1799:
25:
1786:
826:, but by adding a "gap condition" to the definition of the order on trees above, he found a natural variation of the theorem unprovable in this system. Much later, the Robertson–Seymour theorem would give another theorem unprovable by Î
2589:
1514:
1633:
790:, some special cases and variants of the theorem can be expressed in subsystems of second-order arithmetic much weaker than the subsystems where they can be proved. This was first observed by
1558:
2372:
2055:
2018:
1453:
1416:
3093:
Smith, Rick L. (1985), "The consistency strengths of some finite forms of the Higman and
Kruskal theorems", in Harrington, L. A.; Morley, M.; Scedrov, A.; et al. (eds.),
2331:
2284:
2238:
2086:
1858:
1620:
1242:
194:
1945:
1343:
991:
759:
364:
1977:
1375:
2163:
614:
578:
1585:
1017:
719:
2399:
2202:
2116:
1202:
1157:
1120:
1091:
1050:
892:
652:
517:
488:
439:
1303:
951:
1790:
To differentiate the two functions, "TREE" (with all caps) is the big TREE function, and "tree" (with all letters in lowercase) is the weak tree function.
2495:
43:
3084:(1985), "Nonprovability of certain combinatorial properties of finite trees", in Harrington, L. A.; Morley, M.; Scedrov, A.; et al. (eds.),
2937:
4154:
212:
The version given here is that proven by Nash-Williams; Kruskal's formulation is somewhat stronger. All trees we consider are finite.
4199:
4137:
3667:
3123:
3503:
1458:
3984:
1781:{\displaystyle \mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{8}(7)}(7)}(7)}(7)}(7).}
803:
147:
798:
field of reverse mathematics. In the case where the trees above are taken to be unlabeled (that is, in the case where
4120:
3979:
3397:
61:
3974:
3217:
3212:
3207:
3202:
3197:
3192:
3187:
3182:
3610:
2999:
2813:
674:
is well-quasi-ordered under the inf-embeddable order defined above. (That is to say, given any infinite sequence
127:
1519:
3692:
2337:
4204:
4011:
3931:
3432:
3312:
3081:
2424:
158:
3796:
3725:
3605:
3317:
3307:
3247:
2419:
2414:
1164:
787:
154:
91:
4189:
3699:
3687:
3650:
3625:
3600:
3554:
3523:
3262:
3116:
162:
2023:
1986:
1421:
1384:
3996:
3630:
3620:
3496:
3407:
2088:
explodes to a value that is so big that many other "large" combinatorial constants, such as
Friedman's
2841:
2291:
2244:
2212:
2060:
1832:
1594:
1216:
168:
161:, a result that has also proved important in reverse mathematics and leads to the even-faster-growing
3969:
3392:
1910:
1308:
956:
724:
329:
4209:
3901:
3528:
3470:
2875:
Internal finite tree embeddings. Reflections on the foundations of mathematics (Stanford, CA, 1998)
1950:
1348:
39:
4149:
4132:
2632:
2123:
846:
779:
143:
4061:
3677:
3465:
3360:
3109:
783:
583:
547:
4194:
4039:
3874:
3865:
3734:
3615:
3569:
3533:
3489:
3427:
3417:
3355:
2790:
2785:
279:
1563:
996:
698:
4127:
4086:
4076:
4066:
3811:
3774:
3764:
3744:
3729:
3074:
3031:
3011:
2972:
2926:
2882:
2462:) is defined as the length of the longest possible sequence that can be constructed with a
2377:
1802:
A sequence of rooted trees labelled from a set of 3 labels (blue < red < green). The
850:
813:
result with a provably impredicative proof. This case of the theorem is still provable by Î
2980:
2178:
2092:
1178:
1133:
1096:
1067:
1026:
868:
628:
493:
464:
415:
8:
4054:
3965:
3911:
3870:
3860:
3749:
3682:
3645:
3302:
1282:
930:
667:
135:
87:
83:
3015:
1818:
By incorporating labels, Friedman defined a far faster-growing function. For a positive
1053:
4166:
4093:
3946:
3855:
3845:
3786:
3704:
3640:
3350:
3252:
3242:
3035:
2960:
2402:
2167:
1588:
1168:
115:
4006:
3097:, Studies in Logic and the Foundations of Mathematics, North-Holland, pp. 119–136
111:
4103:
4081:
3941:
3926:
3906:
3709:
3267:
3088:, Studies in Logic and the Foundations of Mathematics, North-Holland, pp. 87–117
3066:
3039:
2917:
854:
2877:, Lect. Notes Log., vol. 15, Urbana, IL: Assoc. Symbol. Logic, pp. 60–91,
3916:
3769:
3452:
3272:
3222:
3062:
3019:
2952:
2912:
842:
4098:
3881:
3759:
3754:
3739:
3655:
3564:
3549:
3412:
3070:
3027:
2968:
2922:
2878:
2584:{\displaystyle n(1)=3,n(2)=11,\,{\textrm {and}}\,n(3)>2\uparrow ^{7197}158386}
810:
791:
4016:
4001:
3991:
3850:
3828:
3806:
3442:
3333:
3232:
3047:
2933:
1810:
vertices, and no tree is inf-embeddable within any later tree in the sequence.
119:
3023:
4183:
4115:
4071:
4049:
3921:
3791:
3779:
3584:
3402:
3177:
3157:
3132:
2893:
771:
367:
3936:
3818:
3801:
3719:
3559:
3512:
3437:
2889:
1127:
1061:
4142:
3835:
3714:
3579:
3447:
3387:
2171:
220:
75:
4110:
4044:
3885:
3237:
2964:
107:
4161:
4034:
3840:
3343:
3338:
3172:
2287:. Graham's number, for example, is much smaller than the lower bound
2956:
3956:
3823:
3574:
3277:
1798:
200:
application of the theorem gives the existence of the fast-growing
197:
2938:"Well-quasi-ordering, the tree theorem, and Vazsonyi's conjecture"
3167:
1819:
103:
3101:
1814:
is defined to be the longest possible length of such a sequence.
