1048:
argued based on this that since all science's predictions are in the vocabulary of observation terms, the theoretical vocabulary of science is in principle eliminable. He himself raised two objections to this argument: 1) the new axioms of science are practically unmanageable, and 2) science uses
1053:
argues that this argument is based on a misconception that the sole aim of science is successful prediction. He proposes that the main reason we need theoretical terms is that we wish to talk about theoretical entities (such as viruses, radio stars, and elementary particles).
771:
439:
694:
216:
104:
1121:
1042:
995:
968:
869:
842:
629:
570:
543:
516:
489:
371:
344:
317:
270:
151:
1015:
941:
913:
889:
815:
795:
600:
The proof above shows that for each recursively enumerable set of axioms there is a recursive set of axioms with the same deductive closure. A set of axioms is
590:
462:
290:
239:
124:
604:
if there is a primitive recursive function that decides membership in the set. To obtain a primitive recursive axiomatization, instead of replacing a formula
844:
is in the original recursively enumerable set of axioms. It is possible for a primitive recursive function to parse an expression of form (*) to obtain
705:
444:
Since each formula has finite length, it is checkable whether or not it is of the said form. If it is of the said form and consists of
379:
637:
159:
1116:
1087:
1049:
inductive reasoning and eliminating theoretical terms may alter the inductive relations between observational sentences.
1082:
49:
29:
1099:
892:
63:
37:
601:
45:
33:
1045:
17:
242:
1020:
973:
946:
847:
820:
607:
548:
521:
494:
467:
349:
322:
295:
248:
129:
943:
is a recursively axiomatizable theory and we divide its predicates into two disjoint sets
895:
is primitive recursive, it is possible for a primitive recursive function to verify that
1000:
926:
898:
874:
800:
780:
575:
447:
275:
224:
109:
41:
1071:
1110:
1094:
1050:
1044:
are recursively enumerable, and hence, based on Craig's theorem, axiomatizable.
592:
and then checking symbol-for-symbol whether the expressions are identical.
346:
lends itself according to the following informal reasoning. Each member of
766:{\displaystyle \underbrace {A_{i}\land \dots \land A_{i}} _{f(i)}}
48:
theorem, although both results are named after the same logician,
434:{\displaystyle \underbrace {B_{j}\land \dots \land B_{j}} _{j}.}
106:
be an enumeration of the axioms of a recursively enumerable set
689:{\displaystyle \underbrace {A_{i}\land \dots \land A_{i}} _{i}}
211:{\displaystyle \underbrace {A_{i}\land \dots \land A_{i}} _{i}}
545:. Again, it is checkable whether the conjunct is in fact
1074:". Bulletin of Symbolic Logic, vol. 18, no. 3 (2012).
1023:
1003:
976:
949:
929:
901:
877:
850:
823:
803:
783:
708:
640:
610:
578:
551:
524:
497:
470:
450:
382:
352:
325:
298:
278:
251:
227:
162:
132:
112:
66:
1036:
1009:
989:
962:
935:
907:
883:
863:
836:
809:
789:
765:
688:
623:
584:
572:by going through the enumeration of the axioms of
564:
537:
510:
483:
456:
433:
365:
338:
311:
284:
264:
233:
210:
145:
118:
98:
1072:Vaught's Theorem on Axiomatizability by a Scheme
126:of first-order formulas. Construct another set
44:. This result is not related to the well-known
292:are thus equivalent; the proof will show that
915:is indeed a computation history as required.
817:, returns a computation history showing that
319:is a recursive set. A decision procedure for
8:
1122:Theorems in the foundations of mathematics
1085:. "On Axiomatizability Within a System",
1028:
1022:
1002:
981:
975:
954:
948:
928:
900:
876:
855:
849:
828:
822:
802:
782:
748:
736:
717:
710:
707:
680:
668:
649:
642:
639:
615:
609:
577:
556:
550:
529:
523:
502:
496:
475:
469:
449:
422:
410:
391:
384:
381:
357:
351:
330:
324:
303:
297:
277:
256:
250:
226:
202:
190:
171:
164:
161:
137:
131:
111:
84:
71:
65:
1063:
1103:, Vol. 62, No. 10 (1965), pp. 251.260.
7:
1091:, Vol. 18, No. 1 (1953), pp. 30-32.
