079:+
4076:x
4069:,
4065:N
4058:x
4048:,
4045:x
4039:{
4031:)
4028:x
4025:(
4001:N
3972:.
3970:d
3966:b
3962:d
3958:c
3950:a
3942:b
3923:c
3919:c
3911:c
3891:c
3862:a
3858:a
3854:a
3835:c
3831:a
3827:a
3819:c
3815:c
3796:c
3780:a
3768:a
3760:a
3752:c
3744:a
3732:)
3724:)
3714:c
3706:a
3681:)
3673:)
3663:d
3655:b
3629:K
3627:)
3621:K
3619:)
3611:)
3603:)
3591:K
3589:)
3579:b
3571:b
3565:)
3561:(
3559::
3552:.
3550:d
3546:b
3542:c
3538:a
3534:b
3530:d
3526:b
3522:a
3518:a
3514:a
3510:a
3506:d
3502:c
3498:a
3494:d
3485:b
3481:a
3477:d
3473:c
3465:a
3461:c
3454:.
3452:b
3448:a
3444:c
3440:b
3436:a
3432:b
3428:a
3424:c
3417:.
3415:b
3411:a
3407:a
3403:b
3399:a
3391:a
3387:a
3383:a
3379:a
3375:a
3371:a
3367:a
3363:d
3359:c
3355:c
3351:a
3347:d
3343:c
3339:a
3335:d
3331:c
3323:d
3319:c
3315:c
3311:b
3307:a
3303:a
3299:K
3297:)
3295:d
3291:c
3287:K
3285:)
3283:b
3279:a
3275:b
3271:a
3265:.
3263:d
3259:b
3255:b
3251:a
3247:d
3243:a
3239:c
3235:a
3231:a
3227:d
3223:c
3219:c
3212:a
3208:K
3206:)
3204:b
3200:a
3196:K
3194:)
3192:d
3188:c
3184:d
3180:c
3176:c
3169:a
3165:a
3161:a
3157:a
3153:b
3149:a
3145:a
3141:K
3139:)
3135:(
3129:b
3125:a
3118:b
3114:a
3110:b
3106:a
3096:K
3094:)
3092:d
3088:c
3084:K
3082:)
3076:d
3072:c
3068:c
3064:b
3060:a
3056:a
3052:d
3048:b
3044:c
3040:a
3033::
3025:d
3021:b
3017:c
3013:a
3006:d
3002:c
2998:b
2994:a
2978:b
2974:a
2966:b
2962:a
2910:.
2907:}
2904:}
2901:b
2898:,
2895:1
2892:{
2889:,
2886:}
2883:a
2880:,
2877:0
2874:{
2871:{
2859:)
2855:b
2852:,
2849:a
2846:(
2825:;
2822:}
2819:}
2816:b
2813:,
2810:a
2807:{
2804:,
2801:a
2798:{
2786:)
2782:b
2779:,
2776:a
2773:(
2752:;
2749:}
2746:}
2743:b
2740:,
2737:a
2734:{
2731:,
2728:}
2725:b
2722:{
2719:{
2707:)
2703:b
2700:,
2697:a
2694:(
2671:a
2668:=
2665:b
2645:)
2642:a
2639:,
2636:b
2633:(
2630:=
2627:)
2624:b
2621:,
2618:a
2615:(
2595:)
2592:y
2589:=
2586:b
2583:(
2577:)
2574:x
2571:=
2568:a
2565:(
2559:)
2556:y
2553:,
2550:x
2547:(
2544:=
2541:)
2538:b
2535:,
2532:a
2529:(
2491:2
2464:1
2436:}
2433:}
2430:x
2427:{
2421:a
2416:|
2411:}
2408:y
2405:,
2402:x
2399:{
2393:a
2386:{
2383:=
2380:}
2377:y
2374:{
2354:y
2348:x
2326:.
2323:y
2320:=
2317:}
2314:y
2311:{
2305:=
2301:}
2297:}
2294:x
2291:{
2285:a
2279:}
2276:x
2273:{
2267:}
2264:y
2261:,
2258:x
2255:{
2250:|
2245:}
2242:y
2239:,
2236:x
2233:{
2227:a
2219:{
2212:=
2208:}
2204:p
2195:a
2189:p
2180:p
2172:|
2167:p
2158:a
2150:{
2143:=
2140:)
2137:p
2134:(
2129:2
2102:.
