2030:) of a finite set is equal to the order type. Counting in the everyday sense typically starts from one, so it assigns to each object the size of the initial segment with that object as last element. Note that these numbers are one more than the formal ordinal numbers according to the isomorphic order, because these are equal to the number of earlier objects (which corresponds to counting from zero). Thus for finite
1653:
1640:
1074:
1064:
1034:
1024:
1002:
992:
957:
935:
925:
890:
858:
848:
823:
791:
776:
756:
724:
714:
704:
684:
652:
632:
622:
612:
580:
560:
516:
496:
486:
449:
424:
414:
382:
357:
320:
290:
258:
196:
161:
2721:) imply the axiom of choice and hence a well order of the reals. Nonetheless, it is possible to show that the ZFC+GCH axioms alone are not sufficient to prove the existence of a definable (by a formula) well order of the reals. However it is consistent with ZFC that a definable well ordering of the reals exists—for example, it is consistent with ZFC that
3418:
Subsets that are unbounded by themselves but bounded in the whole set; they have no maximum, but a supremum outside the subset; if the subset is non-empty this supremum is a limit point of the subset and hence also of the whole set; if the subset is empty this supremum is the minimum of the whole
2165:
2374:
2624:
1384:
1313:
2533:
1628:
1242:
1513:
1469:
1425:
2026:, to find the ordinal number of a particular object, or to find the object with a particular ordinal number, corresponds to assigning ordinal numbers one by one to the objects. The size (number of elements,
3203:
3289:
1650:
indicates that the property is not guaranteed in general (it might, or might not, hold). For example, that every equivalence relation is symmetric, but not necessarily antisymmetric, is indicated by
1583:
1480:
1436:
1395:
1324:
1253:
1197:
3215:
within the set. Within the set of real numbers, either with the ordinary topology or the order topology, 0 is also a limit point of the set. It is also a limit point of the set of limit points.
3107:
2087:
3034:
2868:
2297:
2081:
Another well ordering of the natural numbers is given by defining that all even numbers are less than all odd numbers, and the usual ordering applies within the evens and the odds:
2544:
1186:
1572:
2701:
3400:
2903:
2941:
2754:
2429:
1319:
2022:
of the well-ordered set. The position of each element within the ordered set is also given by an ordinal number. In the case of a finite set, the basic operation of
1726:
1248:
1807:
1778:
1752:
1540:
1157:
1687:
3415:
by themselves); this can be an isolated point or a limit point of the whole set; in the latter case it may or may not be also a limit point of the subset.
4280:
3368:— this type does not occur in finite sets, and may or may not occur in an infinite set; the infinite sets without limit point are the sets of order type
61:
3507:
Bonnet, Rémi; Finkel, Alain; Haddad, Serge; Rosa-Velardo, Fernando (2013). "Ordinal theory for expressiveness of well-structured transition systems".
2439:
1578:
4263:
1192:
3793:
3302:
of this set, 1 is a limit point of the set, despite being separated from the only limit point 0 under the ordinary topology of the real numbers.
3629:
1475:
3597:
1431:
4110:
1390:
4246:
4105:
3119:
3221:
4100:
2718:
3736:
3818:
2870:
whose elements are nonempty and disjoint intervals. Each such interval contains at least one rational number, so there is an
54:
3430:
in the whole set if and only if it is unbounded in the whole set or it has a maximum that is also maximum of the whole set.
2953:
is countable. On the other hand, a countably infinite subset of the reals may or may not be a well order with the standard
4137:
4057:
3731:
2160:{\displaystyle {\begin{matrix}0&2&4&6&8&\dots &1&3&5&7&9&\dots \end{matrix}}}
3922:
3851:
3051:
3825:
3813:
3776:
3751:
3726:
3680:
3649:
3328:
Every strictly decreasing sequence of elements of the set must terminate after only finitely many steps (assuming the
3756:
3746:
1954:. The distinction between strict and non-strict well orders is often ignored since they are easily interconvertible.
4122:
3622:
4095:
3761:
3442:
3329:
2038:-th element" of a well-ordered set requires context to know whether this counts from zero or one. In a notation "
47:
2971:
2369:{\displaystyle {\begin{matrix}0&1&2&3&4&\dots &-1&-2&-3&\dots \end{matrix}}}
2078:
is a well ordering and has the additional property that every non-zero natural number has a unique predecessor.
1947:
well ordering, then < is a strict well ordering. A relation is a strict well ordering if and only if it is a
4323:
4318:
4027:
3654:
2804:
2053:, but not conversely: well-ordered sets of a particular cardinality can have many different order types (see
1908:, has a unique successor (next element), namely the least element of the subset of all elements greater than
4275:
4258:
3551:
2714:
2619:{\displaystyle {\begin{matrix}0&-1&1&-2&2&-3&3&-4&4&\dots \end{matrix}}}
1993:
4187:
3803:
2645:
2183:. Every element has a successor (there is no largest element). Two elements lack a predecessor: 0 and 1.
1091:
84:
1162:
4313:
4165:
4000:
3991:
3860:
3695:
3659:
3615:
3434:
3427:
3322:
1982:
1978:
1970:
1948:
1101:
94:
3741:
2728:
An uncountable subset of the real numbers with the standard ordering ≤ cannot be a well order: Suppose
2945:
to the natural numbers (which could be chosen to avoid hitting zero). Thus there is an injection from
1546:
4253:
4212:
4202:
4192:
3937:
3900:
3890:
3870:
3855:
3589:
2656:
1977:, states that every set can be well ordered. If a set is well ordered (or even if it merely admits a
1667:
878:
153:
4180:
4091:
4037:
3996:
3986:
3875:
3808:
3771:
3380:
2883:
1690:
1126:
537:
119:
36:
2924:
2737:
2725:, and it follows from ZFC+V=L that a particular formula well orders the reals, or indeed any set.
1379:{\displaystyle {\begin{aligned}a\neq {}&b\Rightarrow \\aRb{\text{ or }}&bRa\end{aligned}}}
4219:
4072:
3981:
3971:
3912:
3830:
3478:
2871:
2401:
2058:
1951:
1121:
1116:
1096:
1086:
1012:
114:
109:
89:
79:
4292:
4132:
3766:
1308:{\displaystyle {\begin{aligned}aRb{\text{ and }}&bRa\\\Rightarrow a={}&b\end{aligned}}}
4229:
4207:
4067:
4052:
4032:
3835:
3593:
3462:
3345:
1929:
1836:
2919:, which could be mapped to zero later). And it is well known that there is an injection from
1646:
indicates that the column's property is always true the row's term (at the very left), while
4042:
3895:
3560:
3542:
3516:
3473:
2011:
1958:
1944:
1905:
1696:
811:
744:
3528:
4224:
4007:
3885:
3880:
3865:
3690:
3675:
3524:
3483:
3373:
2710:
2207:
2200:
2027:
1989:
1974:
1783:
672:
469:
40:
3781:
1810:
A term's definition may require additional properties that are not listed in this table.
