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Frege's startling discovery, of which he may or may not have been fully aware and which has been lost to view since the discovery of
Russell's paradox, was that
83:
119:
1575:
142:. However, not only did Basic Law V fail to be a logical proposition, but the resulting system proved to be inconsistent, because it was subject to
1550:
1529:
165:) of the fundamental propositions of arithmetic from a single consistent principle." This achievement has become known as Frege's theorem.
1432:
1467:
1167:
104:(vol. 1, 1893; vol. 2, 1903), Frege attempted to derive all of the laws of arithmetic from axioms he asserted as logical (see
1372:
96:
58:
1506:
1458:. Edited by Richard C. Jeffrey, introduction by John P. Burgess. Cambridge, Mass: Harvard University Press. p.
1405:
75:
1157:
1113:
1580:
1508:
Die
Grundlagen der Arithmetik – eine logisch-mathematische Untersuchung über den Begriff der Zahl
1547:
1526:
1339:, the result being shown below its main operator. Column 6 shows that the whole formula evaluates to
1336:
143:
45:
1344:
162:
41:
1473:
1463:
1459:
1117:
74:
in the early 1980s and has since been the focus of significant work. It is at the core of the
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1533:
1485:
110:
21:
71:
1564:
1447:
1388:
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49:
1452:
1427:
1156:
The theorem already holds in one of the weakest logics imaginable, the constructive
1423:
154:
79:
1304:
33:
29:
1348:
37:
1477:
17:
1335:(columns 1, 3, 5), each subformula is evaluated according to the rules for
105:
1484:
arithmetic can be derived in a purely logical system like that of his
1307:
to the right gives a semantic proof. For all possible assignments of
1225:{\displaystyle f\mapsto g\mapsto p\mapsto (f(p)\circ g)(p)}
1343:
in every case, i.e. that it is a tautology. In fact, its
161:"contains all the essential steps of a valid proof (in
1544:(in German). Vol. 2. Jena: Verlag Hermann Pohle.
1523:(in German). Vol. 1. Jena: Verlag Hermann Pohle.
1170:
114:; the one truly new principle was one he called the
130:) is the same as the "value-range" of the function
108:). Most of these axioms were carried over from his
1224:
1514:(in German). Breslau: Verlag von Wilhelm Koebner.
1488:from this consistent principle and from it alone.
1395:I, Jena: Verlag Hermann Pohle, 1893, §§20 and 47.
1376:, Breslau: Verlag von Wilhelm Koebner, 1884, §63.
1428:"Frege's Theorem and Foundations for Arithmetic"
153:overshadowed Frege's achievement: according to
168:
8:
1571:Theorems in the foundations of mathematics
120:axiom schema of unrestricted comprehension
1169:
1162:Brouwer–Heyting–Kolmogorov interpretation
1418:
1416:
1414:
1360:
62:) and proven more formally in his 1893
169:Frege's theorem in propositional logic
48:. It was first proven, informally, by
122:): the "value-range" of the function
70:I). The theorem was re-discovered by
7:
1384:
1382:
1433:Stanford Encyclopedia of Philosophy
1320:
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14:
1351:(column 10) are even equivalent.
1116:, Frege's theorem refers to this
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1576:Theorems in propositional logic
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1373:Die Grundlagen der Arithmetik
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149:The inconsistency in Frege's
97:The Foundations of Arithmetic
59:The Foundations of Arithmetic
54:Die Grundlagen der Arithmetik
1542:Grundgesetze der Arithmetik
1521:Grundgesetze der Arithmetik
1393:Grundgesetze der Arithmetik
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64:Grundgesetze der Arithmetik
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255:
1408:, January 26, 2012, p. 2.
76:philosophy of mathematics
102:Basic Laws of Arithmetic
68:Basic Laws of Arithmetic
1454:Logic, Logic, and Logic
1540:Gottlob Frege (1903).
1519:Gottlob Frege (1893).
1505:Gottlob Frege (1884).
