Georg Cantor: ukucwangcisa theory, biography kanye izibalo usapho

Georg Cantor (photo ibonisa kamva kweli nqaku) - yezibalo German abasungula le ngcamango iiseti waza waqalisa ingqikelelo yamanani transfinite, ngamabona ezinkulu, kodwa zahlukile enye kwenye. Kwakhona wanikela inkcazelo amanani olandelelwano ngokusaphuhliswayo kunye iimpawu wamisa arithmetic yabo.

Georg Cantor: ibhayografi emfutshane

Wazalelwa eSt Petersburg 03.03.1845. Uyise waba Danish ongumProtestanti Georg Waldemar Cantor, wayexakeke kurhwebo, xa Vol. H. Ke istock exchange. Unina, uMariya, Bem BendingumKatolika beza kwimizalwane zeemvumi abadumileyo. Xa ngo1856 uyise uGeorge wagula, intsapho ukhangela imozulu sisiba ngumntu bafudukela kuqala eWiesbaden ke Frankfurt. italente lweMathematika, umfundi uvele phambi kwakhe wokuzalwa 15 ngelixa befunda kwizikolo zabucala nakwizikolo zikarhulumente Darmstadt kunye Wiesbaden. Ekugqibeleni, Georg Cantor bacenga uyise ukuzimisela wakhe ukuba abe yezibalo kunokuba wenjineli.

Emva uqeqesho ngokufutshane kwiYunivesithi yase Zurich ngo 1863. Cantor sakhutshelwa University Berlin ukufunda physics, intanda nakwimathematika. Apho ke wafundisa:

• UKarl Theodor Weierstrass, izifundo zabo kuzo phaya kuhlalutyo, kusenokwenzeka ukuba nempembelelo gqitha George;
• Ernst Kummer, owafundisa izibalo kophakamileyo;
• Leopold Kronecker, kwi nombolo ithiyori specialist, owathi kamva abaphikisa Cantor.

Akuba echithe nonyaka omnye kwi University of Göttingen ngo-1866, kunyaka ozayo George wabhala ithisisi zobugqirha yakhe phantsi kwesihloko "Xa imathematika, ubuchule bokubuza imibuzo lubaluleke ngaphezu zokusombulula iingxaki" ngokuphathelele ingxaki Carl Friedrich GAUSS kungalungiswa wakhe Disquisitiones Arithmeticae (1801) . Emva efundisa kancinane kwi-Berlin School for girls Kantor waqalisa ukusebenza kwiYunivesithi Halle, apho wahlala de kube sekupheleni kobomi bakhe, kuqala ibe ngumhlohli, ukususela 1872 njengoko uprofesa oncedisayo, yaye ukususela ngowe-1879 ngowokuqala njenge ngunjingalwazi.

uphando

Ekuqaleni uthotho imisebenzi-10 ukususela 1869 ukuya 1873, Georg Cantor ingqalelo imfundiso amanani. Lo msebenzi ubonisa angathabatheki ukuze umbandela isifundo kunye nefuthe GAUSS Kronecker. Ecetyiswa Heinrich Eduard Heine, oogxa Cantor kaThixo Halle, yaqaphela italente yakhe zemathematika, yena wajika kwithiyori kuthotho netrigonometri, leyo zandiswa ingqiqo yamanani ngokwenene.

Ngokusekelwe umsebenzi umsebenzi variable oluntsonkothileyo sezibalo German Bernhard Riemann ngo-1854, ngo-1870 Cantor lubonise ukuba loo msebenzi angamelwa ngendlela enye kuphela - ngu series netrigonometri. Ukuqwalaselwa yeseti amanani (lamanqaku), nto leyo ukuziphikisa lo mbono, bemka naye, kwindawo yokuqala, ngowe-1872, ukuya inkcazelo yamanani ezingenangqondo ngokwemigaqo ulandelelwano convergent amanani olwamkelweyo (amaqhezu integers) aze emva koko ekuqaleni lomsebenzi impilo yakhe, theory iseti kunye ingqiqo yamanani transfinite.