1757:
1731:
1519:
1136:
3583:
4142:
4127:
4117:
3976:
3954:
3932:
3579:
3468:
3359:
3349:
3299:
2431:
2383:
2217:
2192:
2075:
2015:
2005:
1962:
1843:
1672:
340:
4307:
4241:
4197:
4175:
4047:
3917:
3905:
3710:
2649:
1862:
1111:
1106:
945:
278:
104:
99:
4062:
3944:
3927:
3845:
3685:
3638:
3488:
1912:. There may be elements, besides the least element, that have no predecessor (see
1985:
can be used to prove that a given statement is true for all elements of the set.
4268:
3961:
3840:
3705:
3450:
3412:
3365:
3312:
3212:
2528:{\displaystyle |x|<|y|\qquad {\text{or}}\qquad |x|=|y|{\text{ and }}x\leq y.}
2050:
1925:
1820:
402:
3036:
has no least element and is therefore not a well order under standard ordering
1623:{\displaystyle {\begin{aligned}aRb\Rightarrow \\{\text{not }}bRa\end{aligned}}}
4236:
4170:
2630:
2171:
2019:
1966:
1237:{\displaystyle {\begin{aligned}&aRb\\\Rightarrow {}&bRa\end{aligned}}}
3520:
2398:
Another relation for well ordering the integers is the following definition:
1873:. In some academic articles and textbooks these terms are instead written as
4287:
4160:
3966:
3565:
3446:
3319:
The set is well ordered. That is, every nonempty subset has a least element.
1851:
17:
4082:
3949:
3700:
2023:
217:
1992:
are well ordered by the usual less-than relation is commonly called the
1508:{\displaystyle {\begin{aligned}a\wedge b\\{\text{exists}}\end{aligned}}}
2196:
1464:{\displaystyle {\begin{aligned}a\vee b\\{\text{exists}}\end{aligned}}}
1854:
1420:{\displaystyle {\begin{aligned}\min S\\{\text{exists}}\end{aligned}}}
2046:
can also be an infinite ordinal, it will typically count from zero.
1900:
Every non-empty well-ordered set has a least element. Every element
3355:
With respect to this topology there can be two kinds of elements:
2961:
The natural numbers are a well order under the standard ordering
2722:
1932:, namely the least element of the subset of all upper bounds of
3611:
3607:
3546:
3547:"Some Applications of the Notions of Forcing and Generic Sets"
2706:
3198:{\displaystyle \{-2^{-n}-2^{-m-n}\,|\,0\leq m,n<\omega \}}
2713:) one can show that there is a well order of the reals. Also
3362:— these are the minimum and the elements with a predecessor.
3284:{\displaystyle \{-2^{-n}\,|\,0\leq n<\omega \}\cup \{1\}}
3437:
if and only if it has order type less than or equal to ω
3585:
Real
Analysis: Modern Techniques and Their Applications
2199:
is not a well ordering, since, for example, the set of
3335:
Every subordering is isomorphic to an initial segment.
2549:
2302:
2092:
3383:
3224:
3122:
3054:
2974:
2927:
2886:
2807:
2740:
2659:
2547:
2442:
2404:
2300:
2090:
2061:
set, the set of possible order types is uncountable.
1786:
1760:
1734:
1699:
1675:
1581:
1549:
1522:
1478:
1434:
1393:
1322:
1251:
1195:
1165:
1139:
4153:
4081:
4020:
3790:
3719:
3668:
3315:, then the following are equivalent to each other:
3411:Subsets with a maximum (that is, subsets that are
3394:
3283:
3197:
3102:{\displaystyle \{-2^{-n}\,|\,0\leq n<\omega \}}
3101:
3028:
2935:
2897:
2862:
2748:
2695:
2618:
2527:
2423:
2368:
2159:
2049:For an infinite set the order type determines the
1801:
1772:
1746:
1720:
1681:
1622:
1566:
1534:
1507:
1463:
1419:
1378:
1307:
1236:
1180:
1151:
2648:is not a well ordering, since, for example, the
2213:is an example of well ordering of the integers:
1398:
3588:. Pure and applied mathematics (2nd ed.).
2538:This well order can be visualized as follows:
3623:
3433:A well-ordered set as topological space is a
2705:does not contain a least element. From the
1663:in the "Antisymmetric" column, respectively.
55:
8:
3422:Subsets that are unbounded in the whole set.
3278:
3272:
3266:
3225:
3192:
3123:
3096:
3055:
3023:
2975:
2857:
2814:
1869:together with the ordering is then called a
2203:integers does not contain a least element.
4281:Positive cone of a partially ordered group
3630:
3616:
3608:
3344:Every well-ordered set can be made into a
3029:{\displaystyle \{1/n\,|\,n=1,2,3,\dots \}}
1916:below for an example). A well-ordered set
62:
48:
31:
3564:
3385:
3384:
3382:
3250:
3245:
3244:
3235:
3223:
3170:
3165:
3164:
3149:
3133:
3121:
3080:
3075:
3074:
3065:
3053:
2995:
2990:
2989:
2981:
2973:
2949:to the natural numbers, which means that
2929:
2928:
2926:
2888:
2887:
2885:
2863:{\displaystyle A=\{(x,s(x))\,|\,x\in X\}}
2847:
2842:
2841:
2806:
2742:
2741:
2739:
2658:
2548:
2546:
2508:
2503:
2495:
2487:
2479:
2473:
2467:
2459:
2451:
2443:
2441:
2412:
2403:
2301:
2299:
2091:
2089:
1904:of a well-ordered set, except a possible
1785:
1759:
1733:
1698:
1674:
1602:
1582:
1580:
1550:
1548:
1521:
1496:
1479:
1477:
1452:
1435:
1433:
1408:
1394:
1392:
1356:
1333:
1323:
1321:
1294:
1265:
1252:
1250:
1217:
1196:
1194:
1164:
1138:
4264:Positive cone of an ordered vector space
3499:
2915:(except possibly for a last element of
2220:one of the following conditions holds:
1172:
45:
3445:), that is, if and only if the set is
2191:Unlike the standard ordering ≤ of the
7:
2709:axioms of set theory (including the
2054:
1913:
1666:All definitions tacitly require the
2010:Every well-ordered set is uniquely
1957:Every well-ordered set is uniquely
1181:{\displaystyle S\neq \varnothing :}
3791:Properties & Types (
25:
4247:Positive cone of an ordered field
3325:works for the entire ordered set.
