2323:
Therefore, there is a way to go from a proof of 0 = 1 to a proof of any basic arithmetic equality, and thus to a proof of any complex arithmetic proposition. Furthermore, to get this result it was not necessary to invoke the Peano axiom that states that 0 is "not" the successor of any
964:
Kolmogorov followed the same lines but phrased his interpretation in terms of problems and solutions. To assert a formula is to claim to know a solution to the problem represented by that formula. For instance
1571:
1255:
1201:
1795:
1858:
1386:
1319:
707:
540:
1151:
572:
356:
284:
143:
1957:
2111:
1992:
1919:
2076:
2046:
1493:
1464:
842:
111:
2250:
252:
1103:
989:
816:
425:
2342:
2227:
1719:
1699:
1517:
1339:
925:
902:
959:
790:
655:
2270:
2207:
2187:
2163:
2143:
1878:
1739:
1679:
1659:
1639:
1619:
1591:
1438:
1414:
1275:
1069:
1049:
1029:
1009:
882:
862:
767:
747:
727:
632:
612:
592:
488:
468:
448:
396:
376:
324:
304:
223:
203:
183:
163:
2414:
does (which is always decidable for natural numbers), otherwise there is no proof. These are then built up by induction into more complex algorithms.
2538:
2548:
2390:. It deals with Heyting arithmetic, where the domain of quantification is the natural numbers and the primitive propositions are of the form
2166:
935:
The interpretation of a primitive proposition is supposed to be known from context. In the context of arithmetic, a proof of the formula
2520:
2434:
2422:
2543:
2376:
76:
1526:
1210:
1156:
2275:
A standard example of absurdity is found in dealing with arithmetic. Assume that 0 = 1, and proceed by
80:
1744:
1113:
2375:
that converts one proof to another, or that converts an element of a domain to a proof. Different versions of
2320: + 1. By induction, 0 is equal to all numbers, and therefore any two natural numbers become equal.
1813:
1347:
1280:
663:
496:
2474:
1118:
2489:
2360:
2276:
1884:
545:
329:
257:
116:
17:
1924:
2511:
2279:: 0 = 0 by the axiom of equality. Now (induction hypothesis), if 0 were equal to a certain
2485:
2470:
40:
2081:
1962:
1889:
2494:
2387:
2345:
28:
2055:
2025:
1472:
1443:
821:
90:
52:
2232:
231:
2291:
1082:
968:
798:
404:
44:
2327:
2212:
1704:
1684:
1502:
1324:
910:
887:
2515:
2418:
938:
772:
637:
2280:
2255:
2192:
2172:
2148:
2128:
2122:
1863:
1724:
1664:
1644:
1624:
1604:
1576:
1423:
1399:
1260:
1054:
1034:
1014:
994:
867:
847:
752:
732:
712:
617:
597:
577:
473:
453:
433:
381:
361:
309:
289:
208:
188:
168:
148:
64:
2532:
2383:
60:
48:
1806:
2421:
as defining the notion of a function, then the BHK interpretation describes the
928:
2371:
The BHK interpretation will depend on the view taken about what constitutes a
1959:, and in general has no proof. According to the interpretation, a proof of
2125:
to have a formal negation operator such that there is a proof of "not"
75:
The interpretation states what is intended to be a proof of a given
2316:), but since 0 = 1, therefore 0 would also be equal to
961:
is a computation reducing the two terms to the same numeral.
1804:
Indeed, the same line of thought provides a proof for the
67:. It is the standard explanation of intuitionistic logic.
2324:
natural number. This makes 0 = 1 suitable as
1797:, proving the law of non-contradiction, no matter what
2330:
2258:
2235:
2215:
2195:
2175:
2151:
2131:
2084:
2058:
2028:
1965:
1927:
1892:
1866:
1816:
1747:
1727:
1707:
1687:
1667:
1647:
1627:
1607:
1579:
1529:
1505:
1475:
1446:
1426:
1402:
1350:
1327:
1283:
1263:
1213:
1159:
1121:
1085:
1057:
1037:
1017:
997:
971:
941:
913:
890:
870:
850:
824:
801:
775:
755:
735:
715:
666:
640:
620:
600:
580:
548:
499:
476:
456:
436:
407:
384:
364:
332:
312:
292:
260:
234:
211:
191:
171:
151:
119:
93:
2475:"History of Constructivism in the Twentieth Century"
2488:(2003). "Constructivism and Proof Theory (draft)".