theory isethi

Georg Cantor, ithiyori leyo kucwangcisa zazivela imbalelwano ne-Technical Institute of Braunschweig kwezezibalo uRichard Dedekind, waba abahlobo kunye naye ukususela ebuntwaneni. Bagqiba ukuba iiseti, anesiphelo okanye ongenasiphelo, i nobuninzi lweziqalelo (umz, amanani {0, ± 1, ± 2 ...}) Yiyiphi na indawo ethile, lo gama kugcinwe ubuntu babo. Kodwa xa Georg Cantor isicelo ukufunda iimpawu zazo imbalelwano nye (umzekelo, {A, B, C} ukuba {1, 2, 3}), wakhawuleza waqonda ukuba zahluke amaqondo ukubandakanya, nokuba kube iiseti ongapheliyo , t. e. piece iseti okanye isethi leyo equka inani elifanayo lezinto njengoko ngokwaso. kungekudala indlela yakhe iziphumo emangalisayo.

Ngowe-1873, Georg Cantor (yezibalo) lwabonisa ukuba amanani obengo, nangona ongenasiphelo, ezi axela, ngenxa yokuba kubekwa-one-to-one imbalelwano kunye zendalo (okt. E. 1, 2, 3 ,. D.). Wabonisa ukuba iseti yamanani yokwenene kuqulethe engenasiphelo esinengqiqo ezingenangqondo, ezibalwayo nezingabalwayo. Yintoni imangalisayo, Cantor wabonisa ukuba isethi yonke amanani aljibra iqulethe izinto ezininzi njengoko isethi yonke integers, nokuba amanani ngokudlul izinto ezo aljibra, zona elucwangciso olusezantsi lwe amanani ezingenangqondo yi ezingabalekiyo kwaye kungoko inani labo mkhulu ngaphezu integers kwaye kufuneka luthathwe ongapheliyo.

Abaphikisi kunye nabaxhasi

Kodwa lo msebenzi Cantor, apho kuqala ukubeka phambili iziphumo, akazange yapapashwa "Krell" iphephancwadi njengomnye kubahleli, Kronecker wayechasa. Kodwa emva kokuba ungenelelo lwe Dedekind yanyatheliswa 1874 phantsi kwesihloko esithi "iimpawu onke amanani yokwenene algebra."

Science and impilo yakho

Kulo nyaka, ngexesha honeymoon kunye nomfazi wakhe, Valli Gutman e Interlaken, Switzerland, Cantor wadibana Dedekind ngubani ngobubele wathi kwi imfundiso yakhe entsha. umvuzo George yayincinane, kodwa imali nguyise wakhe, yena wafa ngowe-1863, ukuba wakha umfazi nabantwana bakhe abahlanu ekhaya. Uninzi imisebenzi yakhe ziye zapapashwa eSweden kulindixesha elitsha ACTA Mathematica, umhleli umsunguli leyo Gösta Mittag-Leffler, phakathi kwabantu bokuqala ukuqonda italente sezibalo waseJamani.

Unxibelelwano kunye ezikhona

Theory Cantor waba isifundo esitsha ngokupheleleyo zophando enxulumene izibalo engenasiphelo (umz, ukulandelelana 1, 2, 3 ,. D., ewumise ezintsonkothileyo), nto leyo ixhomekeke kakhulu-one-to-one imbalelwano. Cantor Development of iindlela ezintsha kusetwa imibuzo ngokuphathelele inkqubela Infinity wandiboleka izifundo zakhe entle.

Xa waqiqa ukuba zikho ngokwenene amanani ongapheliyo, Wajika wakhangela bulumko yamandulo aphakathi ngokubhekiselele Infinity kanye ezinokubakho, ngokunjalo kwimfundo zonqulo yokuqala, apho abazali wamnika. Ngowe-1883, kwincwadi yakhe "Iziseko thiyori jikelele iiseti" Kantor zidityanisiwe ingqiqo yakhe ezikhona kaPlato.

Kronecker nabo, yena wathi "kukho" integers kuphela ( "UThixo wadala integers, abanye - umsebenzi womntu"), iminyaka emininzi wenqaba ngamandla iingxoxo zakhe kwaye kwathintela kokuqeshwa kwakhe ukuya University of Berlin.

amanani transfinite

Ngowe-1895-97 gg. Georg Cantor wabumba ngokupheleleyo ingcamango yakhe yokuqhubekeka Infinity, kuquka ulandelelwano kunye namba engaphele ndawo, umsebenzi wakhe idumileyo, epapashwe phantsi kwesihloko "Igalelo kwithiyori amanani transfinite" (1915). Lo msebenzi ubandakanya isisu, apho wayiqhubela umboniso ukuba iseti ongapheliyo zingaziswa kwi-one-to-one imbalelwano nomnye subsets yayo.