2099:x
2096:=
2093:}
2090:x
2087:{
2081:=
2078:p
2069:=
2066:)
2063:p
2060:(
2055:1
2015:.
2012:}
2009:y
2006:,
2003:x
2000:{
1997:=
1994:}
1991:y
1988:,
1985:x
1982:{
1976:}
1973:x
1970:{
1967:=
1962:}
1957:}
1954:y
1951:,
1948:x
1945:{
1942:,
1939:}
1936:x
1933:{
1928:{
1920:=
1917:p
1893:,
1890:}
1887:x
1884:{
1881:=
1878:}
1875:y
1872:,
1869:x
1866:{
1860:}
1857:x
1854:{
1851:=
1846:}
1841:}
1838:y
1835:,
1832:x
1829:{
1826:,
1823:}
1820:x
1817:{
1812:{
1804:=
1801:p
1775:}
1772:}
1769:y
1766:,
1763:x
1760:{
1757:,
1754:}
1751:x
1748:{
1745:{
1742:=
1739:)
1736:y
1733:,
1730:x
1727:(
1724:=
1721:p
1707:2
1704:Y
1700:1
1697:Y
1682:)
1679:)
1674:2
1670:Y
1663:x
1655:1
1651:Y
1644:x
1641:(
1633:2
1629:Y
1620:1
1616:Y
1612::
1609:p
1601:2
1597:Y
1593:,
1588:1
1584:Y
1577:(
1554:.
1551:)
1548:)
1543:2
1539:Y
1532:x
1524:1
1520:Y
1513:x
1510:(
1502:2
1498:Y
1489:1
1485:Y
1481::
1478:p
1470:2
1466:Y
1462:,
1457:1
1453:Y
1446:(
1440:)
1437:Y
1431:x
1428::
1425:p
1419:Y
1413:(
1403:p
1399:x
1385:.
1382:Y
1376:x
1373::
1370:p
1364:Y
1351:p
1347:x
1343:p
1327:}
1324:}
1321:x
1318:{
1315:{
1312:=
1309:}
1306:}
1303:x
1300:{
1294:,
1291:}
1288:x
1285:{
1282:{
1279:=
1276:}
1273:}
1270:x
1264:,
1261:x
1258:{
1255:,
1252:}
1249:x
1246:{
1243:{
1240:=
1235:K
1231:)
1227:x
1221:,
1218:x
1215:(
1195:.
1192:}
1189:}
1186:b
1180:,
1177:a
1174:{
1168:,
1165:}
1162:a
1159:{
1156:{
1144:K
1140:)
1136:b
1130:,
1127:a
1124:(
1114:b
1110:a
1083:}
1079:}
1076:2
1073:,
1070:b
1067:{
1064:,
1061:}
1058:1
1055:,
1052:a
1049:{
1045:{
1038:)
1035:b
1032:,
1029:a
1026:(
994:}
988:,
985:}
982:a
979:{
976:{
966:b
958:b
954:b
915:.
911:}
906:}
902:}
899:b
896:{
892:{
887:,
883:}
875:,
871:}
868:a
865:{
860:{
855:{
847:)
843:b
840:,
837:a
833:(
759:b
755:a
747:b
743:a
739:)
737:b
733:a
731:(
727:b
723:a
707:t
703:n
699:i
695:t
693:(
688:i
681:Ï
677:n
673:p
671:(
668:r
664:Ï
660:p
658:(
655:â
651:Ï
647:p
645:(
643:2
640:Ï
636:p
634:(
632:1
629:Ï
625:p
599:b
596:,
593:a
572:)
570:b
566:a
564:(
557:B
553:A
545:B
541:A
533:B
529:A
525:B
521:A
513:B
509:A
489:.
484:2
480:b
476:=
471:1
467:b
456:2
452:a
448:=
443:1
439:a
430:)
425:2
421:b
417:,
412:2
408:a
404:(
401:=
398:)
393:1
389:b
385:,
380:1
376:a
372:(
340:)
335:2
331:b
327:,
322:2
318:a
314:(
294:)
289:1
285:b
281:,
276:1
272:a
268:(
220:b
212:a
208:b
204:a
197:c
195:,
193:b
189:a
185:c
183:,
181:b
179:,
177:a
173:n
167:n
131:a
127:b
123:b
119:a
109:b
105:a
101:a
97:b
93:b
89:a
85:b
81:a
67:.
63:y
56:4
53:/
49:x
40:y
38:,
36:x
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