802:
has size one), Friedman found that the result was unprovable in
3481:
3422:
3227:
3162:
2981:"Wqo and bqo theory in subsystems of second order arithmetic"
1159:
in Peano arithmetic grows phenomenally fast as a function of
3095:
Harvey
Friedman's Research on the Foundations of Mathematics
3086:
Harvey
Friedman's Research on the Foundations of Mathematics
2894:"What's so special about Kruskal's theorem and the ordinal Γ
2756:
778:, Kruskal's tree theorem can be expressed and proven using
1093:
is true, but Peano arithmetic cannot prove the statement "
3152:
1509:{\displaystyle {\text{tree}}(3)\geq 844,424,930,131,960}
3048:"Proof-theoretic investigations on Kruskal's theorem"
2498:
2380:
2340:
2294:
2247:
2215:
2181:
2126:
2095:
2063:
2026:
1989:
1953:
1913:
1835:
1636:
1597:
1566:
1522:
1461:
1424:
1387:
1351:
1311:
1285:
1219:
1181:
1136:
1099:
1070:
1029:
999:
959:
933:
920:
is a finite sequence of unlabeled rooted trees where
871:
845:
confirms the strength of
Kruskal's theorem, with the
727:
701:
631:
586:
550:
496:
467:
418:
332:
171:
2631:), taking two arguments, is a particular version of
1052:are true as a consequence of Kruskal's theorem and
34:
may be too technical for most readers to understand
2583:
2393:
2366:
2325:
2278:
2232:
2196:
2157:
2110:
2080:
2049:
2012:
1971:
1939:
1852:
1780:
1614:
1579:
1552:
1508:
1447:
1410:
1369:
1337:
1297:
1236:
1196:
1151:
1114:
1085:
1044:
1011:
985:
945:
886:
753:
713:
646:
608:
572:
511:
482:
433:
358:
188:
153:In 2004, the result was generalized from trees to
2945:Transactions of the American Mathematical Society
90:set of labels is itself well-quasi-ordered under
4181:
3045:
2762:
2466:-letter alphabet such that no block of letters x
3002:(1963), "On well-quasi-ordering finite trees",
2849:Ohio State University Department of Mathematics
670:, then the set of rooted trees with labels in
134:). It has since become a prominent example in
3497:
3117:
3046:Rathjen, Michael; Weiermann, Andreas (1993),
2998:
2951:(2), American Mathematical Society: 210–225,
2442:Friedman originally denoted this function by
1204:holds similarly grows extremely quickly with
540:are any two distinct immediate successors of
131:
1622:(where the argument specifies the number of
794:in the early 1980s, an early success of the
2898:? A survey of some results in proof theory"
138:as a statement that cannot be proved in ATR
4155:Positive cone of a partially ordered group
3504:
3490:
3124:
3110:
1553:{\displaystyle {\text{tree}}(4)\gg g_{64}}
300:are rooted trees with vertices labeled in
2916:
2594:
2549:
2541:
1806:th tree in the sequence contains at most
1244:, the weak tree function, as the largest
1126:". Moreover, the length of the shortest
62:Learn how and when to remove this message
46:, without removing the technical details.
4138:Positive cone of an ordered vector space
2872:
2839:
2811:
2783:
2367:{\displaystyle g_{3\uparrow ^{187196}3}}
1797:
3080:
2978:
2932:
2888:
2170:, are extremely small by comparison. A
1890:of rooted trees labelled from a set of
123:
4182:
2840:Friedman, Harvey M. (8 October 1998),
1270:of unlabeled rooted trees, where each
3485:
3105:
3092:
2607:) taking one argument, is defined as
2595:
2479:is a subsequence of any later block x
860:
853:(sometimes confused with the smaller
809:, thus giving the first example of a
44:make it understandable to non-experts
658:Kruskal's tree theorem then states:
243:if the unique path from the root to
18:
2812:Friedman, Harvey M. (1 June 2000),
2768:
2449:
16:Well-quasi-ordering of finite trees
13:
3665:Properties & Types (
2784:Friedman, Harvey (28 March 2006),
2436:
2050:{\displaystyle {\text{TREE}}(2)=3}
2013:{\displaystyle {\text{TREE}}(1)=1}
1712:
1709:
1706:
1703:
1696:
1693:
1690:
1687:
1680:
1677:
1674:
1671:
1664:
1661:
1658:
1655:
1648:
1645:
1642:
1639:
1627:
1448:{\displaystyle {\text{tree}}(2)=5}
1411:{\displaystyle {\text{tree}}(1)=2}
765:
148:arithmetical transfinite recursion
14:
4221:
4121:Positive cone of an ordered field
3398:Indefinite and fictitious numbers
3131:
4200:Theorems in discrete mathematics
3975:Ordered topological vector space
3511:
3055:Annals of Pure and Applied Logic
2814:"Enormous Integers In Real Life"
2326:{\displaystyle A^{A(187196)}(1)}
2279:{\displaystyle A^{A(187196)}(1)}
2233:{\displaystyle {\text{TREE}}(3)}
2081:{\displaystyle {\text{TREE}}(3)}
1853:{\displaystyle {\text{TREE}}(n)}
1793:
1615:{\displaystyle {\text{TREE}}(3)}
1237:{\displaystyle {\text{tree}}(n)}
189:{\displaystyle {\text{TREE}}(3)}
23:
2833:
2450:
1940:{\displaystyle T_{i}\leq T_{j}}
1868:so that we have the following:
1338:{\displaystyle T_{i}\leq T_{j}}
1248:so that we have the following:
986:{\displaystyle T_{i}\leq T_{j}}
754:{\displaystyle T_{i}\leq T_{j}}
359:{\displaystyle T_{1}\leq T_{2}}
126:); a short proof was given by
2805:
2777:
2747:
2738:
2729:
2720:
2711:
2569:
2559:
2553:
2529:
2523:
2508:
2502:
2437:
2350:
2320:
2314:
2309:
2303:
2273:
2267:
2262:
2256:
2227:
2221:
2191:
2185:
2152:
2146:
2141:
2135:
2105:
2099:
2075:
2069:
2038:
2032:
2001:
1995:
1847:
1841:
1772:
1766:
1761:
1755:
1750:
1744:
1739:
1733:
1728:
1722:
1609:
1603:
1534:
1528:
1473:
1467:
1436:
1430:
1399:
1393:
1231:
1225:
1191:
1185:
1146:
1140:
1109:
1103:
1080:
1074:
1039:
1033:
881:
875:
641:
635:
603:
590:
567:
554:
506:
500:
477:
471:
428:
422:
263:if additionally the path from
183:
177:
82:states that the set of finite
1:
3932:Series-parallel partial order
3313:Conway chained arrow notation
2699:
3611:Cantor's isomorphism theorem
3067:10.1016/0168-0072(93)90192-G
2918:10.1016/0168-0072(91)90022-E
2873:Friedman, Harvey M. (2002),
2763:Rathjen & Weiermann 1993
2735:Simpson 1985, Definition 4.1
1972:{\displaystyle i<j\leq m}
1370:{\displaystyle i<j\leq m}
1165:primitive recursive function
849:of the theorem equaling the
207:
7:
3651:Szpilrajn extension theorem
3626:Hausdorff maximal principle
3601:Boolean prime ideal theorem
2408:
2158:{\displaystyle n^{n(5)}(5)}
1516:(about 844 trillion),
691:of rooted trees labeled in
10:
4226:
3997:Topological vector lattice
3408:Largest known prime number
3000:Nash-Williams, C. St.J. A.
