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Takeuti–Feferman–Buchholz ordinal

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1079: 286: 1136: 158: 98: 493: 410: 211: 380: 346: 245: 460: 430: 300: 515: 881: 1021: 160:
in Feferman's theta function, an ordinal collapsing function invented by Solomon Feferman. It is the proof-theoretic ordinal of several formal theories:
1120: 755: 722: 564: 537: 1177: 107: 792: 697: 47: 874: 922: 269: 433: 1113: 961: 867: 465: 101: 250: 1211: 1054: 1201: 1106: 890: 261: 29: 816: 619: 388: 1170: 518: 33: 1216: 1196: 166: 978: 358: 324: 1040: 951: 941: 1206: 218: 1026: 1163: 931: 788: 751: 718: 693: 670: 439: 780: 772: 743: 660: 415: 265: 41: 802: 777:
Iterated Inductive Definitions and Subsystems of Analysis: Recent Proof-Theoretical Studies
500: 901: 798: 349: 299:
Please expand the article to include this information. Further details may exist on the
1147: 1090: 910: 1190: 665: 648: 37: 742:. Lecture Notes in Mathematics (in German). Vol. 500. Springer. pp. 4–25. 738:
Buchholz, W. (1975). "Normalfunktionen und Konstruktive Systeme von Ordinalzahlen".
591: 21: 779:. Lecture Notes in Mathematics. Vol. 897. Springer-Verlag, Berlin-New York. 692:. Studies in Proof Theory, Monographs. Vol. 2. Naples, Italy: Bibliopolis. 353: 1086: 17: 674: 859: 840: 285: 1078: 841:"number theory - Can PA prove very fast growing functions to be total?" 784: 747: 1143: 1135: 36:
and Feferman's theta function. It was named by David Madore, after
863: 153:{\displaystyle \theta _{\varepsilon _{\Omega _{\omega }+1}}(0)} 297:
about the definition of the Takeuti-Feferman-Buchholz ordinal.
279: 93:{\displaystyle \psi _{0}(\varepsilon _{\Omega _{\omega }+1})} 264:
and recursive ordinals, it is still vastly smaller than the
1151: 1094: 256:, the system of ω-times iterated inductive definitions 503: 468: 442: 418: 391: 361: 327: 221: 169: 110: 50: 690:
Proof Theory of Impredicative Subsystems of Analysis
649:"A new system of proof-theoretic ordinal functions" 509: 487: 454: 424: 404: 374: 340: 239: 205: 152: 92: 488:{\displaystyle \alpha \mapsto \omega ^{\alpha }} 1022:the theories of iterated inductive definitions 1171: 1114: 875: 8: 775:; Pohlers, Wolfram; Sieg, Wilfried (1981). 1178: 1164: 1121: 1107: 882: 868: 860: 688:Buchholz, Wilfried; Schütte, Kurt (1988). 32:, which acts as the limit of the range of 664: 502: 479: 467: 441: 417: 396: 390: 366: 360: 332: 326: 231: 226: 220: 179: 174: 168: 125: 120: 115: 109: 73: 68: 55: 49: 213:, a subsystem of second-order arithmetic 44:and Wilfried Buchholz. It is written as 26:Takeuti–Feferman–Buchholz ordinal (TFBO) 529: 247:-comprehension + transfinite induction 405:{\displaystyle \varepsilon _{\beta }} 7: 1132: 1130: 1075: 1073: 717:(2nd ed.). Dover Publications. 586: 584: 559: 557: 104:invented by Wilfried Buchholz, and 363: 329: 223: 206:{\displaystyle \Pi _{1}^{1}-CA+BI} 171: 122: 100:using Buchholz's psi function, an 70: 14: 998:Takeuti–Feferman–Buchholz ordinal 375:{\displaystyle \aleph _{\alpha }} 341:{\displaystyle \Omega _{\alpha }} 260:Despite being one of the largest 1134: 1077: 653:Annals of Pure and Applied Logic 284: 472: 147: 141: 87: 61: 16:In the mathematical fields of 1: 1029: < ω‍ 740:⊨ISILC Proof Theory Symposion 1150:. You can help Knowledge by 1093:. You can help Knowledge by 1020:Proof-theoretic ordinals of 666:10.1016/0168-0072(86)90052-7 240:{\displaystyle \Pi _{1}^{1}} 647:Buchholz, W. (1986-01-01). 