2336:
2264:
2244:
2221:
2201:
2181:
2157:
2137:
2105:
2070:
2040:
1986:
1951:
1913:
1872:
1852:
1789:
1733:
1713:
1693:
1673:
1653:
1633:
1613:
1585:
1565:
1511:
1487:
1458:
1432:
1408:
1380:
1333:
1313:
1269:
1249:
1195:
1145:
1097:
1063:
1043:
1023:
1003:
983:
953:
919:
896:
876:
856:
836:
810:
784:
761:
741:
721:
701:
649:
626:
606:
586:
566:
534:
482:
462:
442:
419:
390:
370:
350:
318:
298:
278:
246:
217:
197:
177:
157:
137:
105:
1031:; to solve it requires a method to solve problem
2359:0). This use of 0 = 1 validates the
2348:(and the Peano axiom is rewritten 0 =
1079:The identity function is a proof of the formula
1566:{\displaystyle (P\wedge (P\to \bot ))\to \bot }
1250:{\displaystyle (P\wedge (P\to \bot ))\to \bot }
1196:{\displaystyle (P\wedge (P\to \bot ))\to \bot }
931:(nontermination in some programming languages).
2169:. The BHK interpretation instead takes "not"
8:
1766:
1754:
561:
549:
345:
333:
273:
261:
132:
120:
1790:{\displaystyle f(\langle a,b\rangle )=b(a)}
2512:"The Development of Intuitionistic Logic"
2493:
2453:
2425:between natural deduction and functions.
2386:theory identifies the functions with the
2329:
2257:
2234:
2214:
2194:
2174:
2150:
2130:
2083:
2057:
2027:
1964:
1926:
1891:
1865:
1815:
1746:
1726:
1706:
1686:
1666:
1646:
1626:
1606:
1578:
1528:
1504:
1474:
1445:
1425:
1401:
1349:
1326:
1282:
1262:
1212:
1158:
1120:
1084:
1056:
1036:
1016:
996:
970:
940:
912:
889:
869:
849:
823:
800:
774:
754:
734:
714:
676:
665:
639:
619:
599:
579:
547:
509:
498:
475:
455:
435:
406:
383:
363:
331:
311:
291:
259:
233:
210:
190:
170:
150:
118:
92:
33:Brouwer–Heyting–Kolmogorov interpretation
18:Brouwer-Heyting-Kolmogorov interpretation
2446:
1853:{\displaystyle (P\wedge (P\to Q))\to Q}
1661:is a function that converts a proof of
1495:is a function that converts a proof of
2121:It is not, in general, possible for a
1381:{\displaystyle (P\wedge (P\to \bot ))}
1314:{\displaystyle (P\wedge (P\to \bot ))}
702:{\displaystyle (\forall x{\in }S)(Px)}
535:{\displaystyle (\exists x{\in }S)(Px)}
1146:{\displaystyle \neg (P\wedge \neg P)}
59:, because of the connection with the
7:
2252:is a function converting a proof of
2145:exactly when there isn't a proof of
1523:Putting it all together, a proof of
2521:Stanford Encyclopedia of Philosophy
2406:is simply the trivial algorithm if
567:{\displaystyle \langle x,a\rangle }
351:{\displaystyle \langle 1,b\rangle }
279:{\displaystyle \langle 0,a\rangle }
138:{\displaystyle \langle a,b\rangle }
2410:evaluates to the same number that
2331:
2236:
2216:
2094:
2065:
2035:
1975:
1952:{\displaystyle P\vee (P\to \bot )}
1943:
1902:
1708:
1688:
1560:
1548:
1506:
1482:
1453:
1369:
1328:
1302:
1244:
1232:
1190:
1178:
1134:
1122:
914:
891:
831:
802:
670:
503:
55:. It is also sometimes called the
25:
844:, so a proof of it is a function
2209:leads to absurdity, designated
2167:Gödel's incompleteness theorems
2510:Van Atten, Mark (4 May 2022).
2106:{\displaystyle P\vee (\neg P)}
2100:
2091:
2062:
2032:
1987:{\displaystyle P\vee (\neg P)}
1981:
1972:
1946:
1940:
1934:
1914:{\displaystyle P\vee (\neg P)}
1908:
1899:
1844:
1841:
1838:
1832:
1826:
1817:
1784:
1778:
1769:
1751:
1557:
1554:
1551:
1545:
1539:
1530:
1479:
1450:
1375:
1372:
1366:
1360:
1351:
1308:
1305:
1299:
1293:
1284:
1241:
1238:
1235:
1229:
1223:
1214:
1187:
1184:
1181:
1175:
1169:
1160:
1140:
1125:
1089:
975:
828:
696:
687:
684:
667:
529:
520:
517:
500:
411:
1:
2539:Dependently typed programming
2549:Constructivism (mathematics)
2379:will diverge on this point.