2851:, pp. 5, 48 (Thm.6.8)
2786:"273:Sigma01/optimal/size"
2744:Simpson 1985, Theorem 5.14
271:contains no other vertex.
97:
4027:
3955:
3894:
3664:
3593:
3542:
3519:
3461:
3393:Extended real number line
3373:
3326:
3308:Knuth's up-arrow notation
3295:
3286:
3139:
3024:10.1017/S0305004100003844
2979:Marcone, Alberto (2001),
2753:Marcone 2001, p. 8–9
2726:Friedman 2002, p. 60
2717:Simpson 1985, Theorem 1.8
2425:Robertson–Seymour theorem
2334:, which is approximately
1983:The TREE sequence begins
1171:, for example. The least
159:Robertson–Seymour theorem
128:Crispin Nash-Williams
3606:Cantor–Bernstein theorem
3318:Steinhaus–Moser notation
2430:
2420:Kanamori–McAloon theorem
2415:Paris–Harrington theorem
788:Paris–Harrington theorem
609:{\displaystyle F(w_{2})}
573:{\displaystyle F(w_{1})}
4150:Partially ordered group
3970:Specialization preorder
2842:"Long Finite Sequences"
2774:Smith 1985, p. 120
847:proof-theoretic ordinal
780:second-order arithmetic
144:second-order arithmetic
3636:Kruskal's tree theorem
3631:Knaster–Tarski theorem
3621:Dushnik–Miller theorem
3361:Fast-growing hierarchy
3004:Proc. Camb. Phil. Soc.
2585:
2395:
2368:
2327:
2280:
2234:
2198:
2159:
2112:
2082:
2051:
2014:
1973:
1947:does not hold for any
1941:
1854:
1815:
1782:
1616:
1581:
1580:{\displaystyle g_{64}}
1554:
1510:
1449:
1412:
1371:
1345:does not hold for any
1339:
1299:
1238:
1198:
1163:, far faster than any
1153:
1116:
1087:
1046:
1013:
1012:{\displaystyle i<j}
987:
947:
888:
763:
755:
715:
714:{\displaystyle i<j}
648:
610:
574:
513:
484:
435:
412:precedes the label of
360:
190:
146:theory with a form of
80:Kruskal's tree theorem
3418:Long and short scales
3356:Grzegorczyk hierarchy
2905:Ann. Pure Appl. Logic
2821:Ohio State University
2791:Ohio State University
2586:
2396:
2394:{\displaystyle g_{x}}
2369:
2328:
2281:
2235:
2209:weak lower bound for
2199:
2160:
2113:
2083:
2052:
2015:
1974:
1942:
1855:
1801:
1783:
1617:
1582:
1555:
1511:
1450:
1413:
1372:
1340:
1300:
1239:
1199:
1154:
1117:
1088:
1047:
1014:
988:
948:
889:
756:
716:
660:
649:
611:
575:
544:, then the path from
514:
485:
436:
373:from the vertices of
361:
280:partially ordered set
223:, and given vertices
191:
4205:Trees (graph theory)
4128:Ordered vector space
2633:Ackermann's function
2496:
2378:
2338:
2292:
2245:
2213:
2197:{\displaystyle n(4)}
2179:
2124:
2111:{\displaystyle n(4)}
2093:
2061:
2024:
1987:
1951:
1911:
1907:vertices, such that
1872:There is a sequence
1833:
1634:
1595:
1564:
1520:
1459:
1422:
1385:
1349:
1309:
1305:vertices, such that
1283:
1252:There is a sequence
1217:
1197:{\displaystyle P(n)}
1179:
1152:{\displaystyle P(n)}
1134:
1115:{\displaystyle P(n)}
1097:
1086:{\displaystyle P(n)}
1068:
1045:{\displaystyle P(n)}
1027:
997:
957:
931:
887:{\displaystyle P(n)}
869:
851:small Veblen ordinal
725:
699:
647:{\displaystyle F(v)}
629:
584:
548:
512:{\displaystyle F(v)}
494:
483:{\displaystyle F(w)}
465:
448:is any successor of
434:{\displaystyle F(v)}
416:
330:
169:
3966:Alexandrov topology
3912:Lexicographic order
3871:Well-quasi-ordering
3433:Orders of magnitude
3303:Scientific notation
3082:Simpson, Stephen G.
3016:1963PCPS...59..833N
2988:Reverse Mathematics
2793:Department of Maths
1894:labels, where each
1298:{\displaystyle i+n}
1023:All the statements
946:{\displaystyle i+n}
784:Goodstein's theorem
382:to the vertices of
257:immediate successor
136:reverse mathematics
4190:Mathematical logic
3947:Transitive closure
3907:Converse/Transpose
3616:Dilworth's theorem
3351:Ackermann function
2581:
2391:
2364:
2323:
2276:
2230:
2194:
2155:
2108:
2078:
2047:
2010:
1969:
1937:
1860:to be the largest
1850:
1816:
1778:
1612:
1577:
1550:
1506:
1445:
1408:
1367:
1335:
1295:
1234:
1194:
1169:Ackermann function
1149:
1112:
1083:
1042:
1009:
983:
943:
894:is the statement:
884:
861:Weak tree function
751:
711:
668:well-quasi-ordered
644:
606:
570:
509:
490:is a successor of
480:
431:
356:
186:
120:Joseph Kruskal
88:well-quasi-ordered
4175:
4174:
4133:Partially ordered
3942:Symmetric closure
3927:Reflexive closure
3670:
3479:
3478:
3369:
3368:
2546:
2403:Graham's function
2219:
2205:, and, hence, an
2067:
2057:, then suddenly,
2030:
1993:
1839:
1630:) is larger than
1601:
1526:
1465:
1428:
1391:
1381:It is known that
1223:
855:Ackermann ordinal
395:For all vertices
175:
72:
71:
64:
4217:
3917:Linear extension
3666:
3646:Mirsky's theorem
3506:
3499:
3492:
3483:
3482:
3293:
3292:
3223:Eddington number
3168:Hundred thousand
3126:
3119:
3112:
3103:
3102:
3098:
3089:
3077:
3052:
3042:
2995:
2985:
2975:
2942:
2929:
2920:
2902:
2890:Gallier, Jean H.
