2243:
222:
While these results originally seemed too simple or too technical to warrant independent interest, they have eventually turned out to be central to the theories in which they occur.
622:
1297:
366:
1380:
521:
97:). However, a lemma can be considered a minor result whose sole purpose is to help prove a more substantial theorem â a step in the direction of proof.
1694:
1852:
640:
1707:
1030:
175:
1292:
422:
105:
Some powerful results in mathematics are known as lemmas, first named for their originally minor purpose. These include, among others:
1712:
1702:
1439:
645:
1190:
636:
2267:
1848:
378:
346:
140:
1945:
1689:
514:
1250:
943:
684:
2206:
1908:
1671:
1666:
1491:
912:
596:
2201:
1984:
1901:
1614:
1545:
1422:
664:
1272:
2126:
1952:
1638:
871:
1277:
1609:
1348:
606:
507:
2004:
1999:
1933:
1523:
917:
885:
576:
650:
2223:
2172:
2069:
1567:
1528:
1005:
150:
81:λáżÎŒÎŒÎ±, (perfect passive ΔጎληΌΌαÎč) something received or taken. Thus something taken for granted in an argument.
2064:
679:
1994:
1533:
1385:
1368:
1091:
571:
1896:
1873:
1834:
1720:
1661:
1307:
1227:
1071:
1015:
628:
110:
2186:
1913:
1891:
1858:
1751:
1597:
1582:
1555:
1506:
1390:
1325:
1150:
1116:
1111:
985:
816:
793:
2116:
1969:
1761:
1479:
1215:
1121:
980:
965:
846:
821:
205:
2242:
490:
165:
65:" or an "auxiliary theorem". In many cases, a lemma derives its importance from the theorem it aims to
61:
which is used as a stepping stone to a larger result. For that reason, it is also known as a "helping
2089:
2051:
1928:
1732:
1572:
1496:
1474:
1302:
1260:
1159:
1126:
990:
778:
689:
262:
144:
2218:
2109:
2094:
2074:
2031:
1918:
1868:
1794:
1739:
1676:
1469:
1464:
1412:
1180:
1169:
841:
741:
669:
660:
656:
591:
252:
170:
115:
2247:
2016:
1979:
1964:
1957:
1940:
1726:
1592:
1518:
1501:
1454:
1267:
1176:
1010:
995:
955:
907:
892:
880:
836:
811:
581:
530:
66:
31:
1744:
1200:
200:
195:
190:
370:
360:
2182:
1989:
1799:
1789:
1681:
1562:
1397:
1373:
1154:
1138:
1043:
1020:
897:
866:
831:
726:
561:
374:
342:
247:
160:
155:
125:
2196:
2191:
2084:
2041:
1863:
1824:
1819:
1804:
1630:
1587:
1484:
1282:
1232:
806:
768:
474:
301:
185:
180:
135:
130:
2272:
2177:
2167:
2121:
2104:
2059:
2021:
1923:
1843:
1650:
1577:
1550:
1538:
1444:
1358:
1332:
1287:
1255:
1056:
858:
801:
751:
716:
674:
215:
120:
2162:
2141:
2099:
2079:
1974:
1829:
1427:
1417:
1407:
1402:
1336:
1210:
1086:
975:
970:
948:
549:
478:
297:
257:
2261:
2136:
1814:
1321:
1106:
1096:
1066:
1051:
721:
78:
326:
2036:
1883:
1784:
1776:
1656:
1604:
1513:
1449:
1432:
1363:
1222:
1081:
783:
566:
210:
69:; however, a lemma can also turn out to be more important than originally thought.
2146:
2026:
1205:
1195:
1142:
826:
746:
731:
611:
556:
38:
1076:
931:
902:
708:
486:
305:
242:
17:
2228:
2131:
1184:
1101:
1061:
1025:
961:
773:
763:
736:
237:
394:
2213:
2011:
1459:
1164:
758:
1809:
601:
277:
272:
94:
90:
62:
58:
499:
423:"What is the difference between a theorem, a lemma, and a corollary?"
267:
1353:
699:
544:
232:
503:
328:"Lemma." Merriam-Webster.com Dictionary, Merriam-Webster.
446:
479:
Opinion 82: A Good Lemma is Worth a
Thousand Theorems
89:
There is no formal distinction between a lemma and a
2155:
2050:
1882:
1775:
1627:
1320:
1243:
1137:
1041:
930:
857:
792:
707:
698:
620:
537:
485:This article incorporates material from Lemma on
362:Handbook of Writing for the Mathematical Sciences
491:Creative Commons Attribution/Share-Alike License
367:Society for Industrial and Applied Mathematics
515:
8:
1341:
936:
704:
522:
508:
500:
27:Theorem for proving more complex theorems
319:
289:
395:"Definition of lemma | Dictionary.com"
7:
416:
414:
46:
25:
341:Lexington Books. March 12, 2018.