2078:is provable then neither is
1051:given a solution to problem
57:realizability interpretation
2435:Curry–Howard correspondence
2286:, then 1 would be equal to
2272:into a proof of absurdity.
991:is the problem of reducing
2565:
2071:{\displaystyle P\to \bot }
2041:{\displaystyle P\to \bot }
1488:{\displaystyle P\to \bot }
1459:{\displaystyle P\to \bot }
837:{\displaystyle P\to \bot }
81:induction on the structure
1593:that converts a pair <
1277:that converts a proof of
864:that converts a proof of
729:that converts an element
450:that converts a proof of
106:{\displaystyle P\wedge Q}
1388:is a pair of proofs <
1114:law of non-contradiction
2245:{\displaystyle \lnot P}
2117:Definition of absurdity
1883:On the other hand, the
247:{\displaystyle P\vee Q}
79:. This is specified by
51:, and independently by
2544:Functional programming
2367:Definition of function
2361:principle of explosion
2355:→ 0 =
2338:
2277:mathematical induction
2266:
2246:
2223:
2203:
2183:
2159:
2139:
2107:
2072:
2042:
1988:
1953:
1915:
1885:law of excluded middle
1874:
1854:
1791:
1741:that does this, where
1735:
1721:. There is a function
1715:
1695:
1675:
1655:
1635:
1615:
1587:
1567:
1513:
1489:
1460:
1434:
1410:
1382:
1335:
1315:
1271:
1251:
1197:
1147:
1099:
1098:{\displaystyle P\to P}
1065:
1045:
1025:
1005:
985:
984:{\displaystyle P\to Q}
955:
921:
898:
878:
858:
838:
812:
811:{\displaystyle \neg P}
786:
763:
743:
723:
703:
651:
628:
608:
588:
568:
536:
484:
464:
444:
421:
420:{\displaystyle P\to Q}
392:
372:
352:
320:
300:
280:
248:
219:
199:
179:
159:
139:
107:
2339:
2337:{\displaystyle \bot }
2267:
2247:
2229:, so that a proof of
2224:
2222:{\displaystyle \bot }
2204:
2184:
2160:
2140:
2108:
2073:
2043:
1989:
1954:
1916:
1875:
1855:
1792:
1736:
1716:
1714:{\displaystyle \bot }
1696:
1694:{\displaystyle \bot }
1676:
1656:
1636:
1616:
1588:
1568:
1514:
1512:{\displaystyle \bot }
1490:
1461:
1435:
1411:
1383:
1336:
1334:{\displaystyle \bot }
1316:
1272:
1252:
1198:
1148:
1100:
1066:
1046:
1026:
1006:
986:
956:
922:
920:{\displaystyle \bot }
907:There is no proof of
899:
897:{\displaystyle \bot }
879:
859:
839:
813:
787:
764:
744:
724:
704:
652:
629:
609:
589:
569:
537:
485:
465:
445:
422:
393:
373:
353:
321:
301:
281:
249:
220:
200:
180:
160:
140:
108:
2388:computable functions
2328:
2256:
2233:
2213:
2193:
2173:
2149:
2129:
2082:
2056:
2026:
1963:
1925:
1890:
1880:is any proposition.
1864:
1814:
1745:
1725:
1705:
1685:
1665:
1645:
1625:
1605:
1577:
1527:
1503:
1473:
1444:
1424:
1400:
1348:
1325:
1281:
1261:
1211:
1157:
1119:
1083:
1055:
1035:
1015:
995:
969:
939:
911:
888:
868:
848:
822:
799:
773:
753:
733:
713:
664:
638:
618:
598:
578:
546:
497:
474:
454:
434:
405:
382:
362:
330:
310:
290:
258:
232:
209:
189:
169:
149:
117:
91:
41:intuitionistic logic
954:{\displaystyle x=y}
927:, the absurdity or
2346:Heyting arithmetic
2334:
2290: + 1, (
2262:
2242:
2219:
2199:
2179:
2155:
2135:
2103:
2068:
2048:. Thus if neither
2038:
1984:
1949:
1911:
1870:
1850:
1787:
1731:
1711:
1701:– into a proof of
1691:
1671:
1651:
1631:
1611:
1583:
1563:
1509:
1485:
1456:
1430:
1406:
1378:
1331:
1311:
1267:
1247:
1193:
1143:
1095:
1061:
1041:
1021:
1001:
981:
951:
917:
894:
874:
854:
834:
808:
785:{\displaystyle Px}
782:
759:
739:
719:
699:
650:{\displaystyle Px}
647:
624:
604:
584:
564:
532:
480:
460:
440:
417:
388:
368:
348:
316:
296:
276:
244:
215:
195:
175:
155:
135:
103:
71:The interpretation
37:BHK interpretation
29:mathematical logic
2265:{\displaystyle P}
2202:{\displaystyle P}
2182:{\displaystyle P}
2158:{\displaystyle P}
2138:{\displaystyle P}
1873:{\displaystyle Q}
1734:{\displaystyle f}
1674:{\displaystyle P}
1654:{\displaystyle b}
1634:{\displaystyle P}
1614:{\displaystyle a}
1586:{\displaystyle f}