102:ordinal collapsing function 1233: 1129: 1072: 1043: ≥ ω‍ 845:Mathematics Stack Exchange 817:"ordinal analysis in nLab" 1055:First uncountable ordinal 897: 923:Feferman–Schütte ordinal 891:Large countable ordinals 565:"Buchholz's ψ functions" 538:"Buchholz's ψ functions" 455:{\displaystyle 1+\beta } 262:large countable ordinals 962:Bachmann–Howard ordinal 713:Takeuti, Gaisi (2013). 519:Buchholz's psi function 348:represent the smallest 266:proof-theoretic ordinal 34:Buchholz's psi function 30:large countable ordinal 1089:-related article is a 902:First infinite ordinal 620:"Collapsingfunktionen" 511: 489: 456: 426: 425:{\displaystyle \beta } 406: 376: 342: 295:is missing information 241: 207: 154: 94: 1142:This article about a 512: 510:{\displaystyle \psi } 490: 457: 427: 407: 377: 343: 242: 208: 155: 95: 1041:Nonrecursive ordinal 952:large Veblen ordinal 942:small Veblen ordinal 771:Buchholz, Wilfried; 627:University of Munich 501: 466: 440: 416: 389: 359: 325: 219: 167: 108: 48: 1027:Computable ordinals 592:"A Zoo of Ordinals" 236: 184: 979:Buchholz's ordinal 785:10.1007/bfb0091894 748:10.1007/BFb0079544 507: 485: 462:th fixed point of 452: 422: 402: 372: 338: 237: 222: 203: 170: 150: 90: 1159: 1158: 1102: 1101: 1067: 1066: 932:Ackermann ordinal 773:Feferman, Solomon 757:978-3-540-07533-2 724:978-0-486-32067-0 318: 317: 1224: 1212:Set theory stubs 1180: 1173: 1166: 1138: 1131: 1123: 1116: 1109: 1081: 1074: 1051: 1050: 1037: 1036: 884: 877: 870: 861: 855: 854: 852: 851: 837: 831: 830: 828: 827: 813: 807: 806: 768: 762: 761: 735: 729: 728: 710: 704: 703: 685: 679: 678: 668: 644: 638: 637: 635: 634: 624: 616: 610: 609: 607: 606: 596: 588: 579: 578: 576: 575: 561: 552: 551: 549: 548: 534: 516: 514: 513: 508: 494: 492: 491: 486: 484: 483: 461: 459: 458: 453: 431: 429: 428: 423: 411: 409: 408: 403: 401: 400: 381: 379: 378: 373: 371: 370: 347: 345: 344: 339: 337: 336: 313: 310: 304: 288: 280: 246: 244: 243: 238: 235: 230: 212: 210: 209: 204: 183: 178: 159: 157: 156: 151: 140: 139: 138: 137: 130: 129: 99: 97: 96: 91: 86: 85: 78: 77: 60: 59: 42:Solomon Feferman 1232: 1231: 1227: 1226: 1225: 1223: 1222: 1221: 1202:Ordinal numbers 1187: 1186: 1185: 1184: 1128: 1127: 1070: 1068: 1063: 1049: 1046: 1045: 1044: 1035: 1032: 1031: 1030: 1016: 1014: 993: 987: 974: 928: 919: 911:Epsilon numbers 893: 888: 858: 849: 847: 839: 838: 834: 825: 823: 815: 814: 810: 795: 770: 769: 765: 758: 737: 736: 732: 725: 712: 711: 707: 700: 687: 686: 682: 646: 645: 641: 632: 630: 622: 618: 617: 613: 604: 602: 594: 590: 589: 582: 573: 571: 563: 562: 555: 546: 544: 536: 535: 531: 527: 499: 498: 475: 464: 463: 438: 437: 436:, equal to the 414: 413: 392: 387: 386: 362: 357: 356: 328: 323: 322: 314: 308: 305: 298: 289: 278: 254: 217: 216: 165: 164: 121: 116: 111: 106: 105: 69: 64: 51: 46: 45: 12: 11: 5: 1230: 1228: 1220: 1219: 1214: 1209: 1204: 1199: 1189: 1188: 1183: 1182: 1175: 1168: 1160: 1157: 1156: 1139: 1126: 1125: 1118: 1111: 1103: 1100: 1099: 1082: 1065: 1064: 1062: 1061: 1052: 1047: 1038: 1033: 1024: 1018: 1010: 1008: 995: 989: 985: 976: 972: 959: 949: 939: 929: 926: 920: 917: 908: 898: 895: 894: 889: 887: 886: 879: 872: 864: 857: 856: 832: 808: 793: 763: 756: 730: 723: 705: 698: 680: 639: 611: 580: 553: 528: 526: 523: 522: 521: 506: 495: 482: 478: 474: 471: 451: 448: 445: 434:epsilon number 421: 412:represent the 399: 395: 383: 369: 365: 335: 331: 316: 315: 292: 290: 283: 277: 274: 258: 257: 252: 248: 234: 229: 225: 214: 202: 199: 196: 193: 190: 187: 182: 177: 173: 149: 146: 143: 136: 133: 128: 124: 119: 114: 89: 84: 81: 76: 72: 67: 63: 58: 54: 13: 10: 9: 6: 4: 3: 2: 1229: 1218: 1215: 1213: 1210: 1208: 1205: 1203: 1200: 1198: 1195: 1194: 1192: 1181: 