2241:
427:David Richeson: Division by Zero
57:) is a generally minor, proven
489:, which is licensed under the
1:
2202:History of mathematical logic
421:Richeson, Dave (2008-09-23).
93:, only one of intention (see
2127:Primitive recursive function
359:Higham, Nicholas J. (1998).
143:(any of several named after
447:"Oxford English Dictionary"
339:Beyond the Problem of Evil.
2289:
1191:SchröderâBernstein theorem
918:Monadic predicate calculus
577:Foundations of mathematics
29:
2237:
2224:Philosophy of mathematics
2173:Automated theorem proving
1344:
1298:Von NeumannâBernaysâGödel
939:
453:. Oxford University Press
2268:Mathematical terminology
30:Not to be confused with
1874:Self-verifying theories
1695:Tarski's axiomatization
646:Tarski's undefinability
641:incompleteness theorems
85:Comparison with theorem
2248:Mathematics portal
1859:Proof of impossibility
1507:propositional variable
817:Propositional calculus
2117:Kolmogorov complexity
2070:Computably enumerable
1970:Model complete theory
1762:Principia Mathematica
822:Propositional formula
651:BanachâTarski paradox
337:Loewen, Nathan R. B.
206:Vitali covering lemma
2065:ChurchâTuring thesis
2052:Computability theory
1261:continuum hypothesis
779:Square of opposition
637:Gödel's completeness
151:Greendlinger's lemma
145:Carl Friedrich Gauss
41:and other fields, a
2219:Mathematical object
2110:P versus NP problem
2075:Computable function
1869:Reverse mathematics
1795:Logical consequence
1672:primitive recursive
1667:elementary function
1440:Free/bound variable
1293:TarskiâGrothendieck
812:Logical connectives
742:Logical equivalence
592:Logical consequence
278:Theorem terminology
253:Inference objection
95:Theorem terminology
2017:Transfer principle
1980:Semantics of logic
1965:Categorical theory
1941:Non-standard model
1455:Logical connective
582:Information theory
531:Mathematical logic
399:www.dictionary.com
166:LovĂĄsz local lemma
32:Lemma (morphology)
2255:
2254:
2187:Abstract category
1990:Theories of truth
1800:Rule of inference
1790:Natural deduction
1771:
1770:
1316:
1315:
1021:Cartesian product
926:
925:
832:Many-valued logic
807:Boolean functions
690:Russell's paradox
665:diagonal argument
562:First-order logic
248:Fundamental lemma
101:Well-known lemmas
16:(Redirected from
2280:
2246:
2245:
2197:History of logic
2192:Category of sets
2085:Decision problem
1864:Ordinal analysis
1805:Sequent calculus
1703:Boolean algebras
1643:
1642:
1617:
1588:logical/constant
1342:
1328:
1251:ZermeloâFraenkel
1002:Set operations:
937:
874:
705:
685:LöwenheimâSkolem
572:Formal semantics
524:
517:
510:
501:
475:Doron Zeilberger
463:
462:
460:
458:
443:
437:
436:
434:
433:
418:
409:
408:
406:
405:
391:
385:
384:
356:
350:
335:
329:
324:
309:
302:argument mapping
294:
176:Poincaré's lemma
171:Nakayama's lemma
116:Burnside's lemma
48:
21:
2288:
2287:
2283:
2282:
2281:
2279:
2278:
2277:
2258:
2257:
2256:
2251:
2240:
2233:
2178:Category theory
2168:Algebraic logic
2151:
2122:Lambda calculus
2060:Church encoding
2046:
2022:Truth predicate
1878:
1844:Complete theory
1767:
1636:
1632:
1628:
1623:
1615:
1335: and
1331:
1326:
1312:
1288:New Foundations
1256:axiom of choice
1239:
1201:Gödel numbering
1141: and
1133:
1037:
922:
872:
853:
802:Boolean algebra
788:
752:Equiconsistency
717:Classical logic
694:
675:Halting problem
663: and
639: and
627: and
626:
621:Theorems (
616:
533:
528:
497:
471:
466:
456:
454:
445:
444:
440:
431:
429:
420:
419:
412:
403:
401:
393:
392:
388:
381:
358:
357:
353:
336:
332:
325:
321:
317:
312:
295:
291:
287:
282:
228:
220:
201:Urysohn's lemma
196:Sperner's lemma
191:Schwarz's lemma
103:
87:
75:
35:
28:
23:
22:
15:
12:
11:
5:
2286:
2284:
2276:
2275:
2270:
2260:
2259:
2253:
2252:
2238:
2235:
2234:
2232:
2231:
2226:
2221:
2216:
2211:
2210:
2209:
2199:
2194:
2189:
2180:
2175:
2170:
2165:
2163:Abstract logic
2159:
2157:
2153:
2152:
2150:
2149:
2144:
2142:Turing machine
2139:
2134:
2129:
2124:
2119:
2114:
2113:
2112:
2107:
2102:
2097:
2092:
2082:
2080:Computable set
2077:
2072:
2067:
2062:
2056:
2054:
2048:
2047:
2045:
2044:
2039:
2034:
2029:
2024:
2019:
2014:
2009:
2008:
2007:
2002:
1997:
1987:
1982:
1977:
1975:Satisfiability
1972:
1967:
1962:
1961:
1960:
1950:
1949:
1948:
1938:
1937:
1936:
1931:
1926:
1921:
1916:
1906:
1905:
1904:
1899:
1892:Interpretation
1888:
1886:
1880:
1879:
1877:
1876:
1871:
1866:
1861:
1856:
1846:
1841:
1840:
1839:
1838:
1837:
1827:
1822:
1812:
1807:
1802:
1797:
1792:
1787:
1781:
1779:
1773:
1772:
1769:
1768:
1766:
1765:
1757:
1756:
1755:
1754:
1749:
1748:
1747:
1742:
1737:
1717:
1716:
1715:
1713:minimal axioms
1710:
1699:
1698:
1697:
1686:
1685:
1684:
1679:
1674:
1669:
1664:
1659:
1646:
1644:
1625:
1624:
1622:
1621:
1620:
1619:
1607:
1602:
1601:
1600:
1595:
1590:
1585:
1575:
1570:
1565:
1560:
1559:
1558:
1553:
1543:
1542:
1541:
1536:
1531:
1526:
1516:
1511:
1510:
1509:
1504:
1499:
1489:
1488:
1487:
1482:
1477:
1472:
1467:
1462:
1452:
1447:
1442:
1437:
1436:
1435:
1430:
1425:
1420:
1410:
1405:
1403:Formation rule
1400:
1395:
1394:
1393:
1388:
1378:
1377:
1376:
1366:
1361:
1356:
1351:
1345:
1339:
1322:Formal systems
1318:
1317:
1314:
1313:
1311:
1310:
1305:
1300:
1295:
1290:
1285:
1280:
1275:
1270:
1265:
1264:
1263:
1258:
1247:
1245:
1241:
1240:
1238:
1237:
1236:
1235:
1225:
1220:
1219:
1218:
1211:Large cardinal
1208:
1203:
1198:
1193:
1188:
1174:
1173:
1172:
1167:
1162:
1147:
1145:
1135:
1134:
1132:
1131:
1130:
1129:
1124:
1119:
1109:
1104:
1099:
1094:
1089:
1084:
1079:
1074:
1069:
1064:
1059:
1054:
1048:
1046:
1039:
1038:
1036:
1035:
1034:
1033:
1028:
1023:
1018:
1013:
1008:
1000:
999:
998:
993:
983:
978:
976:Extensionality
973:
971:Ordinal number
968:
958:
953:
952:
951:
940:
934:
928:
927:
924:
923:
921:
920:
915:
910:
905:
900:
895:
890:
889:
888:
878:
877:
876:
863:
861:
855:
854:
852:
851:
850:
849:
844:
839:
829:
824:
819:
814:
809:
804:
798:
796:
790:
789:
787:
786:
781:
776:
771:
766:
761:
756:
755:
754:
744:
739:
734:
729:
724:
719:
713:
711:
702:
696:
695:
693:
692:
687:
682:
677:
672:
667:
655:Cantor's
653:
648:
643:
633:
631:
618:
617:
615:
614:
609:
604:
599:
594:
589:
584:
579:
574:
569:
564:
559:
554:
553:
552:
541:
539:
535:
534:
529:
527:
526:
519:
512:
504:
482:
481:
470:
469:External links
467:
465:
464:
438:
410:
386:
379:
351:
330:
318:
316:
313:
311:
310:
298:informal logic
288:
286:
283:
281:
280:
275:
270:
265:
260:
258:List of lemmas
255:
250:
245:
240:
235:
229:
227:
224:
219:
218:
213:
211:Yoneda's lemma
208:
203:
198:
193:
188:
183:
178:
173:
168:
163:
161:Jordan's lemma
158:
153:
148:
138:
133:
128:
126:Euclid's lemma
123:
118:
113:
111:BĂ©zout's lemma
107:
102:
99:
86:
83:
74:
71:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2285:
2274:
2271:
2269:
2266:
2265:
2263:
2250:
2249:
2244:
2236:
2230:
2227:
2225:
2222:
2220:
2217:
2215:
2212:
2208:
2205:
2204:
2203:
2200:
2198:
2195:
2193:
2190:
2188:
2184:
2181:
2179:
2176:
2174:
2171:
2169:
2166:
2164:
2161:
2160:
2158:
2154:
2148:
2145:
2143:
2140:
2138:
2137:Recursive set
2135:
2133:
2130:
2128:
2125:
2123:
2120:
2118:
2115:
2111:
2108:
2106:
2103:
2101:
2098:
2096:
2093:
2091:
2088:
2087:
2086:
2083:
2081:
2078:
2076:
2073:
2071:
2068:
2066:
2063:
2061:
2058:
2057:
2055:
2053:
2049:
2043:
2040:
2038:
2035:
2033:
2030:
2028:
2025:
2023:
2020:
2018:
2015:
2013:
2010:
2006:
2003:
2001:
1998:
1996:
1993:
1992:
1991:
1988:
1986:
1983:
1981:
1978:
1976:
1973:
1971:
1968:
1966:
1963:
1959:
1956:
1955:
1954:
1951:
1947:
1946:of arithmetic
1944:
1943:
1942:
1939:
1935:
1932:
1930:
1927:
1925:
1922:
1920:
1917:
1915:
1912:
1911:
1910:
1907:
1903:
1900:
1898:
1895:
1894:
1893:
1890:
1889:
1887:
1885:
1881:
1875:
1872:
1870:
1867:
1865:
1862:
1860:
1857:
1854:
1853:from ZFC
1850:
1847:
1845:
1842:
1836:
1833:
1832:
1831:
1828:
1826:
1823:
1821:
1818:
1817:
1816:
1813:
1811:
1808:
1806:
1803:
1801:
1798:
1796:
1793:
1791:
1788:
1786:
1783:
1782:
1780:
1778:
1774:
1764:
1763:
1759:
1758:
1753:
1752:non-Euclidean
1750:
1746:
1743:
1741:
1738:
1736:
1735:
1731:
1730:
1728:
1725:
1724:
1722:
1718:
1714:
1711:
1709:
1706:
1705:
1704:
1700:
1696:
1693:
1692:
1691:
1687:
1683:
1680:
1678:
1675:
1673:
1670:
1668:
1665:
1663:
1660:
1658:
1655:
1654:
1652:
1648:
1647:
1645:
1640:
1634:
1629:Example
1626:
1618:
1613:
1612:
1611:
1608:
1606:
1603:
1599:
1596:
1594:
1591:
1589:
1586:
1584:
1581:
1580:
1579:
1576:
1574:
1571:
1569:
1566:
1564:
1561:
1557:
1554:
1552:
1549:
1548:
1547:
1544:
1540:
1537:
1535:
1532:
1530:
1527:
1525:
1522:
1521:
1520:
1517:
1515:
1512:
1508:
1505:
1503:
1500:
1498:
1495:
1494:
1493:
1490:
1486:
1483:
1481:
1478:
1476:
1473:
1471:
1468:
1466:
1463:
1461:
1458:
1457:
1456:
1453:
1451:
1448:
1446:
1443:
1441:
1438:
1434:
1431:
1429:
1426:
1424:
1421:
1419:
1416:
1415:
1414:
1411:
1409:
1406:
1404:
1401:
1399:
1396:
1392:
1389:
1387:
1386:by definition
1384:
1383:
1382:
1379:
1375:
1372:
1371:
1370:
1367:
1365:
1362:
1360:
1357:
1355:
1352:
1350:
1347:
1346:
1343:
1340:
1338:
1334:
1329:
1323:
1319:
1309:
1306:
1304:
1301:
1299:
1296:
1294:
1291:
1289:
1286:
1284:
1281:
1279:
1276:
1274:
1273:KripkeâPlatek
1271:
1269:
1266:
1262:
1259:
1257:
1254:
1253:
1252:
1249:
1248:
1246:
1242:
1234:
1231:
1230:
1229:
1226:
1224:
1221:
1217:
1214:
1213:
1212:
1209:
1207:
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181:Riesz's lemma
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131:Farkas' lemma
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79:Ancient Greek
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56:
52:
44:
40:
33:
19:
18:Lemma (logic)
2239:
2037:Ultraproduct
1884:Model theory
1849:Independence
1785:Formal proof
1777:Proof theory
1760:
1733:
1690:real numbers
1662:second-order
1573:Substitution
1450:Metalanguage
1391:conservative
1364:Axiom schema
1308:Constructive
1278:MorseâKelley
1244:Set theories
1223:Aleph number
1216:inaccessible
1122:Grothendieck
1006:intersection
893:Higher-order
881:Second-order
827:Truth tables
784:Venn diagram
586:
567:Formal proof
496:
484:
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455:. Retrieved
450:
441:
430:. Retrieved
426:
402:. Retrieved
398:
389:
361:
354:
338:
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322:
292:
221:
216:Zorn's lemma
121:Dehn's lemma
104:
88:
76:
54:
50:
42:
36:
2147:Type theory
2095:undecidable
2027:Truth value
1914:equivalence
1593:non-logical
1206:Enumeration
1196:Isomorphism
1143:cardinality
1127:Von Neumann
1092:Ultrafilter
1057:Uncountable
991:equivalence
908:Quantifiers
898:Fixed-point
867:First-order
747:Consistency
732:Proposition
709:Traditional
680:Lindström's
670:Compactness
612:Type theory
557:Cardinality
451:www.oed.com
369:. pp.