1268:, then given a reason
1226:
1160:. The proof under the
1158:implicational calculus
100:(1884), and later, in
1252:denote a reason that
1236:denote a reason that
1227:
32:that states that the
1337:material conditional
1276:, we know that both
1168:
1404:Richard Pettigrew,
1347:(column 2) and its
1232:. In words: "Let
1114:propositional logic
1557:in modern notation
1553:2017-08-29 at the
1536:in modern notation
1532:2016-10-21 at the
1406:"Basic set theory"
1222:
163:second-order logic
138:) if and only if ∀
118:(now known as the
42:second-order logic
40:can be derived in
1110:
1109:
144:Russell's paradox
82:(at least of the
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46:Hume's principle
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1555:Wayback Machine
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1534:Wayback Machine
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1260:. Then given a
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111:Begriffsschrift
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84:Scottish School
26:Frege's theorem
22:metamathematics
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72:Crispin Wright
13:
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2:
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1465:
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1429:
1425:
1424:Zalta, Edward
1419:
1417:
1415:
1411:
1407:
1401:
1398:
1394:
1390:
1389:Gottlob Frege
1385:
1383:
1379:
1375:
1374:
1369:
1368:Gottlob Frege
1364:
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1354:
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1240:implies that
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1005:
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805:
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739:
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672:
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610:
605:
603:
593:
591:
584:
579:
572:
570:
560:
553:
551:
550:
544:
539:
534:
532:
522:
520:
515:
510:
505:
503:
493:
491:
484:
479:
472:
470:
460:
453:
451:
450:
444:
439:
434:
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393:
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384:
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372:
370:
360:
353:
351:
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344:
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322:
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303:
293:
291:
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246:
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200:
197:
194:
192:
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147:
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141:
137:
133:
129:
125:
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107:
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99:
98:
89:
87:
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81:
77:
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65:
61:
60:
55:
51:
50:Gottlob Frege
47:
43:
39:
35:
31:
27:
23:
19:
1581:Metatheorems
1541:
1520:
1507:
1483:
1481:
1453:
1442:
1431:
1400:
1392:
1371:
1363:
1340:
1332:
1328:
1324:
1316:
1308:
1302:
1297:
1293:
1289:
1285:
1281:
1277:
1273:
1269:
1265:
1261:
1257:
1253:
1249:
1245:
1241:
1237:
1233:
1155:
1149:
1145:
1141:
1137:
1133:
1129:
1125:
1111:
1093:
1075:
1057:
242:
234:
220:
212:
198:
190:
179:
159:Grundgesetze
158:
155:Edward Zalta
151:Grundgesetze
150:
148:
139:
135:
131:
127:
123:
115:
109:
101:
95:
93:
80:neo-logicism
67:
63:
57:
53:
52:in his 1884
34:Peano axioms
25:
15:
1305:truth table
116:Basic Law V
30:metatheorem
1565:Categories
1499:References
1349:consequent
1345:antecedent
1248:. And let
86:variety).
38:arithmetic
1292:holds by
1284:and that
1280:holds by
1205:∘
1187:↦
1181:↦
1175:↦
1118:tautology
78:known as
18:metalogic
1551:Archived
1546:–
1530:Archived
1525:–
1478:37509971
1450:(1998).
1426:(2013),
1321:✓
1313:✗
1300:holds."
1288:implies
1256:implies
1244:implies
1046:✓
1041:✓
1036:✓
1017:✓
1012:✓
1007:✓
981:✓
946:✗
941:✗
936:✓
917:✓
912:✓
907:✓
881:✗
846:✓
841:✓
836:✓
817:✗
812:✗
807:✓
781:✓
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717:✗
712:✗
707:✓
681:✓
646:✓
641:✓
636:✗
617:✓
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546:✗
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446:✓
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307:✗
281:✓
106:logicism
90:Overview
1548:Edition
1527:Edition
1136:)) → ((
1476:
1466:
1331:, and
1164:reads
1144:) → (
157:, the
1512:(PDF)
1355:Notes
1323:) to
1315:) or
1309:false
1296:. So
44:from
28:is a
1474:OCLC
1464:ISBN
1341:true
1317:true
1303:The
1272:for
20:and
1460:154
1128:→ (
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1106:13
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94:In
66:I (
36:of
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1567::
1480:.
1472:.
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1430:,
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1120::
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1100:11
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203:))
146:.
24:,
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1333:R
1329:Q
1325:P
1319:(
1311:(
1298:R
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1124:(
1088:9
1085:8
1082:7
1076:6
1070:5
1067:4
1064:3
1058:2
1054:1
1029:Y
1000:Y
967:Y
929:N
900:Y
867:N
829:Y
800:Y
767:Y
729:Y
700:Y
667:Y
629:Y
600:Y
567:Y
529:Y
500:Y
467:Y
429:Y
400:Y
367:Y
329:Y
300:Y
267:Y
243:R
239:→
235:P
231:(
228:→
225:)
221:Q
217:→
213:P
206:→
199:R
195:→
191:Q
187:(
184:→
180:P
176:(
140:x
136:x
134:(
132:g
128:x
126:(
124:f
56:(
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