1216:, includes the following topics: almost free abelian groups, partition problems, failure of preservation of chain conditions in Boolean algebras under products, existence of Jónsson algebras, existence of entangled linear orders, equivalently narrow Boolean algebras, and the existence of nonisomorphic models equivalent in certain infinitary logics.
352:
924:
836:
1161:
511:
672:
259:
1096:
748:
570:
127:
405:
244:
962:
607:
450:
992:
150:
176:
860:
788:
347:{\displaystyle \operatorname {pcf} (A)=\left\{\operatorname {cf} \left(\prod A/D\right):D\,\,{\mbox{is an ultrafilter on}}\,\,A\right\}.}
1105:
459:
612:
1047:
699:
528:
85:
1256:
372:
181:
1212:
The theory has found a great deal of applications, besides cardinal arithmetic. The original survey by Shelah,
1219:
In the meantime, many further applications have been found in Set Theory, Model Theory, Algebra and
Topology.
937:
1166:
579:
410:
1334:
999:
1296:
1003:
974:
132:
1306:
1265:
1170:
155:
67:
44:
1318:
1277:
1314:
1273:
47:
998:)). Another consequence is that if λ is singular and no regular cardinal less than λ is
1284:
1251:
1178:
757:
919:{\displaystyle \{B_{\theta }:\theta \in \operatorname {pcf} (A),\theta <\lambda \}}
365:) consists of regular cardinals. Considering ultrafilters concentrated on elements of
50:, and has many more applications as well. The abbreviation "PCF" stands for "possible
1328:
1310:
1196:) has no inaccessible limit point. This is equivalent to the statement that pcf(pcf(
32:
1174:
75:
36:
20:
51:
28:
40:
250:) is the set of cofinalities that occur if we consider all ultrafilters on
1269:
1301:
831:{\displaystyle \operatorname {cf} \left(\prod A/D\right)<\lambda }
682:
is the set of all regular cardinals between two cardinals), then pcf(
1242:
1156:{\displaystyle 2^{\aleph _{\omega _{1}}}<\aleph _{\omega _{2}}}
506:{\displaystyle \{B_{\theta }:\theta \in \operatorname {pcf} (A)\}}
667:{\displaystyle \left|\operatorname {pcf} (A)\right|\leq 2^{|A|}}
1232:, Oxford Logic Guides, vol. 29. Oxford University Press, 1994.
1091:{\displaystyle 2^{\aleph _{\omega }}<\aleph _{\omega _{1}}}
743:{\displaystyle 2^{\aleph _{\omega }}<\aleph _{\omega _{4}}}
1018:
The most notorious conjecture in pcf theory states that |pcf(
565:{\displaystyle \operatorname {cf} \left(\prod A/D\right)}
122:{\displaystyle \operatorname {cf} \left(\prod A/D\right)}
1002:, then also λ is not Jónsson. In particular, there is a
39:. It gives strong upper bounds on the cardinalities of
325:
129:
denote the cofinality of the ordered set of functions
1108:
1050:
977:
940:
863:
791:
702:
615:
582:
531:
462:
413:
375:
262:
184:
158:
135:
88:
1184:
A weaker, still unsolved conjecture states that if |
686:) is also an interval of regular cardinals and |pcf(
1254:(1978), "Jonsson algebras in successor cardinals",
1155:
1090:
986:
956:
918:
830:
742:
666:
601:
564:
505:
444:
400:{\displaystyle A\subseteq \operatorname {pcf} (A)}
399:
346:
238:
170:
144:
121:
934:) there is a sequence of length λ of elements of
430:
456:) has a largest element, and there are subsets
1289:Bulletin of the American Mathematical Society
8:
1287:(1992), "Cardinal arithmetic for skeptics",
913:
864:
500:
463:
239:{\displaystyle \{x\in A:f(x)<g(x)\}\in D}
227:
185:
678:is an interval of regular cardinals (i.e.,
23:theory, introduced by Saharon Shelah (
152:where the ordering is defined as follows:
1300:
1145:
1140:
1123:
1118:
1113:
1107:
1080:
1075:
1060:
1055:
1049:
976:
964:which is both increasing and cofinal mod
948:
939:
871:
862:
809:
790:
732:
727:
712:
707:
701:
657:
649:
648:
614:
587:
581:
549:
530:
470:
461:
422:
414:
412:
374:
332:
331:
324:
323:
322:
303:
261:
183:
157:
134:
106:
87:
1042:is strong limit, then the sharp bound
971:. This implies that the cofinality of
694:|. This implies the famous inequality
24:
1173:) or even from the nonexistence of a
994:under pointwise dominance is max(pcf(
7:
1010:, which settles an old conjecture.