156:ItĂŽ's lemma
59:proposition
39:mathematics
2262:Categories
1958:elementary
1651:arithmetic
1519:Quantifier
1497:functional
1369:Expression
1087:Transitive
1031:identities
1016:complement
949:hereditary
932:Set theory
487:PlanetMath
432:2019-11-28
404:2019-11-28
315:References
306:philosophy
243:Co-premise
2229:Supertask
2132:Recursion
2090:decidable
1924:saturated
1902:of models
1825:deductive
1820:axiomatic
1740:Hilbert's
1727:Euclidean
1708:canonical
1631:axiomatic
1563:Signature
1492:Predicate
1381:Extension
1303:Ackermann
1228:Operation
1107:Universal
1097:Recursive
1072:Singleton
1067:Inhabited
1052:Countable
1042:Types of
1026:power set
996:partition
913:Predicate
859:Predicate
774:Syllogism
764:Soundness
737:Inference
727:Tautology
629:paradoxes
263:Objection
238:Corollary
77:From the
73:Etymology
2214:Logicism
2207:timeline
2183:Concrete
2042:Validity
2012:T-schema
2005:Kripke's
2000:Tarski's
1995:semantic
1985:Strength
1934:submodel
1929:spectrum
1897:function
1745:Tarski's
1734:Elements
1721:geometry
1677:Robinson
1598:variable
1583:function
1556:spectrum
1546:Sentence
1502:variable
1445:Language
1398:Relation
1359:Automata
1349:Alphabet
1333:language
1187:-jection
1165:codomain
1151:Function
1112:Universe
1082:Infinite
986:Relation
769:Validity
759:Argument
657:theorem,
457:26 April
296:Such as
226:See also
2156:Related
1953:Diagram
1851: (
1830:Hilbert
1815:Systems
1810:Theorem
1688:of the
1633:systems
1413:Formula
1408:Grammar
1324: (
1268:General
981:Forcing
966:Element
886:Monadic
661:paradox
602:Theorem
538:General
273:Theorem
91:theorem
63:theorem
55:lemmata
2273:Lemmas
1919:finite
1682:Skolem
1635:
1610:Theory
1578:Symbol
1568:String
1551:atomic
1428:ground
1423:closed
1418:atomic
1374:ground
1337:syntax
1233:binary
1160:domain
1077:Finite
842:finite
700:Logics
659:
607:Theory
377:
345:
304:, and
268:Porism
51:lemmas
1909:Model
1657:Peano
1514:Proof
1354:Arity
1283:Naive
1170:image
1102:Fuzzy
1062:Empty
1011:union
956:Class
597:Model
587:Lemma
545:Axiom
349:p. 47
285:Notes
233:Axiom
67:prove
43:lemma
2032:Type
1835:list
1639:list
1616:list
1605:Term
1539:rank
1433:open
1327:list
1139:Maps
1044:sets
903:Free
873:list
623:list
550:list
459:2023
375:ISBN
343:ISBN
1719:of
1701:of
1649:of
1181:Sur
1155:Map
962:Ur-
944:Set
53:or
47:pl.
37:In
2264::
2105:NP
1729::
1723::
1653::
1330:),
1185:Bi
1177:In
477:,
449:.
425:.
413:^
397:.
373:.
371:16
365:.
300:,
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2185:/
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