3211:. The previous set is the set of
2195:, the standard ordering ≤ of the
4101:Ordered topological vector space
3407:For subsets we can distinguish:
2719:generalized continuum hypothesis
2057:, below, for an example). For a
1651:
1638:
1567:{\displaystyle {\text{not }}aRa}
1072:
1062:
1032:
1022:
1000:
990:
955:
933:
923:
888:
856:
846:
821:
789:
774:
754:
722:
712:
702:
682:
650:
630:
620:
610:
578:
558:
514:
494:
484:
447:
422:
412:
380:
355:
318:
288:
256:
194:
159:
2696:{\displaystyle (0,1)\subseteq }
2644:The standard ordering ≤ of any
2478:
2472:
2291:can be visualized as follows:
2074:The standard ordering ≤ of the
27:Class of mathematical orderings
3246:
3166:
3076:
2991:
2843:
2838:
2835:
2829:
2817:
2690:
2678:
2672:
2660:
2504:
2496:
2488:
2480:
2468:
2460:
2452:
2444:
2170:This is a well-ordered set of
1659:in the "Symmetric" column and
1595:
1340:
1285:
1214:
1:
4058:Series-parallel partial order
3395:{\displaystyle \mathbb {N} .}
2898:{\displaystyle \mathbb {Q} .}
1973:, which is equivalent to the
1969:of the well-ordered set. The
1850:with the property that every
1660:
1647:
1057:
1052:
1047:
1042:
1017:
985:
980:
975:
970:
965:
950:
918:
913:
908:
903:
898:
883:
871:
866:
841:
836:
831:
816:
804:
799:
784:
769:
764:
749:
737:
732:
697:
692:
677:
665:
660:
645:
640:
605:
593:
588:
573:
568:
553:
548:
543:
529:
524:
509:
504:
479:
474:
462:
457:
442:
437:
432:
407:
395:
390:
375:
370:
365:
350:
345:
333:
328:
313:
308:
303:
298:
283:
271:
266:
251:
246:
241:
236:
231:
226:
209:
204:
189:
184:
179:
174:
169:
3737:Cantor's isomorphism theorem
2936:{\displaystyle \mathbb {Q} }
2749:{\displaystyle \mathbb {R} }
2065:Examples and counterexamples
3777:Szpilrajn extension theorem
3752:Hausdorff maximal principle
3727:Boolean prime ideal theorem
3509:Information and Computation
2907:There is an injection from
2424:{\displaystyle x\leq _{z}y}
4340:
4123:Topological vector lattice
2717:proved that ZF + GCH (the
2003:
1981:), the proof technique of
1920:contains for every subset
1865:in this ordering. The set
3645:
3330:axiom of dependent choice
3044:Examples of well orders:
1988:The observation that the
3732:Cantor–Bernstein theorem
3521:10.1016/j.ic.2012.11.003
3348:by endowing it with the
4276:Partially ordered group
4096:Specialization preorder
3566:10.4064/fm-56-3-325-345
3552:Fundamenta Mathematicae
3307:Equivalent formulations
2797:is the last element of
2268:are both negative, and
2249:are both positive, and
1996:(for natural numbers).
1994:well-ordering principle
3762:Kruskal's tree theorem
3757:Knaster–Tarski theorem
3747:Dushnik–Miller theorem
3396:
3285:
3199:
3103:
3030:
2937:
2899:
2864:
2750:
2697:
2620:
2529:
2425:
2370:
2161:
2055:§ Natural numbers
1914:§ Natural numbers
1803:
1774:
1748:
1722:
1721:{\displaystyle a,b,c,}
1683:
1624:
1568:
1536:
1509:
1465:
1421:
1380:
1309:
1238:
1182:
1153:
3435:first-countable space
3397:
3323:Transfinite induction
3286:
3200:
3104:
3031:
2938:
2900:
2865:
2751:
2698:
2621:
2530:
2426:
2382:is isomorphic to the
2371:
2162:
1983:transfinite induction
1979:well-founded relation
1971:well-ordering theorem
1804:
1775:
1749:
1723:
1684:
1625:
1569:
1537:
1510:
1466:
1422:
1381:
1310:
1239:
1183:
1154:
1133:Definitions, for all
4254:Ordered vector space
3449:or has the smallest
3381:
3222:
3120:
3052:
2972:
2925:
2884:
2805:
2781:be the successor of
2738:
2657:
2545:
2440:
2402:
2298:
2088:
1802:{\displaystyle aRc.}
1784:
1758:
1732:
1697:
1673:
1668:homogeneous relation
1579:
1547:
1520:
1476:
1432:
1391:
1320:
1249:
1193:
1163:
1137:
879:Strict partial order
154:Equivalence relation
4092:Alexandrov topology
4038:Lexicographic order
3997:Well-quasi-ordering
3592:. pp. 4–6, 9.
3218:The set of numbers
3116:The set of numbers
3048:The set of numbers
2042:-th element" where
1833:well-order relation
1773:{\displaystyle bRc}
1747:{\displaystyle aRb}
1535:{\displaystyle aRa}
1152:{\displaystyle a,b}
538:Well-quasi-ordering
4073:Transitive closure
4033:Converse/Transpose
3742:Dilworth's theorem
3580:Folland, Gerald B.