857:is the ideal generated by the sets
763:If λ is an infinite cardinal, then
1137:
1115:
1072:
1057:
957:{\displaystyle \prod B_{\lambda }}
724:
709:
14:
1214:Cardinal arithmetic for skeptics
602:{\displaystyle B_{\theta }\in D}
1311:10.1090/s0273-0979-1992-00261-6
517:such that for each ultrafilter
898:
892:
658:
650:
633:
627:
572:is the least element θ of pcf(
497:
491:
445:{\displaystyle |A|<\min(A)}
439:
433:
423:
415:
394:
388:
275:
269:
224:
218:
209:
203:
1:
1257:Israel Journal of Mathematics
1038:). This would imply that if ℵ
674:. Shelah also proved that if
838:holds for every ultrafilter
1100:holds. The analogous bound
1030:of regular cardinals with |
1351:
770:is the following ideal on
407:. Shelah proved, that if
930:, i.e., for every λ∈pcf(
987:{\displaystyle \prod A}
145:{\displaystyle \prod A}
27:), that deals with the
1157:
1092:
1026:| holds for every set
988:
958:
920:
832:
744:
668:
603:
566:
507:
446:
401:
348:
240:
172:
171:{\displaystyle f<g}
146:
123:
66:is an infinite set of
1158:
1093:
989:
959:
921:
833:
745:
669:
604:
567:
508:
447:
402:
349:
241:
173:
147:
124:
1106:
1048:
975:
938:
861:
789:
700:
613:
580:
529:
460:
411:
373:
327:is an ultrafilter on
260:
182:
156:
133:
86:
1230:Cardinal Arithmetic
1270:10.1007/BF02760829
1167:Chang's conjecture
1153:
1088:
984:
954:
916:
828:
740:
664:
599:
562:
503:
442:
397:
344:
329:
236:
168:
142:
119:
1243:Menachem Kojman:
1014:Unsolved problems
328:
68:regular cardinals
19:is the name of a
1342:
1321:
1304:
1280:
1228:Saharon Shelah,
1162:
1160:
1159:
1154:
1152:
1151:
1150:
1149:
1132:
1131:
1130:
1129:
1128:
1127:
1097:
1095:
1094:
1089:
1087:
1086:
1085:
1084:
1067:
1066:
1065:
1064:
993:
991:
990:
985:
963:
961:
960:
955:
953:
952:
925:
923:
922:
917:
876:
875:
837:
835:
834:
829:
821:
817:
813:
749:
747:
746:
741:
739:
738:
737:
736:
719:
718:
717:
716:
673:
671:
670:
665:
663:
662:
661:
653:
640:
636:
609:. Consequently,
608:
606:
605:
600:
592:
591:
571:
569:
568:
563:
561:
557:
553:
512:
510:
509:
504:
475:
474:
451:
449:
448:
443:
426:
418:
406:
404:
403:
398:
353:
351:
350:
345:
340:
336:
330:
326:
315:
311:
307:
245:
243:
242:
237:
177:
175:
174:
169:
151:
149:
148:
143:
128:
126:
125:
120:
118:
114:
110:
58:Main definitions
1350:
1349:
1345:
1344:
1343:
1341:
1340:
1339:
1325:
1324:
1285:Shelah, Saharon
1283:
1252:Shelah, Saharon
1250:
1239:
1225:
1210:
1163:
1141:
1136:
1119:
1114:
1109:
1104:
1103:
1098:
1076:
1071:
1056:
1051:
1046:
1045:
1041:
1016:
1009:
1004:Jónsson algebra
973:
972:
970:
944:
936:
935:
867:
859:
858:
856:
802:
798:
787:
786:
784:
769:
755:
752:assuming that ℵ
750:
728:
723:
708:
703:
698:
697:
644:
620:
616:
611:
610:
583:
578:
577:
542:
538:
527:
526:
466:
458:
457:
409:
408:
371:
370:
361:Obviously, pcf(
359:
354:
296:
292:
285:
281:
258:
257:
180:
179:
154:
153:
131:
130:
99:
95:
84:
83:
60:
12:
11:
5:
1348:
1346:
1338:
1337:
1327:
1326:
1323:
1322:
1295:(2): 197–210,
1291:, New Series,
1281:
1248:
1238:
1237:External links
1235:
1234:
1233:
1224:
1221:
1209:
1206:
1148:
1144:
1139:
1135:
1126:
1122:
1117:
1112:
1102:
1083:
1079:
1074:
1070:
1063:
1059:
1054:
1044:
1039:
1015:
1012:
1007:
983:
980:
968:
951:
947:
943:
926:. There exist
915:
912:
909:
906:
903:
900:
897:
894:
891:
888:
885:
882:
879:
874:
870:
866:
854:
827:
824:
820:
816:
812:
808:
805:
801:
797:
794:
782:
767:
753:
735:
731:
726:
722:
715:
711:
706:
696:
660:
656:
652:
647:
643:
639:
635:
632:
629:
626:
623:
619:
598:
595:
590:
586:
560:
556:
552:
548:
545:
541:
537:
534:
502:
499:
496:
493:
490:
487:
484:
481:
478:
473:
469:
465:
441:
438:
435:
432:
429:
425:
421:
417:
396:
393:
390:
387:
384:
381:
378:
369:, we get that
358:
355:
343:
339:
335:
321:
318:
314:
310:
306:
302:
299:
295:
291:
288:
284:
280:
277:
274:
271:
268:
265:
256:
235:
232:
229:
226:
223:
220:
217:
214:
211:
208:
205:
202:
199:
196:
193:
190:
187:
167:
164:
161:
141:
138:
117:
113:
109:
105:
102:
98:
94:
91:
82:, then we let
59:
56:
13:
10:
9:
6:
4:
3:
2:
1347:
1336:
1333:
1332:
1330:
1320:
1316:
1312:
1308:
1303:
1298:
1294:
1290:
1286:
1282:
1279:
1275:
1271:
1267:
1263:
1259:
1258:
1253:
1249:
1247:
1246:
1241:
1240:
1236:
1231:
1227:
1226:
1222:
1220:
1217:
1215:
1207:
1205:
1203:
1199:
1195:
1191:
1187:
1182:
1180:
1176:
1172:
1168:
1165:follows from
1146:
1142:
1133:
1124:
1120:
1110:
1101:
1081:
1077:
1068:
1061:
1052:
1043:
1037:
1033:
1029:
1025:
1021:
1013:
1011:
1005:
1001:
997:
981:
978:
967:
949:
945:
941:
933:
929:
910:
907:
904:
901:
895:
889:
886:
883:
880:
877:
872:
868:
853:
849:
845:
841:
825:
822:
818:
814:
810:
806:
803:
799:
795:
792:
781:
777:
773:
766:
761:
759:
733:
729:
720:
713:
704:
695:
693:
689:
685:
681:
677:
654:
645:
641:
637:
630:
624:
621:
617:
596:
593:
588:
584:
575:
558:
554:
550:
546:
543:
539:
535:
532:
524:
520:
516:
494:
488:
485:
482:
479:
476:
471:
467:
455:
436:
427:
419:
391:
385:
382:
379:
376:
368:
364:
356:
341:
337:
333:
319:
316:
312:
308:
304:
300:
297:
293:
289:
286:
282:
278:
272:
266:
263:
255:
253:
249:
233:
230:
221:
215:
212:
206:
200:
197:
194:
191:
188:
165:
162:
159:
139:
136:
115:
111:
107:
103:
100:
96:
92:
89:
81:
77:
73:
69:
65:
57:
55:
53:
49:
46:
42:
38:
34:
33:ultraproducts
30:
26:
22:
18:
1302:math/9201251
1292:
1288:
1264:(1): 57–64,
1261:
1255:
1244:
1229:
1218:
1213:
1211:
1208:Applications
1201:
1197:
1193:
1192:), then pcf(
1189:
1185:
1183:
1164:
1099:
1035:
1031:
1027:
1023:
1019:
1017:
995:
965:
931:
927:
851:
847:
843:
839:
779:
775:
771:
764:
762:
758:strong limit
751:
691:
687:
683:
679:
675:
576:) such that
573:
522:
518:
514:
453:
366:
362:
360:
357:Main results
251:
247:
79:
71:
63:
61:
52:cofinalities
37:ordered sets
21:mathematical
16:
15:
1175:Kurepa tree
452:, then pcf(
254:, that is,
76:ultrafilter
1335:Set theory
1245:PCF Theory
1223:References
41:power sets
29:cofinality
17:PCF theory
1188:|<min(
1143:ω
1138:ℵ
1121:ω
1116:ℵ
1078:ω
1073:ℵ
1062:ω
1058:ℵ
1034:|<min(
979:∏
950:λ
942:∏
911:λ
905:θ
890:
884:∈
881:θ
873:θ
826:λ
804:∏
796:
730:ω
725:ℵ
714:ω
710:ℵ
642:≤
625:
594:∈
589:θ
544:∏
536:
489:
483:∈
480:θ
472:θ
386:
380:⊆
298:∏
290:
267:
231:∈
192:∈
137:∏
101:∏
93:
48:cardinals
1329:Category
45:singular
1319:1112424
1278:0505434
1200:))=pcf(
1171:Magidor
1000:Jónsson
850:. Then
690:)|<|
31:of the
1317:
1276:
1179:Shelah
928:scales
246:. pcf(
74:is an
1297:arXiv
969:<λ
855:<λ
842:with
783:<λ
768:<λ
1134:<
1069:<
1022:)|=|
1006:on ℵ
908:<
823:<
721:<
428:<
213:<
163:<
25:1978
1307:doi
1266:doi
1204:).
1181:).
1008:ω+1
887:pcf
785:if
756:is
622:pcf
521:on
513:of
486:pcf
431:min
383:pcf
264:pcf
178:if
78:on
62:If
54:".
43:of
35:of
1331::
1315:MR
1313:,
1305:,
1293:26
1274:MR
1272:,
1262:30
1260:,
793:cf
774:.
760:.
533:cf
525:,
287:cf
90:cf
70:,
1309::
1299::
1268::
1202:A
1198:A
1194:A
1190:A
1186:A
1177:(
1169:(
1147:2
1125:1
1111:2
1082:1
1053:2
1040:ω
1036:A
1032:A
1028:A
1024:A
1020:A
996:A
982:A
966:J
946:B
932:A
914:}
902:,
899:)
896:A
893:(
878::
869:B
865:{
852:J
848:D
846:∈
844:B
840:D
819:)
815:D
811:/
807:A
800:(
780:J
778:∈
776:B
772:A
765:J
754:ω
734:4
705:2
692:A
688:A
684:A
680:A
676:A
659:|
655:A
651:|
646:2
638:|
634:)
631:A
628:(
618:|
597:D
585:B
574:A
559:)
555:D
551:/
547:A
540:(
523:A
519:D
515:A
501:}
498:)
495:A
492:(
477::
468:B
464:{
454:A
440:)
437:A
434:(
424:|
420:A
416:|
395:)
392:A
389:(
377:A
367:A
363:A
342:.
338:}
334:A
320:D
317::
313:)
309:D
305:/
301:A
294:(
283:{
279:=
276:)
273:A
270:(
252:A
248:A
234:D
228:}
225:)
222:x
219:(
216:g
210:)
207:x
204:(
201:f
198::
195:A
189:x
186:{
166:g
160:f
140:A
116:)
112:D
108:/
104:A
97:(
80:A
72:D
64:A
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