2885:
2860:
2859:
2858:
2856:
2846:
2837:
2831:
2830:
2829:
2827:
2818:
2809:
2803:
2802:
2801:
2799:
2781:
2775:
2772:
2766:
2760:
2754:
2751:
2745:
2742:
2736:
2733:
2727:
2724:
2718:
2715:
2599:
2590:
2588:
2587:
2582:
2577:
2576:
2548:
2547:
2544:
2454:
2441:
2400:
2398:
2397:
2392:
2390:
2389:
2373:
2371:
2370:
2365:
2363:
2362:
2358:
2357:
2332:
2330:
2329:
2324:
2313:
2312:
2285:
2283:
2282:
2277:
2266:
2265:
2239:
2237:
2236:
2231:
2220:
2217:
2203:
2201:
2200:
2195:
2164:
2162:
2161:
2156:
2145:
2144:
2117:
2115:
2114:
2109:
2087:
2085:
2084:
2079:
2068:
2065:
2056:
2054:
2053:
2048:
2031:
2028:
2019:
2017:
2016:
2011:
1994:
1991:
1978:
1976:
1975:
1970:
1946:
1944:
1943:
1938:
1936:
1935:
1923:
1922:
1906:
1901:
1893:
1889:
1865:
1859:
1857:
1856:
1851:
1840:
1837:
1826:
1813:
1809:
1805:
1787:
1785:
1784:
1779:
1765:
1764:
1754:
1753:
1743:
1742:
1732:
1731:
1721:
1720:
1715:
1699:
1683:
1667:
1651:
1621:
1619:
1618:
1613:
1602:
1599:
1586:
1584:
1583:
1578:
1576:
1575:
1559:
1557:
1556:
1551:
1549:
1548:
1527:
1524:
1515:
1513:
1512:
1507:
1466:
1463:
1454:
1452:
1451:
1446:
1429:
1426:
1417:
1415:
1414:
1409:
1392:
1389:
1376:
1374:
1373:
1368:
1344:
1342:
1341:
1336:
1334:
1333:
1321:
1320:
1304:
1302:
1301:
1296:
1277:
1269:
1247:
1243:
1241:
1240:
1235:
1224:
1221:
1208:
1203:
1201:
1200:
1195:
1174:
1162:
1158:
1156:
1155:
1150:
1125:
1122:is true for all
1121:
1119:
1118:
1113:
1092:
1090:
1089:
1084:
1062:Peano arithmetic
1059:
1051:
1049:
1048:
1043:
1018:
1016:
1015:
1010:
992:
990:
989:
984:
982:
981:
969:
968:
952:
950:
949:
944:
926:
919:
901:
893:
891:
890:
885:
843:Ordinal analysis
834:
833:
821:
820:
801:
797:
782:. However, like
777:
760:
758:
757:
752:
750:
749:
737:
736:
720:
718:
717:
712:
695:, there is some
694:
690:
673:
665:
653:
651:
650:
645:
624:
615:
613:
612:
607:
602:
601:
579:
577:
576:
571:
566:
565:
543:
539:
530:
518:
516:
515:
510:
489:
487:
486:
481:
460:
451:
447:
440:
438:
437:
432:
411:
407:
398:
390:
381:
372:
365:
363:
362:
357:
355:
354:
342:
341:
325:
312:
303:
299:
290:
277:
270:
266:
262:
254:
250:
246:
242:
234:
230:
226:
218:
195:
193:
192:
187:
176:
173:
67:
60:
56:
53:
47:
27:
26:
19:
4225:
4224:
4220:
4219:
4218:
4216:
4215:
4214:
4210:Wellfoundedness
4180:
4179:
4176:
4171:
4167:Young's lattice
4023:
3951:
3890:
3740:Heyting algebra
3688:Boolean algebra
3660:
3641:Laver's theorem
3589:
3555:Boolean algebra
3550:Binary relation
3538:
3515:
3510:
3480:
3475:
3457:
3413:List of numbers
3381:
3379:
3377:
3375:
3365:
3322:
3288:
3282:
3253:Graham's number
3243:Skewes's number
3145:
3143:
3141:
3135:
3130:
3050:
2983:
2957:10.2307/1993287
2940:
2900:
2897:
2864:
2863:
2854:
2852:
2844:
2838:
2834:
2825:
2823:
2816:
2810:
2806:
2797:
2795:
2782:
2778:
2773:
2769:
2761:
2757:
2752:
2748:
2743:
2739:
2734:
2730:
2725:
2721:
2716:
2712:
2702:
2572:
2568:
2543:
2542:
2497:
2494:
2493:
2491:
2484:
2478:
2471:
2433:
2411:
2385:
2381:
2379:
2376:
2375:
2353:
2349:
2345:
2341:
2339:
2336:
2335:
2299:
2295:
2293:
2290:
2289:
2252:
2248:
2246:
2243:
2242:
2216:
2214:
2211:
2210:
2180:
2177:
2176:
2168:Graham's number
2131:
2127:
2125:
2122:
2121:
2094:
2091:
2090:
2064:
2062:
2059:
2058:
2027:
2025:
2022:
2021:
1990:
1988:
1985:
1984:
1952:
1949:
1948:
1931:
1927:
1918:
1914:
1912:
1909:
1908:
1904:
1900:
1896:
1891:
1888:
1879:
1873:
1866:
1863:
1836:
1834:
1831:
1830:
1827:
1824:
1811:
1807:
1803:
1796:
1716:
1702:
1701:
1700:
1686:
1685:
1684:
1670:
1669:
1668:
1654:
1653:
1652:
1638:
1637:
1635:
1632:
1631:
1598:
1596:
1593:
1592:
1589:Graham's number
1571:
1567:
1565:
1562:
1561:
1544:
1540:
1523:
1521:
1518:
1517:
1462:
1460:
1457:
1456:
1425:
1423:
1420:
1419:
1388:
1386:
1383:
1382:
1350:
1347:
1346:
1329:
1325:
1316:
1312:
1310:
1307:
1306:
1284:
1281:
1280:
1276:
1272:
1268:
1259:
1253:
1245:
1220:
1218:
1215:
1214:
1206:
1180:
1177:
1176:
1172:
1160:
1135:
1132:
1131:
1123:
1098:
1095:
1094:
1069:
1066:
1065:
1064:can prove that
1057:
1028:
1025:
1024:
998:
995:
994:
977:
973:
964:
960:
958:
955:
954:
953:vertices, then
932:
929:
928:
925:
921:
918:
909:
903:
899:
870:
867:
866:
863:
838:
832:
829:
828:
827:
825:
819:
816:
815:
814:
807:
799:
795:
792:Harvey Friedman
775:
768:
766:Friedman's work
745:
741:
732:
728:
726:
723:
722:
700:
697:
696:
692:
688:
681:
675:
671:
663:
630:
627:
626:
623:
617:
597:
593:
585:
582:
581:
561:
557:
549:
546:
545:
541:
538:
532:
529:
523:
495:
492:
491:
466:
463:
462:
459:
453:
449:
445:
417:
414:
413:
409:
408:, the label of
406:
400:
396:
389:
383:
380:
374:
370:
366:if there is an
350:
346:
337:
333:
331:
328:
327:
324:
318:
311:
305:
301:
298:
292:
289:
283:
275:
268:
264:
260:
252:
248:
244:
240:
232:
228:
224:
216:
210:
172:
170:
167:
166:
165:, which dwarfs
141:
112:Andrew Vázsonyi
100:
68:
57:
51:
48:
40:help improve it
37:
28:
24:
17:
12:
11:
5:
4223:
4213:
4212:
4207:
4202:
4197:
4192:
4173:
4172:
4170:
4169:
4164:
4159:
4158:
4157:
4147:
4146:
4145:
4140:
4135:
4125:
4124:
4123:
4113:
4108:
4107:
4106:
4101:
4094:Order morphism
4091:
4090:
4089:
4079:
4074:
4069:
4064:
4059:
4058:
4057:
4047:
4042:
4037:
4031:
4029:
4025:
4024:
4022:
4021:
4020:
4019:
4014:
4012:Locally convex
4009:
4004:
3994:
3992:Order topology
3989:
3988:
3987:
3985:Order topology
3982:
3972:
3962:
3960:
3953:
3952:
3950:
3949:
3944:
3939:
3934:
3929:
3924:
3919:
3914:
3909:
3904:
3898:
3896:
3892:
3891:
3889:
3888:
3878:
3868:
3863:
3858:
3853:
3848:
3843:
3838:
3833:
3832:
3831:
3821:
3816:
3815:
3814:
3809:
3804:
3799:
3797:Chain-complete
3789:
3784:
3783:
3782:
3777:
3772:
3767:
3762:
3752:
3747:
3742:
3737:
3732:
3722:
3717:
3712:
3707:
3702:
3697:
3696:
3695:
3685:
3680:
3674:
3672:
3662:
3661:
3659:
3658:
3653:
3648:
3643:
3638:
3633:
3628:
3623:
3618:
3613:
3608:
3603:
3597:
3595:
3591:
3590:
3588:
3587:
3582:
3577:
3572:
3567:
3562:
3557:
3552:
3546:
3544:
3540:
3539:
3537:
3536:
3531:
3526:
3520:
3517:
3516:
3509:
3508:
3501:
3494:
3486:
3477:
3476:
3474:
3473:
3468:
3462:
3459:
3458:
3456:
3455:
3450:
3445:
3443:Power of three
3440:
3435:
3430:
3425:
3423:Number systems
3420:
3415:
3410:
3405:
3400:
3395:
3390:
3384:
3382:
3378:(alphabetical
3371:
3370:
3367:
3366:
3364:
3363:
3358:
3353:
3348:
3347:
3346:
3341:
3334:Hyperoperation
3330:
3328:
3324:
3323:
3321:
3320:
3315:
3310:
3305:
3299:
3297:
3290:
3284:
3283:
3281:
3280:
3275:
3270:
3265:
3260:
3255:
3250:
3248:Moser's number
3245:
3240:
3235:
3233:Shannon number
3230:
3225:
3220:
3215:
3210:
3205:
3200:
3195:
3190:
3185:
3180:
3175:
3170:
3165:
3160:
3155:
3149:
3147:
3137:
3136:
3129:
3128:
3121:
3114:
3106:
3100:
3099:
3090:
3078:
3043:
3010:(4): 833–835,
2996:
2976:
2934:Kruskal, J. B.
2930:
2911:(3): 199–260,
2895:
2886:
2862:
2861:
2832:
2804:
2776:
2767:
2755:
2746:
2737:
2728:
2719:
2709:
2708:
2701:
2698:
2697:
2696:
2592:
2580:
2575:
2571:
2567:
2564:
2561:
2558:
2555:
2552:
2540:
2537:
2534:
2531:
2528:
2525:
2522:
2519:
2516:
2513:
2510:
2507:
2504:
2501:
2486:
2480:
2473:
2467:
2447:
2432:
2429:
2428:
2427:
2422:
2417:
2410:
2407:
2388:
2384:
2361:
2356:
2352:
2348:
2344:
2322:
2319:
2316:
2311:
2308:
2305:
2302:
2298:
2275:
2272:
2269:
2264:
2261:
2258:
2255:
2251:
2229:
2226:
2223:
2193:
2190:
2187:
2184:
2154:
2151:
2148:
2143:
2140:
2137:
2134:
2130:
2107:
2104:
2101:
2098:
2077:
2074:
2071:
2046:
2043:
2040:
2037:
2034:
2009:
2006:
2003:
2000:
1997:
1981:
1980:
1968:
1965:
1962:
1959:
1956:
1934:
1930:
1926:
1921:
1917:
1898:
1884:
1877:
1862:
1849:
1846:
1843:
1823:
1795:
1792:
1777:
1774:
1771:
1768:
1763:
1760:
1757:
1752:
1749:
1746:
1741:
1738:
1735:
1730:
1727:
1724:
1719:
1714:
1711:
1708:
1705:
1698:
1695:
1692:
1689:
1682:
1679:
1676:
1673:
1666:
1663:
1660:
1657:
1650:
1647:
1644:
1641:
1611:
1608:
1605:
1574:
1570:
1547:
1543:
1539:
1536:
1533:
1530:
1505:
1502:
1499:
1496:
1493:
1490:
1487:
1484:
1481:
1478:
1475:
1472:
1469:
1444:
1441:
1438:
1435:
1432:
1407:
1404:
1401:
1398:
1395:
1379:
1378:
1366:
1363:
1360:
1357:
1354:
1332:
1328:
1324:
1319:
1315:
1294:
1291:
1288:
1274:
1264:
1257:
1233:
1230:
1227:
1193:
1190:
1187:
1184:
1148:
1145:
1142:
1139:
1111:
1108:
1105:
1102:
1082:
1079:
1076:
1073:
1041:
1038:
1035:
1032:
1021:
1020:
1008:
1005:
1002:
980:
976:
972:
967:
963:
942:
939:
936:
923:
914:
907:
898:There is some
883:
880:
877:
874:
862:
859:
836:
830:
823:
817:
805:
767:
764:
748:
744:
740:
735:
731:
710:
707:
704:
686:
679:
656:
655:
643:
640:
637:
634:
621:
605:
600:
596:
592:
589:
569:
564:
560:
556:
553:
536:
527:
520:
508:
505:
502:
499:
479:
476:
473:
470:
457:
442:
430:
427:
424:
421:
404:
387:
378:
353:
349:
345:
340:
336:
322:
315:inf-embeddable
309:
304:, we say that
296:
287:
209:
206:
185:
182:
179:
139:
99:
96:
70:
69:
31:
29:
22:
15:
9:
6:
4:
3:
2:
4222:
4211:
4208:
4206:
4203:
4201:
4198:
4196:
4193:
4191:
4188:
4187:
4185:
4178:
4168:
4165:
4163:
4160:
4156:
4153:
4152:
4151:
4148:
4144:
4141:
4139:
4136:
4134:
4131:
4130:
4129:
4126:
4122:
4119:
4118:
4117:
4116:Ordered field
4114:
4112:
4109:
4105:
4102:
4100:
4097:
4096:
4095:
4092:
4088:
4085:
4084:
4083:
4080:
4078:
4075:
4073:
4072:Hasse diagram
4070:
4068:
4065:
4063:
4060:
4056:
4053:
4052:
4051:
4050:Comparability
4048:
4046:
4043:
4041:
4038:
4036:
4033:
4032:
4030:
4026:
4018:
4015:
4013:
4010:
4008:
4005:
4003:
4000:
3999:
3998:
3995:
3993:
3990:
3986:
3983:
3981:
3978:
3977:
3976:
3973:
3971:
3967:
3964:
3963:
3961:
3958:
3954:
3948:
3945:
3943:
3940:
3938:
3935:
3933:
3930:
3928:
3925:
3923:
3922:Product order
3920:
3918:
3915:
3913:
3910:
3908:
3905:
3903:
3900:
3899:
3897:
3895:Constructions
3893:
3887:
3883:
3879:
3876:
3872:
3869:
3867:
3864:
3862:
3859:
3857:
3854:
3852:
3849:
3847:
3844:
3842:
3839:
3837:
3834:
3830:
3827:
3826:
3825:
3822:
3820:
3817:
3813:
3810:
3808:
3805:
3803:
3800:
3798:
3795:
3794:
3793:
3792:Partial order
3790:
3788:
3785:
3781:
3780:Join and meet
3778:
3776:
3773:
3771:
3768:
3766:
3763:
3761:
3758:
3757:
3756:
3753:
3751:
3748:
3746:
3743:
3741:
3738:
3736:
3733:
3731:
3727:
3723:
3721:
3718:
3716:
3713:
3711:
3708:
3706:
3703:
3701:
3698:
3694:
3691:
3690:
3689:
3686:
3684:
3681:
3679:
3678:Antisymmetric
3676:
3675:
3673:
3669:
3663:
3657:
3654:
3652:
3649:
3647:
3644:
3642:
3639:
3637:
3634:
3632:
3629:
3627:
3624:
3622:
3619:
3617:
3614:
3612:
3609:
3607:
3604:
3602:
3599:
3598:
3596:
3592:
3586:
3585:Weak ordering
3583:
3581:
3578:
3576:
3573:
3571:
3570:Partial order
3568:
3566:
3563:
3561:
3558:
3556:
3553:
3551:
3548:
3547:
3545:
3541:
3535:
3532:
3530:
3527:
3525:
3522:
3521:
3518:
3514:
3507:
3502:
3500:
3495:
3493:
3488:
3487:
3484:
3472:
3469:
3467:
3464:
3463:
3460:
3454:
3451:
3449:
3446:
3444:
3441:
3439:
3436:
3434:
3431:
3429:
3426:
3424:
3421:
3419:
3416:
3414:
3411:
3409:
3406:
3404:
3403:Infinitesimal
3401:
3399:
3396:
3394:
3391:
3389:
3386:
3385:
3383:
3372:
3362:
3359:
3357:
3354:
3352:
3349:
3345:
3342:
3340:
3337:
3336:
3335:
3332:
3331:
3329:
3325:
3319:
3316:
3314:
3311:
3309:
3306:
3304:
3301:
3300:
3298:
3294:
3291:
3285:
3279:
3276:
3274:
3273:Rayo's number
3271:
3269:
3266:
3264:
3261:
3259:
3256:
3254:
3251:
3249:
3246:
3244:
3241:
3239:
3236:
3234:
3231:
3229:
3226:
3224:
3221:
3219:
3216:
3214:
3211:
3209:
3206:
3204:
3201:
3199:
3196:
3194:
3191:
3189:
3186:
3184:
3181:
3179:
3176:
3174:
3171:
3169:
3166:
3164:
3161:
3159:
3156:
3154:
3151:
3150:
3148:
3138:
3134:
3133:Large numbers
3127:
3122:
3120:
3115:
3113:
3108:
3107:
3104:
3096:
3091:
3087:
3083:
3079:
3076:
3072:
3068:
3064:
3060:
3056:
3049:
3044:
3041:
3037:
3033:
3029:
3025:
3021:
3017:
3013:
3009:
3005:
3001:
2997:
2993:
2989:
2982:
2977:
2974:
2970:
2966:
2962:
2958:
2954:
2950:
2946:
2939:
2935:
2931:
2928:
2924:
2919:
2914:
2910:
2906:
2899:
2891:
2887:
2884:
2880:
2876:
2871:
2870:
2869:
2868:
2850:
2843:
2836:
2822:
2815:
2808:
2794:
2792:
2787:
2780:
2771:
2764:
2759:
2750:
2741:
2732:
2723:
2714:
2710:
2707:
2706:
2694:
2690:
2686:
2682:
2678:
2674:
2670:
2666:
2662:
2658:
2654:
2650:
2646:
2642:
2638:
2634:
2630:
2626:
2622:
2618:
2614:
2610:
2606:
2602:
2598:
2597:
2593:
2578:
2573:
2565:
2562:
2556:
2550:
2538:
2535:
2532:
2526:
2520:
2517:
2514:
2511:
2505:
2499:
2490:
2483:
2477:
2470:
2465:
2461:
2457:
2453:
2452:
2448:
2445:
2440:
2439:
2435:
2434:
2426:
2423:
2421:
2418:
2416:
2413:
2412:
2406:
2404:
2386:
2382:
2359:
2354:
2346:
2342:
2333:
2317:
2306:
2300:
2296:
2286:
2270:
2259:
2253:
2249:
2224:
2208:
2204:
2188:
2182:
2173:
2169:
2165:
2149:
2138:
2132:
2128:
2118:
2102:
2096:
2072:
2044:
2041:
2035:
2007:
2004:
1998:
1966:
1963:
1960:
1957:
1954:
1932:
1928:
1924:
1919:
1915:
1902:
1887:
1883:
1876:
1871:
1870:
1869:
1867:
1844:
1828:
1821:
1800:
1794:TREE function
1791:
1788:
1775:
1769:
1758:
1747:
1736:
1725:
1717:
1629:
1625:
1606:
1590:
1572:
1568:
1545:
1541:
1537:
1531:
1503:
1500:
1497:
1494:
1491:
1488:
1485:
1482:
1479:
1476:
1470:
1442:
1439:
1433:
1405:
1402:
1396:
1364:
1361:
1358:
1355:
1352:
1330:
1326:
1322:
1317:
1313:
1292:
1289:
1286:
1278:
1267:
1263:
1256:
1251:
1250:
1249:
1228:
1211:
1209:
1188:
1182:
1170:
1166:
1143:
1137:
1129:
1106:
1100:
1077:
1071:
1063:
1055:
1054:KĹ‘nig's lemma
1036:
1030:
1006:
1003:
1000:
978:
974:
970:
965:
961:
940:
937:
934:
917:
913:
906:
902:such that if
897:
896:
895:
878:
872:
865:Suppose that
858:
856:
852:
848:
844:
840:
812:
808:
793:
789:
785:
781:
773:
762:
746:
742:
738:
733:
729:
708:
705:
702:
685:
678:
669:
659:
638:
632:
620:
598:
594:
587:
562:
558:
551:
535:
526:
521:
503:
497:
474:
468:
456:
443:
425:
419:
403:
394:
393:
392:
386:
377:
369:
368:injective map
351:
347:
343:
338:
334:
321:
316:
308:
295:
286:
281:
272:
258:
238:
222:
215:Given a tree
213:
205:
203:
202:TREE function
199:
180:
164:
163:SSCG function
160:
156:
151:
149:
145:
137:
133:
129:
125:
121:
117:
113:
109:
105:
95:
93:
89:
85:
81:
77:
66:
63:
55:
45:
41:
35:
32:This article
30:
21:
20:
4195:Order theory
4177:
3959:& Orders
3937:Star product
3866:Well-founded
3819:Prefix order
3775:Distributive
3765:Complemented
3735:Foundational
3700:Completeness
3656:Zorn's lemma
3635:
3560:Cyclic order
3543:Key concepts
3513:Order theory
3438:Power of two
3428:Number names
3257:
3163:Ten thousand
3094:
3085:
3061:(1): 49–88,
3058:
3054:
3007:
3003:
2991:
2987:
2948:
2944:
2936:(May 1960),
2908:
2904:
2874:
2867:Bibliography
2866:
2865:
2853:, retrieved
2848:
2835:
2824:, retrieved
2820:
2807:
2796:, retrieved
2789:
2779:
2770:
2758:
2749:
2740:
2731:
2722:
2713:
2704:
2703:
2692:
2688:
2684:
2680:
2676:
2672:
2668:
2664:
2660:
2656:
2652:
2648:
2644:
2640:
2636:
2635:defined as:
2628:
2624:
2620:
2616:
2612:
2608:
2604:
2600:
2596:
2488:
2481:
2475:
2468:
2463:
2459:
2455:
2451:
2443:
2438:
2288:
2241:
2206:
2175:
2120:
2089:
1982:
1903:has at most
1895:
1885:
1881:
1874:
1861:
1822:
1817:
1789:
1623:
1380:
1279:has at most
1271:
1265:
1261:
1254:
1212:
1205:
1022:
915:
911:
904:
864:
841:
796:then-nascent
769:
683:
676:
661:
657:
618:
533:
524:
454:
401:
384:
375:
319:
314:
306:
293:
284:
273:
256:
236:
214:
211:
201:
152:
101:
94:embedding.
92:homeomorphic
79:
73:
58:
49:
33:
4143:Riesz space
4104:Isomorphism
3980:Normal cone
3902:Composition
3836:Semilattice
3745:Homogeneous
3730:Equivalence
3580:Total order
3448:Power of 10
3388:Busy beaver
3193:Quintillion
3188:Quadrillion
2172:lower bound
1056:. For each
811:predicative
391:such that:
251:, and call
108:conjectured
76:mathematics
4184:Categories
4111:Order type
4045:Cofinality
3886:Well-order
3861:Transitive
3750:Idempotent
3683:Asymmetric
3453:Sagan Unit
3287:Expression
3238:Googolplex
3203:Septillion
3198:Sextillion
3144:numerical
2700:References
1175:for which
774:label set
326:and write
4162:Upper set
4099:Embedding
4035:Antichain
3856:Tolerance
3846:Symmetric
3841:Semiorder
3787:Reflexive
3705:Connected
3344:Pentation
3339:Tetration
3327:Operators
3296:Notations
3218:Decillion
3213:Nonillion
3208:Octillion
3140:Examples
3040:251095188
2994:: 303–330
2705:Citations
2655:+1, 1) =
2619:), where
2570:↑
2351:↑
2207:extremely
1964:≤
1925:≤
1538:≫
1477:≥
1362:≤
1323:≤
993:for some
971:≤
772:countable
739:≤
625:contains
344:≤
247:contains
237:successor
208:Statement
52:June 2024
3957:Topology
3824:Preorder
3807:Eulerian
3770:Complete
3720:Directed
3710:Covering
3575:Preorder
3534:Category
3529:Glossary
3376:articles
3374:Related
3278:Infinity
3183:Trillion
3158:Thousand
2892:(1991),
2855:8 August
2826:8 August
2798:8 August
2409:See also
2374:, where
721:so that
278:to be a
198:finitary
4062:Duality
4040:Cofinal
4028:Related
4007:Fréchet
3884:)
3760:Bounded
3755:Lattice
3728:)
3726:Partial
3594:Results
3565:Lattice
3471:History
3289:methods
3263:SSCG(3)
3258:TREE(3)
3178:Billion
3173:Million
3153:Hundred
3075:1212407
3032:0153601
3012:Bibcode
2973:0111704
2965:1993287
2927:1129778
2883:1943303
1880:, ...,
1829:, take
1820:integer
1812:TREE(3)
1591:), and
1560:(where
1260:, ...,
1213:Define
1167:or the
910:, ...,
786:or the
461:, then
231:, call
219:with a
157:as the
130: (
122: (
104:theorem
98:History
86:over a
38:Please
4087:Subnet
4067:Filter
4017:Normed
4002:Banach
3968:&
3875:Better
3812:Strict
3802:Graded
3693:topics
3524:Topics
3380:order)
3228:Googol
3073:
3038:
3030:
2971:
2963:
2925:
2881:
2675:+1) =
2663:, 1),
2579:158386
2485:,...,x
2472:,...,x
2355:187196
2307:187196
2260:187196
2166:, and
1626:; see
1624:labels
770:For a
155:graphs
116:proved
4077:Ideal
4055:Graph
3851:Total
3829:Total
3715:Dense
3466:Names
3268:BH(3)
3146:order
3051:(PDF)
3036:S2CID
2984:(PDF)
2961:JSTOR
2941:(PDF)
2901:(PDF)
2845:(PDF)
2817:(PDF)
2643:) = 2
2431:Notes
2240:, is
1628:below
1128:proof
519:; and
282:. If
274:Take
84:trees
3668:list
2857:2017
2828:2017
2800:2017
2691:+1,
2671:+1,
2639:(1,
2574:7197
2563:>
2218:TREE
2174:for
2066:TREE
2029:TREE
1992:TREE
1958:<
1838:TREE
1600:TREE
1525:tree
1464:tree
1427:tree
1390:tree
1356:<
1222:tree
1004:<
927:has
706:<
221:root
196:. A
174:TREE
132:1963
124:1960
114:and
106:was
102:The
4082:Net
3882:Pre
3063:doi
3020:doi
2953:doi
2913:doi
2695:)).