1176: 1174: 1169: 1167: 1162: 1161: 1155: 1153: 1149: 1145: 1140: 1137: 1133: 1124: 1119: 1117: 1112: 1110: 1105: 1104: 1098: 1096: 1092: 1088: 1083: 1080: 1076: 1071: 1060: 1056: 1053: 1042: 1039: 1028: 1025: 1023: 1019: 1013: 1007: 1003: 999: 996: 992: 984: 980: 977: 971: 967: 963: 960: 957: 953: 950: 947: 943: 940: 937: 933: 930: 924: 921: 916: 912: 909: 907: 903: 900: 899: 896: 892: 885: 880: 878: 873: 871: 866: 865: 862: 846: 842: 836: 833: 822: 818: 812: 809: 804: 800: 796: 794:3-540-11170-0 790: 786: 782: 778: 774: 767: 764: 759: 753: 749: 745: 741: 734: 731: 726: 720: 716: 709: 706: 701: 699:88-7088-166-0 695: 691: 684: 681: 676: 672: 667: 662: 658: 654: 650: 643: 640: 628: 621: 615: 612: 600: 593: 587: 585: 581: 570: 569:cantors-attic 566: 560: 558: 554: 543: 542:cantors-attic 539: 533: 530: 524: 520: 504: 496: 480: 476: 469: 449: 446: 443: 435: 419: 397: 393: 384: 367: 355: 352:ordinal with 351: 333: 320: 319: 312: 302: 296: 293:This article 291: 287: 282: 281: 275: 273: 271: 267: 263: 255: 249: 232: 227: 215: 200: 197: 194: 191: 188: 185: 180: 175: 163: 162: 161: 144: 134: 131: 126: 117: 112: 103: 82: 79: 74: 65: 56: 52: 43: 39: 38:Gaisi Takeuti 35: 31: 27: 23: 19: 1217:Number stubs 1197:Proof theory 1152:expanding it 1141: 1095:expanding it 1084: 1069: 1058: 1011: 1005: 1001: 997: 990: 982: 969: 965: 955: 945: 935: 914: 905: 848:. Retrieved 844: 835: 824:. Retrieved 820: 811: 776: 766: 739: 733: 715:Proof Theory 714: 708: 689: 683: 656: 652: 642: 631:. Retrieved 626: 614: 603:. Retrieved 601:. 2017-07-29 598: 572:. Retrieved 568: 545:. Retrieved 541: 532: 306: 294: 259: 25: 22:proof theory 15: 821:ncatlab.org 659:: 195–207. 354:cardinality 350:uncountable 1207:Set theory 1191:Categories 1087:set theory 850:2021-08-17 826:2021-08-28 633:2021-08-10 605:2021-08-10 574:2021-08-17 547:2021-08-10 525:References 517:represent 309:April 2024 276:Definition 18:set theory 675:0168-0072 505:ψ 481:α 477:ω 473:↦ 470:α 450:β 420:β 398:β 394:ε 368:α 364:ℵ 334:α 330:Ω 301:talk page 224:Π 186:− 172:Π 127:ω 123:Ω 118:ε 113:θ 75:ω 71:Ω 66:ε 53:ψ 925: Γ 803:0655036 1144:number 1057:  1000:  981:  964:  954:  944:  934:  913:  904:  801:  791:  754:  721:  696:  673:  629:. 1981 599:Madore 24:, the 1146:is a 1085:This 623:(PDF) 595:(PDF) 28:is a 1148:stub 1091:stub 789:ISBN 752:ISBN 719:ISBN 694:ISBN 671:ISSN 497:Let 385:Let 321:Let 20:and 973:Ω+1 958:(Ω) 948:(Ω) 938:(Ω) 781:doi 744:doi 661:doi 432:th 270:ZFC 268:of 1193:: 1015:+1 988:(Ω 843:. 819:. 799:MR 797:. 787:. 750:. 669:. 657:32 655:. 651:. 625:. 597:. 583:^ 567:. 556:^ 540:. 272:. 251:ID 40:, 1179:e 1172:t 1165:v 1154:. 1122:e 1115:t 1108:v 1097:. 1059:Ω 1048:1 1034:1 1017:) 1012:ω 1009:Ω 1006:ε 1004:( 1002:ψ 994:) 991:ω 986:0 983:ψ 975:) 970:ε 968:( 966:ψ 956:θ 946:θ 936:θ 927:0 918:0 915:ε 906:ω 883:e 876:t 869:v 853:. 829:. 805:. 783:: 760:. 746:: 727:. 702:. 677:. 663:: 636:. 608:. 577:. 550:. 447:+ 444:1 382:. 311:) 307:( 303:. 253:ω 233:1 228:1 201:I 198:B 195:+ 192:A 189:C 181:1 176:1 148:) 145:0 142:( 135:1 132:+ 88:) 83:1 80:+ 62:( 57:0

Index

set theory
proof theory
large countable ordinal
Buchholz's psi function
Gaisi Takeuti
Solomon Feferman
ordinal collapsing function
IDω
large countable ordinals
proof-theoretic ordinal
ZFC

talk page
uncountable
cardinality
epsilon number
Buchholz's psi function
"Buchholz's ψ functions"


"Buchholz's ψ functions"


"A Zoo of Ordinals"
"Collapsingfunktionen"
"A new system of proof-theoretic ordinal functions"
doi
10.1016/0168-0072(86)90052-7
ISSN
0168-0072

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