1433:{\displaystyle b}
1409:{\displaystyle a}
1270:{\displaystyle f}
1105:, no matter what
1064:{\displaystyle P}
1044:{\displaystyle Q}
1024:{\displaystyle P}
1004:{\displaystyle Q}
877:{\displaystyle P}
857:{\displaystyle f}
762:{\displaystyle S}
742:{\displaystyle x}
722:{\displaystyle f}
627:{\displaystyle a}
607:{\displaystyle S}
594:is an element of
587:{\displaystyle x}
483:{\displaystyle Q}
463:{\displaystyle P}
443:{\displaystyle f}
391:{\displaystyle Q}
371:{\displaystyle b}
319:{\displaystyle P}
299:{\displaystyle a}
218:{\displaystyle Q}
198:{\displaystyle b}
178:{\displaystyle P}
158:{\displaystyle a}
83:of that formula:
53:Andrey Kolmogorov
16:(Redirected from
2556:
2525:
2516:Zalta, Edward N.
2499:
2497:
2481:
2479:
2457:
2451:
2343:
2341:
2340:
2335:
2271:
2269:
2268:
2263:
2251:
2249:
2248:
2243:
2228:
2226:
2225:
2220:
2208:
2206:
2205:
2200:
2188:
2186:
2185:
2180:
2164:
2162:
2161:
2156:
2144:
2142:
2141:
2136:
2112:
2110:
2109:
2104:
2077:
2075:
2074:
2069:
2047:
2045:
2044:
2039:
1993:
1991:
1990:
1985:
1958:
1956:
1955:
1950:
1920:
1918:
1917:
1912:
1879:
1877:
1876:
1871:
1859:
1857:
1856:
1851:
1796:
1794:
1793:
1788:
1740:
1738:
1737:
1732:
1720:
1718:
1717:
1712:
1700:
1698:
1697:
1692:
1681:into a proof of
1680:
1678:
1677:
1672:
1660:
1658:
1657:
1652:
1640:
1638:
1637:
1632:
1620:
1618:
1617:
1612:
1592:
1590:
1589:
1584:
1572:
1570:
1569:
1564:
1518:
1516:
1515:
1510:
1499:into a proof of
1494:
1492:
1491:
1486:
1465:
1463:
1462:
1457:
1439:
1437:
1436:
1431:
1415:
1413:
1412:
1407:
1387:
1385:
1384:
1379:
1340:
1338:
1337:
1332:
1321:into a proof of
1320:
1318:
1317:
1312:
1276:
1274:
1273:
1268:
1256:
1254:
1253:
1248:
1202:
1200:
1199:
1194:
1152:
1150:
1149:
1144:
1104:
1102:
1101:
1096:
1070:
1068:
1067:
1062:
1050:
1048:
1047:
1042:
1030:
1028:
1027:
1022:
1010:
1008:
1007:
1002:
990:
988:
987:
982:
960:
958:
957:
952:
926:
924:
923:
918:
903:
901:
900:
895:
884:into a proof of
883:
881:
880:
875:
863:
861:
860:
855:
843:
841:
840:
835:
817:
815:
814:
809:
791:
789:
788:
783:
769:into a proof of
768:
766:
765:
760:
748:
746:
745:
740:
728:
726:
725:
720:
708:
706:
705:
700:
680:
656:
654:
653:
648:
633:
631:
630:
625:
613:
611:
610:
605:
593:
591:
590:
585:
573:
571:
570:
565:
541:
539:
538:
533:
513:
489:
487:
486:
481:
470:into a proof of
469:
467:
466:
461:
449:
447:
446:
441:
430:
426:
424:
423:
418:
397:
395:
394:
389:
377:
375:
374:
369:
357:
355:
354:
349:
325:
323:
322:
317:
305:
303:
302:
297:
285:
283:
282:
277:
253:
251:
250:
245:
224:
222:
221:
216:
204:
202:
201:
196:
184:
182:
181:
176:
164:
162:
161:
156:
144:
142:
141:
136:
112:
110:
109:
104:
45:L. E. J. Brouwer
43:was proposed by
21:
2564:
2563:
2559:
2558:
2557:
2555:
2554:
2553:
2529:
2528:
2509:
2506:
2484:
2477:
2469:
2466:
2461:
2460:
2452:
2448:
2443:
2431:
2419:lambda calculus
2369:
2326:
2325:
2308:if and only if
2254:
2253:
2231:
2230:
2211:
2210:
2191:
2190:
2171:
2170:
2147:
2146:
2127:
2126:
2119:
2080:
2079:
2054:
2053:
2024:
2023:
1961:
1960:
1923:
1922:
1888:
1887:
1862:
1861:
1860:as well, where
1812:
1811:
1743:
1742:
1723:
1722:
1703:
1702:
1683:
1682:
1663:
1662:
1643:
1642:
1623:
1622:
1603:
1602:
1575:
1574:
1525:
1524:
1501:
1500:
1471:
1470:
1442:
1441:
1422:
1421:
1398:
1397:
1346:
1345:
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2454:Van Atten 2022
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2423:correspondence
2377:constructivism
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2281:natural number
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2010:is a proof of
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2504:External link
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2486:Troelstra, A.