2545:and
2401:is
1587:is
1504:960
1498:131
1492:930
1486:424
1480:844
1130:of
857:).
835:-CA
822:-CA
804:ATR
689:, …
666:is
662:If
616:in
580:to
522:If
452:in
444:If
399:of
317:in
313:is
267:to
259:of
255:an
239:of
150:).
142:(a
118:by
110:by
74:In
42:to
4186::
3142:in
3071:MR
3069:,
3059:60
3057:,
3053:,
3034:,
3028:MR
3026:,
3018:,
3008:59
3006:,
2992:21
2990:,
2986:,
2969:MR
2967:,
2959:,
2949:95
2947:,
2943:,
2923:MR
2921:,
2909:53
2907:,
2903:,
2879:MR
2847:,
2819:,
2788:,
2683:,
2647:,
2627:,
2615:,
2536:11
2492:.
2444:TR
2405:.
2119:,
2020:,
1573:64
1546:64
1455:,
1418:,
1210:.
1060:,
839:.
761:.)
682:,
531:,
291:,
235:a
227:,
204:.
78:,
3880:(
3877:)
3873:(
3724:(
3671:)
3505:e
3498:t
3491:v
3125:e
3118:t
3111:v
3065::
3022::
3014::
2955::
2915::
2896:0
2765:.
2693:n
2689:k
2687:(
2685:A
2681:k
2679:(
2677:A
2673:n
2669:k
2667:(
2665:A
2661:k
2659:(
2657:A
2653:k
2651:(
2649:A
2645:n
2641:n
2637:A
2629:n
2625:k
2623:(
2621:A
2617:x
2613:x
2611:(
2609:A
2605:x
2603:(
2601:A
2591:.
2566:2
2560:)
2557:3
2554:(
2551:n
2539:,
2533:=
2530:)
2527:2
2524:(
2521:n
2518:,
2515:3
2512:=
2509:)
2506:1
2503:(
2500:n
2489:j
2487:2
2482:j
2476:i
2474:2
2469:i
2464:k
2460:k
2458:(
2456:n
2446:.
2387:x
2383:g
2360:3
2347:3
2343:g
2321:)
2318:1
2315:(
2310:)
2304:(
2301:A
2297:A
2274:)
2271:1
2268:(
2263:)
2257:(
2254:A
2250:A
2228:)
2225:3
2222:(
2192:)
2189:4
2186:(
2183:n
2153:)
2150:5
2147:(
2142:)
2139:5
2136:(
2133:n
2129:n
2106:)
2103:4
2100:(
2097:n
2076:)
2073:3
2070:(
2045:3
2042:=
2039:)
2036:2
2033:(
2008:1
2005:=
2002:)
1999:1
1996:(
1979:.
1967:m
1961:j
1955:i
1933:j
1929:T
1920:i
1916:T
1905:i
1899:i
1897:T
1892:n
1886:m
1882:T
1878:1
1875:T
1864:m
1848:)
1845:n
1842:(
1825:n
1808:n
1804:n
1776:.
1773:)
1770:7
1767:(
1762:)
1759:7
1756:(
1751:)
1748:7
1745:(
1740:)
1737:7
1734:(
1729:)
1726:7
1723:(
1718:8
1713:e
1710:e
1707:r
1704:t
1697:e
1694:e
1691:r
1688:t
1681:e
1678:e
1675:r
1672:t
1665:e
1662:e
1659:r
1656:t
1649:e
1646:e
1643:r
1640:t
1610:)
1607:3
1604:(
1569:g
1542:g
1535:)
1532:4
1529:(
1501:,
1495:,
1489:,
1483:,
1474:)
1471:3
1468:(
1443:5
1440:=
1437:)
1434:2
1431:(
1406:2
1403:=
1400:)
1397:1
1394:(
1377:.
1365:m
1359:j
1353:i
1331:j
1327:T
1318:i
1314:T
1293:n
1290:+
1287:i
1275:i
1273:T
1266:m
1262:T
1258:1
1255:T
1246:m
1232:)
1229:n
1226:(
1207:n
1192:)
1189:n
1186:(
1183:P
1173:m
1161:n
1147:)
1144:n
1141:(
1138:P
1124:n
1110:)
1107:n
1104:(
1101:P
1081:)
1078:n
1075:(
1072:P
1058:n
1040:)
1037:n
1034:(
1031:P
1019:.
1007:j
1001:i
979:j
975:T
966:i
962:T
941:n
938:+
935:i
924:i
922:T
916:m
912:T
908:1
905:T
900:m
882:)
879:n
876:(
873:P
837:0
831:1
824:0
818:1
806:0
800:X
776:X
747:j
743:T
734:i
730:T
709:j
703:i
693:X
687:2
684:T
680:1
677:T
672:X
664:X
654:.
642:)
639:v
636:(
633:F
622:2
619:T
604:)
599:2
595:w
591:(
588:F
568:)
563:1
559:w
555:(
552:F
542:v
537:2
534:w
528:1
525:w
507:)
504:v
501:(
498:F
478:)
475:w
472:(
469:F
458:1
455:T
450:v
446:w
441:;
429:)
426:v
423:(
420:F
410:v
405:1
402:T
397:v
388:2
385:T
379:1
376:T
371:F
352:2
348:T
339:1
335:T
323:2
320:T
310:1
307:T
302:X
297:2
294:T
288:1
285:T
276:X
269:w
265:v
261:v
253:w
249:v
245:w
241:v
233:w
229:w
225:v
217:T
184:)
181:3
178:(
140:0
65:)
59:(
54:)
50:(
36:.
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