Elona nani lincinane iimpawu transfinite wayethetha yombane yaso nasiphi na iseti, leyo kubekwa-one-to-one imbalelwano kunye amanani zendalo. Kantor wachaza aleph-zero yakhe. Ukuba kwisininzi ezinkulu transfinite Alef-ezinyuliweyo enye, ezimbini okanye Aleph-t. D. ngakumbi ordinals arithmetic, nto leyo efana izibalo ezisikelwe. Ngenxa yoko, yena alusincedanga ingqiqo yokungapheli.

Inkcaso nazo, yaye nexesha olisebenzisileyo ukuqinisekisa ukuba izimvo zakhe elamkeleke ngokupheleleyo, wachaza yobunzima nokuvavanywa ngokutsha umbuzo yamandulo yintoni na inani. Kantor wabonisa ukuba isethi amanqaku kumgca ngaba inobunzima ngaphezulu Aleph-zero. Oku kwakhokelela ekubeni le ngxaki ezaziwayo-kakuhle ezingenamandla ze hypothesis okuqhubeka - akukho Iikhadinali phakathi aleph-zero kwaye akukho manqaku namandla emgceni. Le ngxaki kwisiqingatha sokuqala nesesibini kwinkulungwane yama-20 umdla omkhulu yaye sele zihlolwa zezibalo baninzi, Vol. H. Kurt Gödel kunye Paul Cohen.

ukudangala

Biography Georga Kantora ukususela 1884 lwaba yokugula kwakhe incipient yengqondo, kodwa waqhubeka isebenze. Ngowe-1897 wanceda ukubamba International Congress yokuqala Mathematicians e Zurich. Okunye kuba wamelana i Kronecker, wayesoloko sympathized kunye zezibalo abaselula thomo kwaye bazama ukufumana indlela eya kubasindisa nabo abahlukunyezwa ngootitshala uyoyikiswa iingcamango ezintsha.

ukuqaphela

Ekuveleni kwenkulungwane umsebenzi wakhe awubona ngokupheleleyo njengesiseko kwithiyori imisebenzi, uhlalutyo kunye yethopholoji efanayo. Ukongezelela, incwadi Kantora Georga waba negalelo kuphuhliso olungaphezulu yesikolo formalist kunye intuitionist kweziseko ubhalo lemathematika. Oku kutshintshile kakhulu ukuba inkqubo yokufundisa kunye unxulunyaniswa i "izibalo omtsha."

Ngowe-1911, Cantor wayephakathi abo bamenyiweyo ekubhiyozeleni isikhumbuzo 500th le University of St Andrews eScotland. Waya apho befuna ukudibana Bertrand Russell, owathi umsebenzi wakhe ezipapashwe kutshanje Principia Mathematica wabiza ukuya yezibalo waseJamani, kodwa loo nto ayizange yenzeke. University inikezelwe Cantor ngomlinganiselo yekhethelo, kodwa ngenxa yokugula kwakhe ayikwazanga ukwamkela iimbasa umntu.

Cantor umhlala ngo-1913 yaye ehlala ubuhlwempu yindlala ngethuba leMfazwe yokuQala yeHlabathi. Imibhiyozo imbeko wakhe wokuzalwa-70 ngo-1915 iye yarhoxiswa ngenxa yemfazwe, kodwa yayibambe umsitho encinane ekhayeni lakhe. Yena wafa 06.01.1918, e Galle, kwisibhedlele sabagula ngengqondo, apho wachitha iminyaka yokugqibela yobomi bakhe.

Georg Cantor: A Biography. usapho

August 9, 1874, sezibalo waseJamani watshata Valli Gutman. Esi sibini wayenoonyana 4 neentombi-2. Umntwana yokugqibela wazalwa ngo-1886 e-Cantor wathenga ikhaya elitsha. Xhasa usapho wanceda legacy kayise. Impilo Cantor bachaphazeleka kakhulu kukufa, unyana wakhe omncinane ngo-1899 - kuba abazange zokudakumba.