3479:Well partial order
3392:
3372:, for example the
3281:
3195:
3099:
3026:
2933:
2895:
2872:injective function
2860:
2746:
2693:
2616:
2614:
2525:
2421:
2366:
2364:
2157:
2155:
2059:countably infinite
2034:, the expression "
1952:strict total order
1799:
1770:
1744:
1718:
1679:
1620:
1618:
1564:
1532:
1505:
1503:
1461:
1459:
1417:
1415:
1376:
1374:
1305:
1303:
1234:
1232:
1178:
1149:
1013:Strict total order
4301:
4300:
4259:Partially ordered
4068:Symmetric closure
4053:Reflexive closure
3796:
3599:978-0-471-31716-6
3463:Tree (set theory)
3346:topological space
2715:Wacław Sierpiński
2511:
2476:
2235:is positive, and
1930:least upper bound
1815:
1814:
1682:{\displaystyle R}
1633:
1632:
1605:
1553:
1499:
1455:
1411:
1359:
1268:
946:Strict weak order
132:Total, Semiconnex
16:(Redirected from
4331:
4043:Linear extension
3792:
3772:Mirsky's theorem
3632:
3625:
3618:
3609:
3603:
3571:
3570:
3568:
3539:
3533:
3532:
3504:
3474:Well-founded set
3465:, generalization
3403:
3401:
3399:
3398:
3393:
3388:
3371:
3297:
3290:
3288:
3287:
3282:
3249:
3243:
3242:
3210:
3204:
3202:
3201:
3196:
3169:
3163:
3162:
3141:
3140:
3112:
3108:
3106:
3105:
3100:
3079:
3073:
3072:
3039:
3035:
3033:
3032:
3027:
2994:
2985:
2964:
2956:
2952:
2948:
2944:
2942:
2940:
2939:
2934:
2932:
2918:
2914:
2910:
2906:
2904:
2902:
2901:
2896:
2891:
2877:
2869:
2867:
2866:
2861:
2846:
2800:
2796:
2792:
2788:
2784:
2780:
2769:
2765:
2761:
2758:well ordered by
2757:
2755:
2753:
2752:
2747:
2745:
2731:
2704:
2702:
2700:
2699:
2694:
2635:
2625:
2623:
2622:
2617:
2615:
2534:
2532:
2531:
2526:
2512:
2509:
2507:
2499:
2491:
2483:
2477:
2474:
2471:
2463:
2455:
2447:
2430:
2428:
2427:
2422:
2417:
2416:
2394:
2381:
2375:
2373:
2372:
2367:
2365:
2290:
2283:
2281:
2275:
2267:
2263:
2258:
2248:
2244:
2238:
2234:
2229:
2216:
2212:
2182:
2166:
2164:
2163:
2158:
2156:
2045:
2041:
2037:
2033:
2012:order isomorphic
1959:order isomorphic
1939:
1935:
1923:
1919:
1911:
1906:greatest element
1903:
1871:well-ordered set
1868:
1860:
1849:
1841:
1808:
1806:
1805:
1800:
1779:
1777:
1776:
1771:
1753:
1751:
1750:
1745:
1727:
1725:
1724:
1719:
1688:
1686:
1685:
1680:
1662:
1658:
1655:
1654:
1649:
1645:
1642:
1641:
1629:
1627:
1626:
1621:
1619:
1606:
1603:
1573:
1571:
1570:
1565:
1554:
1551:
1541:
1539:
1538:
1533:
1514:
1512:
1511:
1506:
1504:
1500:
1497:
1470:
1468:
1467:
1462:
1460:
1456:
1453:
1426:
1424:
1423:
1418:
1416:
1412:
1409:
1385:
1383:
1382:
1377:
1375:
1360:
1357:
1334:
1314:
1312:
1311:
1306:
1304:
1295:
1269:
1266:
1243:
1241:
1240:
1235:
1233:
1218:
1199:
1187:
1185:
1184:
1179:
1158:
1156:
1155:
1150:
1079:
1076:
1075:
1069:
1066:
1065:
1059:
1054:
1049:
1044:
1039:
1036:
1035:
1029:
1026:
1025:
1019:
1007:
1004:
1003:
997:
994:
993:
987:
982:
977:
972:
967:
962:
959:
958:
952:
940:
937:
936:
930:
927:
926:
920:
915:
910:
905:
900:
895:
892:
891:
885:
873:
868:
863:
860:
859:
853:
850:
849:
843:
838:
833:
828:
825:
824:
818:
812:Meet-semilattice
806:
801:
796:
793:
792:
786:
781:
778:
777:
771:
766:
761:
758:
757:
751:
745:Join-semilattice
739:
734:
729:
726:
725:
719:
716:
715:
709:
706:
705:
699:
694:
689:
686:
685:
679:
667:
662:
657:
654:
653:
647:
642:
637:
634:
633:
627:
624:
623:
617:
614:
613:
607:
595:
590:
585:
582:
581:
575:
570:
565:
562:
561:
555:
550:
545:
540:
531:
526:
521:
518:
517:
511:
506:
501:
498:
497:
491:
488:
487:
481:
476:
464:
459:
454:
451:
450:
444:
439:
434:
429:
426:
425:
419:
416:
415:
409:
397:
392:
387:
384:
383:
377:
372:
367:
362:
359:
358:
352:
347:
335:
330:
325:
322:
321:
315:
310:
305:
300:
295:
292:
291:
285:
273:
268:
263:
260:
259:
253:
248:
243:
238:
233:
228:
223:
221:
211:
206:
201:
198:
197:
191:
186:
181:
176:
171:
166:
163:
162:
156:
74:
73:
64:
57:
50:
43:
41:binary relations
32:
21:
4339:
4338:
4334:
4333:
4332:
4330:
4329:
4328:
4324:Wellfoundedness
4319:Ordinal numbers
4304:
4303:
4302:
4297:
4293:Young's lattice
4149:
4077:
4016:
3866:Heyting algebra
3814:Boolean algebra
3786:
3767:Laver's theorem
3715:
3681:Boolean algebra
3676:Binary relation
3664:
3641:
3636:
3606:
3600:
3578:
3574:
3541:
3540:
3536:
3506:
3505:
3501:
3497:
3484:Prewellordering
3459:
3440:
3379:
3378:
3376:
3374:natural numbers
3369:
3360:isolated points
3342:
3313:totally ordered
3309:
3292:
3291:has order type
3231:
3220:
3219:
3206:
3205:has order type
3145:
3129:
3118:
3117:
3110:
3109:has order type
3061:
3050:
3049:
3037:
2970:
2969:
2962:
2957:. For example,
2954:
2950:
2946:
2923:
2922:
2920:
2916:
2912:
2908:
2882:
2881:
2879:
2875:
2803:
2802:
2798:
2794:
2790:
2786:
2782:
2771:
2767:
2763:
2759:
2736:
2735:
2733:
2732:is a subset of
2729:
2711:axiom of choice
2655:
2654:
2652:
2642:
2633:
2613:
2612:
2607:
2602:
2594:
2589:
2581:
2576:
2568:
2563:
2555:
2543:
2542:
2510: and
2438:
2437:
2408:
2400:
2399:
2386:
2379:
2363:
2362:
2357:
2349:
2341:
2333:
2328:
2323:
2318:
2313:
2308:
2296:
2295:
2288:
2277:
2276:| ≤ |
2271:
2269:
2265:
2261:
2250:
2246:
2242:
2236:
2232:
2224:
2214:
2210:
2208:binary relation
2193:natural numbers
2189:
2174:
2154:
2153:
2148:
2143:
2138:
2133:
2128:
2123:
2118:
2113:
2108:
2103:
2098:
2086:
2085:
2076:natural numbers
2072:
2070:Natural numbers
2067:
2043:
2039:
2035:
2031:
2028:cardinal number
2008:
2002:
2000:Ordinal numbers
1990:natural numbers
1975:axiom of choice
1937:
1933:
1921:
1917:
1909:
1901:
1866:
1858:
1847:
1839:
1817:
1816:
1809:
1782:
1781:
1756:
1755:
1730:
1729:
1695:
1694:
1671:
1670:
1664:
1656:
1652:
1643:
1639:
1617:
1616:
1599:
1598:
1577:
1576:
1545:
1544:
1518:
1517:
1502:
1501:
1493:
1492:
1474:
1473:
1458:
1457:
1449:
1448:
1430:
1429:
1414:
1413:
1405:
1404:
1389:
1388:
1373:
1372:
1361:
1344:
1343:
1335:
1318:
1317:
1302:
1301:
1296:
1282:
1281:
1270:
1267: and
1247:
1246:
1231:
1230:
1219:
1211:
1210:
1191:
1190:
1161:
1160:
1135:
1134:
1077:
1073:
1067:
1063:
1037:
1033:
1027:
1023:
1005:
1001:
995:
991:
960:
956:
938:
934:
928:
924:
893:
889:
861:
857:
851:
847:
826:
822:
794:
790:
779:
775:
759:
755:
727:
723:
717:
713:
707:
703:
687:
683:
655:
651:
635:
631:
625:
621:
615:
611:
583:
579:
563:
559:
536:
519:
515:
499:
495:
489:
485:
470:Prewellordering
452:
448:
427:
423:
417:
413:
385:
381:
360:
356:
323:
319:
293:
289:
261:
257:
219:
216:
199:
195:
164:
160:
152:
144:
68:
35:
28:
23:
22:
15:
12:
11:
5:
4337:
4335:
4327:
4326:
4321:
4316:
4306:
4305:
4299:
4298:
4296:
4295:
4290:
4285:
4284:
4283:
4273:
4272:
4271:
4266:
4261:
4251:
4250:
4249:
4239:
4234:
4233:
4232:
4227:
4220:Order morphism
4217:
4216:
4215:
4205:
4200:
4195:
4190:
4185:
4184:
4183:
4173:
4168:
4163:
4157:
4155:
4151:
4150:
4148:
4147:
4146:
4145:
4140:
4138:Locally convex
4135:
4130:
4120:
4118:Order topology
4115:
4114:
4113:
4111:Order topology
4108:
4098:
4088:
4086:
4079:
4078:
4076:
4075:
4070:
4065:
4060:
4055:
4050:
4045:
4040:
4035:
4030:
4024:
4022:
4018:
4017:
4015:
4014:
4004:
3994:
3989:
3984:
3979:
3974:
3969:
3964:
3959:
3958:
3957:
3947:
3942:
3941:
3940:
3935:
3930:
3925:
3923:Chain-complete
3915:
3910:
3909:
3908:
3903:
3898:
3893:
3888:
3878:
3873:
3868:
3863:
3858:
3848:
3843:
3838:
3833:
3828:
3823:
3822:
3821:
3811:
3806:
3800:
3798:
3788:
3787:
3785:
3784:
3779:
3774:
3769:
3764:
3759:
3754:
3749:
3744:
3739:
3734:
3729:
3723:
3721:
3717:
3716:
3714:
3713:
3708:
3703:
3698:
3693:
3688:
3683:
3678:
3672:
3670:
3666:
3665:
3663:
3662:
3657:
3652:
3646:
3643:
3642:
3637:
3635:
3634:
3627:
3620:
3612:
3605:
3604:
3598:
3575:
3573:
3572:
3559:(3): 325–345.
3534:
3498:
3496:
3493:
3492:
3491:
3486:
3481:
3476:
3471:
3469:Ordinal number
3466:
3458:
3455:
3438:
3424:
3423:
3420:
3416:
3405:
3404:
3391:
3387:
3363:
3350:order topology
3341:
3340:Order topology
3338:
3337:
3336:
3333:
3326:
3320:
3308:
3305:
3304:
3303:
3300:order topology
3280:
3277:
3274:
3271:
3268:
3265:
3262:
3259:
3256:
3253:
3248:
3241:
3238:
3234:
3230:
3227:
3216:
3194:
3191:
3188:
3185:
3182:
3179:
3176:
3173:
3168:
3161:
3158:
3155:
3152:
3148:
3144:
3139:
3136:
3132:
3128:
3125:
3114:
3098:
3095:
3092:
3089:
3086:
3083:
3078:
3071:
3068:
3064:
3060:
3057:
3042:
3041:
3025:
3022:
3019:
3016:
3013:
3010:
3007:
3004:
3001:
2998:
2993:
2988:
2984:
2980:
2977:
2966:
2931:
2894:
2890:
2859:
2856:
2853:
2850:
2845:
2840:
2837:
2834:
2831:
2828:
2825:
2822:
2819:
2816:
2813:
2810:
2744:
2692:
2689:
2686:
2683:
2680:
2677:
2674:
2671:
2668:
2665:
2662:
2641:
2638:
2627:
2626:
2611:
2608:
2606:
2603:
2601:
2598:
2595:
2593:
2590:
2588:
2585:
2582:
2580:
2577:
2575:
2572:
2569:
2567:
2564:
2562:
2559:
2556:
2554:
2551:
2550:
2536:
2535:
2524:
2521:
2518:
2515:
2506:
2502:
2498:
2494:
2490:
2486:
2482:
2470:
2466:
2462:
2458:
2454:
2450:
2446:
2432:if and only if
2420:
2415:
2411:
2407:
2384:ordinal number
2377:
2376:
2361:
2358:
2356:
2353:
2350:
2348:
2345:
2342:
2340:
2337:
2334:
2332:
2329:
2327:
2324:
2322:
2319:
2317:
2314:
2312:
2309:
2307:
2304:
2303:
2287:This relation
2285:
2284:
2259:
2240:
2230:
2218:if and only if
2206:The following
2188:
2185:
2168:
2167:
2152:
2149:
2147:
2144:
2142:
2139:
2137:
2134:
2132:
2129:
2127:
2124:
2122:
2119:
2117:
2114:
2112:
2109:
2107:
2104:
2102:
2099:
2097:
2094:
2093:
2071:
2068:
2066:
2063:
2016:ordinal number
2006:Ordinal number
2004:Main article:
2001:
1998:
1963:ordinal number
1844:total ordering
1813:
1812:
1798:
1795:
1792:
1789:
1769:
1766:
1763:
1743:
1740:
1737:
1717:
1714:
1711:
1708:
1705:
1702:
1678:
1635:
1634:
1631:
1630:
1615:
1612:
1609:
1601:
1600:
1597:
1594:
1591:
1588:
1585:
1584:
1574:
1563:
1560:
1557:
1542:
1531:
1528:
1525:
1515:
1495:
1494:
1491:
1488:
1485:
1482:
1481:
1471:
1451:
1450:
1447:
1444:
1441:
1438:
1437:
1427:
1407:
1406:
1403:
1400:
1397:
1396:
1386:
1371:
1368:
1365:
1362:
1358: or
1355:
1352:
1349:
1346:
1345:
1342:
1339:
1336:
1332:
1329:
1326:
1325:
1315:
1300:
1297:
1293:
1290:
1287:
1284:
1283:
1280:
1277:
1274:
1271:
1264:
1261:
1258:
1255:
1254:
1244:
1229:
1226:
1223:
1220:
1216:
1213:
1212:
1209:
1206:
1203:
1200:
1198:
1188:
1177:
1174:
1171:
1168:
1148:
1145:
1142:
1130:
1129:
1124:
1119:
1114:
1109:
1104:
1099:
1094:
1089:
1084:
1081:
1080:
1070:
1060:
1055:
1050:
1045:
1040:
1030:
1020:
1015:
1009:
1008:
998:
988:
983:
978:
973:
968:
963:
953:
948:
942:
941:
931:
921:
916:
911:
906:
901:
896:
886:
881:
875:
874:
869:
864:
854:
844:
839:
834:
829:
819:
814:
808:
807:
802:
797:
787:
782:
772:
767:
762:
752:
747:
741:
740:
735:
730:
720:
710:
700:
695:
690:
680:
675:
669:
668:
663:
658:
648:
643:
638:
628:
618:
608:
603:
597:
596:
591:
586:
576:
571:
566:
556:
551:
546:
541:
533:
532:
527:
522:
512:
507:
502:
492:
482:
477:
472:
466:
465:
460:
455:
445:
440:
435:
430:
420:
410:
405:
399:
398:
393:
388:
378:
373:
368:
363:
353:
348:
343:
341:Total preorder
337:
336:
331:
326:
316:
311:
306:
301:
296:
286:
281:
275:
274:
269:
264:
254:
249:
244:
239:
234:
229:
224:
213:
212:
207:
202:
192:
187:
182:
177:
172:
167:
157:
149:
148:
146:
141:
139:
137:
135:
133:
130:
128:
126:
123:
122:
117:
112:
107:
102:
97:
92:
87:
82:
77:
70:
69:
67:
66:
59:
52:
44:
30:
29:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4336:
4325:
4322:
4320:
4317:
4315:
4312:
4311:
4309:
4294:
4291:
4289:
4286:
4282:
4279:
4278:
4277:
4274:
4270:
4267:
4265:
4262:
4260:
4257:
4256:
4255:
4252:
4248:
4245:
4244:
4243:
4242:Ordered field
4240:
4238:
4235:
4231:
4228:
4226:
4223:
4222:
4221:
4218:
4214:
4211:
4210:
4209:
4206:
4204:
4201:
4199:
4198:Hasse diagram
4196:
4194:
4191:
4189:
4186:
4182:
4179:
4178:
4177:
4176:Comparability
4174:
4172:
4169:
4167:
4164:
4162:
4159:
4158:
4156:
4152:
4144:
4141:
4139:
4136:
4134:
4131:
4129:
4126:
4125:
4124:
4121:
4119:
4116:
4112:
4109:
4107:
4104:
4103:
4102:
4099:
4097:
4093:
4090:
4089:
4087:
4084:
4080:
4074:
4071:
4069:
4066:
4064:
4061:
4059:
4056:
4054:
4051:
4049:
4048:Product order
4046:
4044:
4041:
4039:
4036:
4034:
4031:
4029:
4026:
4025:
4023:
4021:Constructions
4019:
4013:
4009:
4005:
4002:
3998:
3995:
3993:
3990:
3988:
3985:
3983:
3980:
3978:
3975:
3973:
3970:
3968:
3965:
3963:
3960:
3956:
3953:
3952:
3951:
3948:
3946:
3943:
3939:
3936:
3934:
3931:
3929:
3926:
3924:
3921:
3920:
3919:
3918:Partial order
3916:
3914:
3911:
3907:
3906:Join and meet
3904:
3902:
3899:
3897:
3894:
3892:
3889:
3887:
3884:
3883:
3882:
3879:
3877:
3874:
3872:
3869:
3867:
3864:
3862:
3859:
3857:
3853:
3849:
3847:
3844:
3842:
3839:
3837:
3834:
3832:
3829:
3827:
3824:
3820:
3817:
3816:
3815:
3812:
3810:
3807:
3805:
3804:Antisymmetric
3802:
3801:
3799:
3795:
3789:
3783:
3780:
3778:
3775:
3773:
3770:
3768:
3765:
3763:
3760:
3758:
3755:
3753:
3750:
3748:
3745:
3743:
3740:
3738:
3735:
3733:
3730:
3728:
3725:
3724:
3722:
3718:
3712:
3711:Weak ordering
3709:
3707:
3704:
3702:
3699:
3697:
3696:Partial order
3694:
3692:
3689:
3687:
3684:
3682:
3679:
3677:
3674:
3673:
3671:
3667:
3661:
3658:
3656:
3653:
3651:
3648:
3647:
3644:
3640:
3633:
3628:
3626:
3621:
3619:
3614:
3613:
3610:
3601:
3595:
3591:
3587:
3586:
3581:
3577:
3576:
3567:
3562:
3558:
3554:
3553:
3548:
3544:
3538:
3535:
3530:
3526:
3522:
3518:
3514:
3510:
3503:
3500:
3494:
3490:
3487:
3485:
3482:
3480:
3477:
3475:
3472:
3470:
3467:
3464:
3461:
3460:
3456:
3454:
3452:
3448:
3444:
3436:
3431:
3429:
3421:
3417:
3414:
3410:
3409:
3408:
3389:
3375:
3367:
3364:
3361:
3358:
3357:
3356:
3353:
3351:
3347:
3339:
3334:
3331:
3327:
3324:
3321:
3318:
3317:
3316:
3314:
3306:
3301:
3295:
3275:
3269:
3263:
3260:
3257:
3254:
3251:
3239:
3236:
3232:
3228:
3217:
3214:
3209:
3189:
3186:
3183:
3180:
3177:
3174:
3171:
3159:
3156:
3153:
3150:
3146:
3142:
3137:
3134:
3130:
3126:
3115:
3093:
3090:
3087:
3084:
3081:
3069:
3066:
3062:
3058:
3047:
3046:
3045:
3020:
3017:
3014:
3011:
3008:
3005:
3002:
2999:
2996:
2986:
2982:
2978:
2967:
2960:
2959:
2958:
2892:
2873:
2854:
2851:
2848:
2832:
2826:
2823:
2820:
2811:
2808:
2778:
2774:
2726:
2724:
2720:
2716:
2712:
2708:
2687:
2684:
2681:
2675:
2669:
2666:
2663:
2651:
2650:open interval
2647:
2646:real interval
2639:
2637:
2632:
2629:This has the
2609:
2604:
2599:
2596:
2591:
2586:
2583:
2578:
2573:
2570:
2565:
2560:
2557:
2552:
2541:
2540:
2539:
2522:
2519:
2516:
2513:
2500:
2492:
2484:
2464:
2456:
2448:
2436:
2435:
2434:
2433:
2418:
2413:
2409:
2405:
2396:
2393:
2389:
2385:
2359:
2354:
2351:
2346:
2343:
2338:
2335:
2330:
2325:
2320:
2315:
2310:
2305:
2294:
2293:
2292:
2280:
2274:
2260:
2257:
2253:
2241:
2231:
2227:
2223:
2222:
2221:
2219:
2209:
2204:
2202:
2198:
2194:
2186:
2184:
2181:
2177:
2173:
2150:
2145:
2140:
2135:
2130:
2125:
2120:
2115:
2110:
2105:
2100:
2095:
2084:
2083:
2082:
2079:
2077:
2069:
2064:
2062:
2060:
2056:
2052:
2047:
2029:
2025:
2021:
2018:, called the
2017:
2013:
2007:
1999:
1997:
1995:
1991:
1986:
1984:
1980:
1976:
1972:
1968:
1965:, called the
1964:
1960:
1955:
1953:
1950:
1946:
1941:
1931:
1927:
1915:
1907:
1898:
1896:
1895:well ordering
1892:
1888:
1884:
1880:
1876:
1872:
1864:
1863:least element
1856:
1853:
1845:
1838:
1834:
1830:
1829:well-ordering
1826:
1822:
1811:
1796:
1793:
1790:
1787:
1767:
1764:
1761:
1741:
1738:
1735:
1715:
1712:
1709:
1706:
1703:
1700:
1692:
1676:
1669:
1637:
1636:
1613:
1610:
1607:
1592:
1589:
1586:
1575:
1561:
1558:
1555:
1543:
1529:
1526:
1523:
1516:
1489:
1486:
1483:
1472:
1445:
1442:
1439:
1428:
1401:
1387:
1369:
1366:
1363:
1353:
1350:
1347:
1337:
1330:
1327:
1316:
1298:
1291:
1288:
1278:
1275:
1272:
1262:
1259:
1256:
1245:
1227:
1224:
1221:
1207:
1204:
1201:
1189:
1175:
1169:
1166:
1146:
1143:
1140:
1132:
1131:
1128:
1125:
1123:
1120:
1118:
1115:
1113:
1110:
1108:
1105:
1103:
1100:
1098:
1095:
1093:
1092:Antisymmetric
1090:
1088:
1085:
1083:
1082:
1071:
1061:
1056:
1051:
1046:
1041:
1031:
1021:
1016:
1014:
1011:
1010:
999:
989:
984:
979:
974:
969:
964:
954:
949:
947:
944:
943:
932:
922:
917:
912:
907:
902:
897:
887:
882:
880:
877:
876:
870:
865:
855:
845:
840:
835:
830:
820:
815:
813:
810:
809:
803:
798:
788:
783:
773:
768:
763:
753:
748:
746:
743:
742:
736:
731:
721:
711:
701:
696:
691:
681:
676:
674:
671:
670:
664:
659:
649:
644:
639:
629:
619:
609:
604:
602:
601:Well-ordering
599:
598:
592:
587:
577:
572:
567:
557:
552:
547:
542:
539:
535:
534:
528:
523:
513:
508:
503:
493:
483:
478:
473:
471:
468:
467:
461:
456:
446:
441:
436:
431:
421:
411:
406:
404:
401:
400:
394:
389:
379:
374:
369:
364:
354:
349:
344:
342:
339:
338:
332:
327:
317:
312:
307:
302:
297:
287:
282:
280:
279:Partial order
277:
276:
270:
265:
255:
250:
245:
240:
235:
230:
225:
222:
215:
214:
208:
203:
193:
188:
183:
178:
173:
168:
158:
155:
151:
150:
147:
142:
140:
138:
136:
134:
131:
129:
127:
125:
124:
121:
118:
116:
113:
111:
108:
106:
103:
101:
98:
96:
93:
91:
88:
86:
85:Antisymmetric
83:
81:
78:
76:
75:
72:
71:
65:
60:
58:
53:
51:
46:
42:
38:
34:
33:
19:
4314:Order theory
4085:& Orders
4063:Star product
4011:
3992:Well-founded
3945:Prefix order
3901:Distributive
3891:Complemented
3861:Foundational
3826:Completeness
3782:Zorn's lemma
3686:Cyclic order
3669:Key concepts
3639:Order theory
3584:
3556:
3550:
3543:Feferman, S.
3537:
3512:
3508:
3502:
3489:Directed set
3453:order type.
3432:
3426:A subset is
3425:
3406:
3366:limit points
3354:
3343:
3311:If a set is
3310:
3293:
3213:limit points
3207:
3043:
2789:ordering on
2776:
2772:
2727:
2643:
2628:
2537:
2397:
2391:
2387:
2378:
2286:
2278:
2272:
2255:
2251:
2225:
2205:
2190:
2179:
2175:
2169:
2080:
2073:
2048:
2014:to a unique
2009:
1987:
1961:to a unique
1956:
1949:well-founded
1942:
1899:
1894:
1891:well ordered
1890:
1886:
1883:wellordering
1882:
1878:
1874:
1870:
1832:
1828:
1824:
1818:
1665:
1102:Well-founded
600:
220:(Quasiorder)
95:Well-founded
4269:Riesz space
4230:Isomorphism
4106:Normal cone
4028:Composition
3962:Semilattice
3871:Homogeneous
3856:Equivalence
3706:Total order
3451:uncountable
3298:. With the
2762:. For each
2239:is negative
2051:cardinality
1926:upper bound
1879:wellordered
1821:mathematics
1122:Irreflexive
403:Total order
115:Irreflexive
4308:Categories
4237:Order type
4171:Cofinality
4012:Well-order
3987:Transitive
3876:Idempotent
3809:Asymmetric
3495:References
2631:order type
2172:order type
2020:order type
1967:order type
1945:non-strict
1943:If ≤ is a
1887:well order
1825:well-order
1693:: for all
1691:transitive
1127:Asymmetric
120:Asymmetric
37:Transitive
18:Well order
4288:Upper set
4225:Embedding
4161:Antichain
3982:Tolerance
3972:Symmetric
3967:Semiorder
3913:Reflexive
3831:Connected
3447:countable
3443:omega-one
3270:∪
3264:ω
3255:≤
3237:−
3229:−
3190:ω
3175:≤
3157:−
3151:−
3143:−
3135:−
3127:−
3094:ω
3085:≤
3067:−
3059:−
3021:…
2852:∈
2676:⊆
2610:…
2597:−
2584:−
2571:−
2558:−
2517:≤
2410:≤
2360:…
2352:−
2344:−
2336:−
2331:…
2151:…
2121:…
1875:wellorder
1852:non-empty
1604:not
1596:⇒
1552:not
1487:∧
1443:∨
1341:⇒
1331:≠
1286:⇒
1215:⇒
1173:∅
1170:≠
1117:Reflexive
1112:Has meets
1107:Has joins
1097:Connected
1087:Symmetric
218:Preorder
145:reflexive
110:Reflexive
105:Has meets
100:Has joins
90:Connected
80:Symmetric
4083:Topology
3950:Preorder
3933:Eulerian
3896:Complete
3846:Directed
3836:Covering
3701:Preorder
3660:Category
3655:Glossary
3582:(1999).