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2471:Troelstra, A.
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2417:If one takes
2415:
2413:
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2398:. A proof of
2397:
2393:
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2385:
2384:realizability
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1601:> – where
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62:
61:realizability
58:
54:
50:
49:Arend Heyting
46:
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34:
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19:
2519:
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2120:
2049:
2019:
2015:
2011:
2007:
2003:
1999:
1995:
1882:
1807:modus ponens
1805:
1803:
1798:
1598:
1594:
1522:
1496:
1417:
1396:>, where
1393:
1389:
1111:
1106:
1078:
963:
934:
795:The formula
74:
56:
36:
32:
26:
2292:Peano axiom
2002:> where
1921:expands to
1469:A proof of
1344:A proof of
1207:A proof of
1153:expands to
929:bottom type
660:A proof of
493:A proof of
401:A proof of
228:A proof of
87:A proof of
2533:Categories
2464:References
542:is a pair
254:is either
113:is a pair
63:theory of
2490:CiteSeerX
2382:Kleene's
2332:⊥
2237:¬
2217:⊥
2095:¬
2089:∨
2066:⊥
2063:→
2036:⊥
2033:→
2018:is 1 and
2006:is 0 and
1976:¬
1970:∨
1944:⊥
1941:→
1932:∨
1903:¬
1897:∨
1845:→
1833:→
1824:∧
1767:⟩
1755:⟨
1709:⊥
1689:⊥
1561:⊥
1558:→
1549:⊥
1546:→
1537:∧
1507:⊥
1483:⊥
1480:→
1454:⊥
1451:→
1370:⊥
1367:→
1358:∧
1329:⊥
1303:⊥
1300:→
1291:∧
1245:⊥
1242:→
1233:⊥
1230:→
1221:∧
1191:⊥
1188:→
1179:⊥
1176:→
1167:∧
1135:¬
1132:∧
1123:¬
1090:→
976:→
915:⊥
892:⊥
832:⊥
829:→
803:¬
678:∈
671:∀
562:⟩
550:⟨
511:∈
504:∃
412:→
346:⟩
334:⟨
274:⟩
262:⟨
239:∨
133:⟩
121:⟨
98:∧
2473:(1991).
2429:See also
2373:function
1998:,
1597:,
1392:,
1075:Examples
429:function
2518:(ed.).
77:formula
2492:
2165:; see
1641:, and
1420:, and
574:where
358:where
286:where
145:where
31:, the
2514:. In
2478:(PDF)
2441:Notes
2014:, or
1810:rule
427:is a
39:, of
35:, or
2052:nor
1801:is.
1112:The
1109:is.
614:and
185:and
47:and
2344:in
1011:to
749:of
326:or
27:In
2535::
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2412:y
2408:x
2404:y
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2353:n
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2318:n
2314:n
2310:m
2306:n
2303:S
2299:m
2296:S
2288:n
2284:n
2260:P
2240:P
2197:P
2177:P
2153:P
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2101:)
2098:P
2092:(
2086:P
2060:P
2050:P
2030:P
2020:b
2016:a
2012:P
2008:b
2004:a
2000:b
1996:a
1982:)
1979:P
1973:(
1967:P
1947:)
1938:P
1935:(
1929:P
1909:)
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1900:(
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1818:(
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1782:a
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340:,
337:1
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294:a
271:a
268:,
265:0
242:Q
236:P
225:.
213:Q
193:b
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153:a
130:b
127:,
124:a
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20:)
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