596:Primitive recursive axiomatizations
99:{\displaystyle A_{1},A_{2},\dots }
14:
491:if the (reoccurring) conjunct is
758:
752:
699:one instead replaces it with
1:
1088:The Journal of Symbolic Logic
40:is (primitively) recursively
1017:that are in the vocabulary
1138:
919:Philosophical implications
797:is a function that, given
221:for each positive integer
30:recursively enumerable set
1100:The Journal of Philosophy
997:, then those theorems of
518:; otherwise it is not in
56:Recursive axiomatization
1038:
1011:
991:
964:
937:
909:
885:
865:
838:
811:
791:
767:
690:
625:
586:
566:
539:
512:
485:
458:
435:
367:
340:
313:
286:
266:
235:
212:
147:
120:
100:
1097:. "Craig's Theorem",
1039:
1037:{\displaystyle V_{A}}
1012:
992:
990:{\displaystyle V_{B}}
965:
963:{\displaystyle V_{A}}
938:
910:
886:
866:
864:{\displaystyle A_{i}}
839:
837:{\displaystyle A_{i}}
812:
792:
768:
691:
626:
624:{\displaystyle A_{i}}
587:
567:
565:{\displaystyle A_{n}}
540:
538:{\displaystyle T^{*}}
513:
511:{\displaystyle A_{j}}
486:
484:{\displaystyle T^{*}}
459:
436:
368:
366:{\displaystyle T^{*}}
341:
339:{\displaystyle T^{*}}
314:
312:{\displaystyle T^{*}}
287:
267:
265:{\displaystyle T^{*}}
236:
213:
148:
146:{\displaystyle T^{*}}
121:
101:
1117:Computability theory
1021:
1001:
974:
947:
927:
899:
893:Kleene's T predicate
875:
848:
821:
801:
781:
706:
638:
608:
576:
549:
522:
495:
468:
464:conjuncts, it is in
448:
380:
350:
323:
296:
276:
249:
225:
160:
130:
110:
64:
38:first-order language
34:well-formed formulas
602:primitive recursive
46:Craig interpolation
1034:
1007:
987:
960:
933:
905:
881:
861:
834:
807:
787:
763:
762:
746:
686:
685:
678:
621:
582:
562:
535:
508:
481:
454:
431:
427:
420:
363:
336:
309:
282:
262:
243:deductive closures
231:
208:
207:
200:
143:
116:
96:
28:) states that any
18:mathematical logic
1010:{\displaystyle T}
936:{\displaystyle T}
908:{\displaystyle j}
884:{\displaystyle j}
810:{\displaystyle i}
790:{\displaystyle f}
711:
709:
643:
641:
585:{\displaystyle T}
457:{\displaystyle j}
385:
383:
285:{\displaystyle T}
234:{\displaystyle i}
165:
163:
119:{\displaystyle T}
1129:
1075:
1068:
1043:
1041:
1040:
1035:
1033:
1032:
1016:
1014:
1013:
1008:
996:
994:
993:
988:
986:
985:
969:
967:
966:
961:
959:
958:
942:
940:
939:
934:
914:
912:
911:
906:
891:. Then, because
890:
888:
887:
882:
870:
868:
867:
862:
860:
859:
843:
841:
840:
835:
833:
832:
816:
814:
813:
808:
796:
794:
793:
788:
772:
770:
769:
764:
761:
747:
742:
741:
740:
722:
721:
695:
693:
692:
687:
684:
679:
674:
673:
672:
654:
653:
630:
628:
627:
622:
620:
619:
591:
589:
588:
583:
571:
569:
568:
563:
561:
560:
544:
542:
541:
536:
534:
533:
517:
515:
514:
509:
507:
506:
490:
488:
487:
482:
480:
479:
463:
461:
460:
455:
440:
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437:
432:
426:
421:
416:
415:
414:
396:
395:
372:
370:
369:
364:
362:
361:
345:
343:
342:
337:
335:
334:
318:
316:
315:
310:
308:
307:
291:
289:
288:
283:
271:
269:
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263:
261:
260:
240:
238:
237:
232:
217:
215:
214:
209:
206:
201:
196:
195:
194:
176:
175:
152:
150:
149:
144:
142:
141:
125:
123:
122:
117:
105:
103:
102:
97:
89:
88:
76:
75:
1137:
1136:
1132:
1131:
1130:
1128:
1127:
1126:
1107:
1106:
1079:
1078:
1069:
1065:
1060:
1024:
1019:
1018:
999:
998:
977:
972:
971:
950:
945:
944:
925:
924:
921:
897:
896:
873:
872:
851:
846:
845:
824:
819:
818:
799:
798:
779:
778:
732:
713:
712:
704:
703:
664:
645:
644:
636:
635:
611:
606:
605:
598:
574:
573:
552:
547:
546:
525:
520:
519:
498:
493:
492:
471:
466:
465:
446:
445:
406:
387:
386:
378:
377:
373:is of the form
353:
348:
347:
326:
321:
320:
299:
294:
293:
274:
273:
252:
247:
246:
223:
222:
186:
167:
166:
158:
157:
133:
128:
127:
108:
107:
80:
67:
62:
61:
58:
24:(also known as
22:Craig's theorem
12:
11:
5:
1135:
1133:
1125:
1124:
1119:
1109:
1108:
1105:
1104:
1092:
1077:
1076:
1062:
1061:
1059:
1056:
1046:Carl G. Hempel
1031:
1027:
1006:
984:
980:
957:
953:
932:
920:
917:
904:
880:
858:
854:
831:
827:
806:
786:
775:
774:
760:
757:
754:
751:
745:
739:
735:
731:
728:
725:
720:
716:
697:
696:
683:
677:
671:
667:
663:
660:
657:
652:
648:
618:
614:
597:
594:
581:
559:
555:
532:
528:
505:
501:
478:
474:
453:
442:
441:
430:
425:
419:
413:
409:
405:
402:
399:
394:
390:
360:
356:
333:
329:
306:
302:
281:
259:
255:
230:
219:
218:
205:
199:
193:
189:
185:
182:
179:
174:
170:
153:consisting of
140:
136:
115:
95:
92:
87:
83:
79:
74:
70:
57:
54:
13:
10:
9:
6:
4:
3:
2:
1134:
1123:
1120:
1118:
1115:
1114:
1112:
1102:
1101:
1096:
1095:Hilary Putnam
1093:
1090:
1089:
1084:
1083:William Craig
1081:
1080:
1073:
1067:
1064:
1057:
1055:
1052:
1051:Hilary Putnam
1047:
1029:
1025:
1004:
982:
978:
955:
951:
930:
918:
916:
902:
894:
878:
856:
852:
829:
825:
804:
784:
755:
749:
743:
737:
733:
729:
726:
723:
718:
714:
702:
701:
700:
681:
675:
669:
665:
661:
658:
655:
650:
646:
634:
633:
632:
616:
612:
603:
595:
593:
579:
557:
553:
530:
526:
503:
499:
476:
472:
451:
428:
423:
417:
411:
407:
403:
400:
397:
392:
388:
376:
375:
374:
358:
354:
331:
327:
304:
300:
279:
257:
253:
244:
228:
203:
197:
191:
187:
183:
180:
177:
172:
168:
156:
155:
154:
138:
134:
113:
93:
90:
85:
81:
77:
72:
68:
55:
53:
51:
50:William Craig
47:
43:
42:axiomatizable
39:
35:
31:
27:
26:Craig's trick
23:
19:
1098:
1086:
1070:A. Visser, "
1066:
922:
776:
698:
599:
443:
220:
59:
25:
21:
15:
1111:Categories
1058:References
744:⏟
730:∧
727:⋯
724:∧
676:⏟
662:∧
659:⋯
656:∧
531:∗
477:∗
418:⏟
404:∧
401:⋯
398:∧
359:∗
332:∗
305:∗
258:∗
198:⏟
184:∧
181:⋯
178:∧
139:∗
94:…
777:where
631:with
241:. The
36:of a
970:and
871:and
272:and
60:Let
923:If
773:(*)
245:of
32:of
16:In
1113::
52:.
20:,
1030:A
1026:V
1005:T
983:B
979:V
956:A
952:V
931:T
903:j
879:j
857:i
853:A
830:i
826:A
805:i
785:f
759:)
756:i
753:(
750:f
738:i
734:A
719:i
715:A
682:i
670:i
666:A
651:i
647:A
617:i
613:A
580:T
558:n
554:A
527:T
504:j
500:A
473:T
452:j
429:.
424:j
412:j
408:B
393:j
389:B
355:T
328:T
301:T
280:T
254:T
229:i
204:i
192:i
188:A
173:i
169:A
135:T
114:T
91:,
86:2
82:A
78:,
73:1
69:A
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