3545:(1964).
3515:: 1–22.
3457:See also
2968:The set
2793:(unless
2201:negative
2197:integers
2187:Integers
2024:counting
1924:with an
1661:✗
1648:✗
1058:✗
1053:✗
1048:✗
1043:✗
1018:✗
986:✗
981:✗
976:✗
971:✗
966:✗
951:✗
919:✗
914:✗
909:✗
904:✗
899:✗
884:✗
872:✗
867:✗
842:✗
837:✗
832:✗
817:✗
805:✗
800:✗
785:✗
770:✗
765:✗
750:✗
738:✗
733:✗
698:✗
693:✗
678:✗
666:✗
661:✗
646:✗
641:✗
606:✗
594:✗
589:✗
574:✗
569:✗
554:✗
549:✗
544:✗
530:✗
525:✗
510:✗
505:✗
480:✗
475:✗
463:✗
458:✗
443:✗
438:✗
433:✗
408:✗
396:✗
391:✗
376:✗
371:✗
366:✗
351:✗
346:✗
334:✗
329:✗
314:✗
309:✗
304:✗
299:✗
284:✗
272:✗
267:✗
252:✗
247:✗
242:✗
237:✗
232:✗
227:✗
210:✗
205:✗
190:✗
185:✗
180:✗
175:✗
170:✗
4188:Duality
4166:Cofinal
4154:Related
4133:Fréchet
4010:)
3886:Bounded
3881:Lattice
3854:)
3852:Partial
3720:Results
3691:Lattice
3529:3016456
3428:cofinal
3413:bounded
3402:
3377:
2943:
2921:
2905:
2880:
2801:). Let
2756:
2734:
2703:
2653:
1835:) on a
673:Lattice
4213:Subnet
4193:Filter
4143:Normed
4128:Banach
4094:&
4001:Better
3938:Strict
3928:Graded
3819:topics
3650:Topics
3596:
3527:
2770:, let
2282:|
2270:|
1893:, and
1881:, and
1861:has a
1855:subset
1498:exists
1454:exists
1410:exists
39:
4203:Ideal
4181:Graph
3977:Total
3955:Total
3841:Dense
3590:Wiley
2874:from
2640:Reals
2215:x R y
1842:is a
1780:then
143:Anti-
3794:list
3594:ISBN
3419:set.
3261:<
3187:<
3091:<
2457:<
2264:and
2245:and
1827:(or
1823:, a
1754:and
1159:and
4208:Net
4008:Pre
3561:doi
3517:doi
3513:224
3296:+ 1
2911:to
2878:to
2785:in
2766:in
2723:V=L
2707:ZFC
2228:= 0
1936:in
1885:or
1857:of
1846:on
1837:set
1831:or
1819:In
1728:if
1689:be
1399:min
4310::
3557:56
3555:.
3549:.
3525:MR
3523:.
3511:.
3352:.
3332:).
2636:.
2475:or
2395:.
2390:+
2254:≤
2178:+
1940:.
1928:a
1897:.
1889:,
1877:,
4006:(
4003:)
3999:(
3850:(
3797:)
3631:e
3624:t
3617:v
3602:.
3569:.
3563::
3531:.
3519::
3441:(
3439:1
3390:.
3386:N
3370:ω
3294:ω
3279:}
3276:1
3273:{
3267:}
3258:n
3252:0
3247:|
3240:n
3233:2
3226:{
3208:ω
3193:}
3184:n
3181:,
3178:m
3172:0
3167:|
3160:n
3154:m
3147:2
3138:n
3131:2
3124:{
3113:.
3111:ω
3097:}
3088:n
3082:0
3077:|
3070:n
3063:2
3056:{
3040:.
3038:≤
3024:}
3018:,
3015:3
3012:,
3009:2
3006:,
3003:1
3000:=
2997:n
2992:|
2987:n
2983:/
2979:1
2976:{
2965:.
2963:≤
2955:≤
2951:X
2947:X
2930:Q
2917:X
2913:A
2909:X
2893:.
2889:Q
2876:A
2858:}
2855:X
2849:x
2844:|
2839:)
2836:)
2833:x
2830:(
2827:s
2824:,
2821:x
2818:(
2815:{
2812:=
2809:A
2799:X
2795:x
2791:X
2787:≤
2783:x
2779:)
2777:x
2775:(
2773:s
2768:X
2764:x
2760:≤
2743:R
2730:X
2691:]
2688:1
2685:,
2682:0
2679:[
2673:)
2670:1
2667:,
2664:0
2661:(
2634:ω
2605:4
2600:4
2592:3
2587:3
2579:2
2574:2
2566:1
2561:1
2553:0
2523:.
2520:y
2514:x
2505:|
2501:y
2497:|
2493:=
2489:|
2485:x
2481:|
2469:|
2465:y
2461:|
2453:|
2449:x
2445:|
2419:y
2414:z
2406:x
2392:ω
2388:ω
2380:R
2355:3
2347:2
2339:1
2326:4
2321:3
2316:2
2311:1
2306:0
2289:R
2279:y
2273:x
2266:y
2262:x
2256:y
2252:x
2247:y
2243:x
2237:y
2233:x
2226:x
2211:R
2180:ω
2176:ω
2146:9
2141:7
2136:5
2131:3
2126:1
2116:8
2111:6
2106:4
2101:2
2096:0
2044:β
2040:β
2036:n
2032:n
1938:S
1934:T
1922:T
1918:S
1910:s
1902:s
1867:S
1859:S
1848:S
1840:S
1797:.
1794:c
1791:R
1788:a
1768:c
1765:R
1762:b
1742:b
1739:R
1736:a
1716:,
1713:c
1710:,
1707:b
1704:,
1701:a
1677:R
1657:Y
1644:Y
1614:a
1611:R
1608:b
1593:b
1590:R
1587:a
1562:a
1559:R
1556:a
1530:a
1527:R
1524:a
1490:b
1484:a
1446:b
1440:a
1402:S
1370:a
1367:R
1364:b
1354:b
1351:R
1348:a
1338:b
1328:a
1299:b
1292:=
1289:a
1279:a
1276:R
1273:b
1263:b
1260:R
1257:a
1228:a
1225:R
1222:b
1208:b
1205:R
1202:a
1176::
1167:S
1147:b
1144:,
1141:a
1078:Y
1068:Y
1038:Y
1028:Y
1006:Y
996:Y
961:Y
939:Y
929:Y
894:Y
862:Y
852:Y
827:Y
795:Y
780:Y
760:Y
728:Y
718:Y
708:Y
688:Y
656:Y
636:Y
626:Y
616:Y
584:Y
564:Y
520:Y
500:Y
490:Y
453:Y
428:Y
418:Y
386:Y
361:Y
324:Y
294:Y
262:Y
200:Y
165:Y
63:e
56:t
49:v
20:)
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