Georg Cantor: teev kev tshawb xav, biography thiab tsev neeg kev ua lej

Georg Cantor (yees duab qhia tau hais tias tom qab nyob rau hauv tsab xov xwm) - German mathematician uas tsim cov kev tshawb xav ntawm poob lawm thiab nkag tau lub tswvyim ntawm transfinite xov tooj, infinitely loj, tab sis txawv los ntawm txhua lwm yam. Nws kuj tau muab ib tug txhais ntawm ordinal thiab cardinal xov tooj thiab tsim lawv cov kev laij zauv.

Georg Cantor: ib tug luv luv biography

Yug nyob rau hauv St. Petersburg 03.03.1845. Nws txiv yog ib tug Danish Protestant Georg Waldemar Cantor, twb koom nyob rau hauv cov lag luam tawm, nyob rau hauv Vol. H. Thiab nyob rau hauv Tshuag txauv. Nws niam, Mary, Bem yog ib tug Catholic thiab tuaj ntawm ib tsev neeg ntawm muaj koob musicians. Thaum nyob rau hauv 1856 nws txiv George los ua mob, tsev neeg nyob rau hauv kev tshawb fawb ntawm ib tug yam kev nyab xeeb tsiv thawj Wiesbaden ces mus rau Frankfurt. Lej txuj ci, tus tub los tshwm ua ntej nws 15th lub hnub yug thaum kawm nyob rau hauv tsev kawm ntawv ntiav thiab pej xeem cov tsev kawm ntawv nyob rau hauv Darmstadt thiab Wiesbaden. Nyob rau hauv lub kawg, Georg Cantor ntxias kom nws txiv nyob rau hauv nws cov kev txiav txim mus ua ib tug mathematician es ib tug engineer.

Tom qab ib tug luv luv kev kawm nyob rau University of Zurich nyob rau hauv 1863. Cantor twb pauv mus rau Berlin University kawm physics, philosophy thiab kev kawm zauv. Muaj nws twb qhia hais tias:

• Karl Theodor Weierstrass, uas nws specialization nyob rau hauv lub tsom xam, tej zaum muaj tus loj tshaj cawv rau George;
• Ernst Kummer, uas tau qhia rau ntau tshaj xam;
• Leopold Kronecker, nyob rau kev tshawb xav tooj kws, uas tom qab txwv Cantor.

Thaum siv ib tug semester ntawm lub University of Göttingen nyob rau hauv 1866, lub xyoo tom ntej George sau nws doctoral thesis nyob rau hauv lub title "Nyob rau hauv kev kawm txog zauv, kos duab rau ntawm cov lus nug yog ntau nqi tshaj daws teeb meem" hais txog qhov teeb meem uas Carl Friedrich Gauss tshuav teeb meem nyob rau hauv nws Disquisitiones Arithmeticae (1801) . Tom qab sib tham me ntsis qhia ntawv rau hauv lub Berlin Tsev Kawm Ntawv rau cov ntxhais Kantor pib ua hauj lwm nyob rau University of Halle, nyob qhov twg nws tseem mus txog rau thaum kawg ntawm nws lub neej, thawj raws li ib tug lecturer, txij li thaum 1872 raws li ib qho kev pab xib fwb, thiab txij li thaum xyoo 1879, Cov thawj raws li ib tug xib fwb.

kev tshawb fawb

Thaum pib ntawm ib tug series ntawm 10 tej hauj lwm los ntawm 1869 mus 1873, Georg Cantor xam tau tias yog cov kev tshawb xav ntawm cov zauv. Lub chaw ua hauj lwm qhia tus mob siab rau rau cov kev kawm ntawm nws txoj kev kawm thiab cov nyhuv ntawm Gauss Kronecker. Thaum lub suggestion ntawm Heinrich Eduard Heine, Cantor tus lug txhawb cov miv ntawm Halle, uas pom tau hais tias nws xaam txuj ci, nws tig mus rau lub hom phiaj ntawm trigonometric series, uas txhab lub tswvyim ntawm cov zauv tiag tiag.

Raws li cov kev ua hauj lwm muaj nuj nqi ntawm ib tug complex nce mus nce los ntawm lub German mathematician Bernhard Riemann nyob rau hauv 1854, nyob rau hauv 1870 Cantor lawm hais tias xws li ib tug muaj nuj nqi yuav tsum muaj tus sawv cev nyob rau hauv tsuas yog ib qho txoj kev - los ntawm trigonometric series. Kev saib xyuas ntawm lub teeb ntawm tus xov tooj (cov ntsiab lus), uas yuav tsis nqe no saib, coj nws, nyob rau hauv thawj qhov chaw, nyob rau hauv 1872, rau lub ntsiab txhais ntawm irrational xov tooj nyob rau hauv cov nqe lus ntawm convergent sequences ntawm rational numbers (fractions zauv) thiab ces mus rau thaum pib ntawm kev ua hauj lwm nyob rau hauv nws lub neej txoj kev ua hauj lwm, txheej kev tshawb xav thiab lub tswvyim ntawm transfinite xov tooj.

txheej kev tshawb xav

Georg Cantor, lub hom phiaj uas teev originated nyob rau hauv xov xwm nrog rau Kev koom haum ntawm Braunschweig mathematician Richard Dedekind, yog cov phooj ywg nrog nws txij thaum yau. Lawv xaus lus tias tus poob lawm, finite los yog infinite, yog ib tug plurality hais (e.g., tus xov tooj {0, ± 1, ± 2 ...}) Uas muaj ib yam khoom, thaum ceev lawv tus kheej. Tab sis thaum Georg Cantor thov mus kawm lawv tus yam ntxwv ib tug xov xwm (piv txwv li, {A, B, C} mus {1, 2, 3}), nws sai sai pom tau hais tias lawv txawv nyob rau hauv lawv cov degree ntawm koom, txawm yog hais tias nws twb infinite poob lawm , t. e. set daim los yog ib tug subset ntawm uas muaj xws li cov tib tooj ntawm cov khoom raws li nws yog nws tus kheej. Nws txoj kev sai muab amazing tau.

Nyob rau hauv 1873, Georg Cantor (mathematician) pom tias rational numbers, txawm hais tias infinite, yog suav, vim hais tias lawv yuav tsum tau muab tso rau hauv ib tug-rau-ib tug xov xwm nrog ntuj (piv txwv li. E. 1, 2, 3 ,. D.). Nws pom hais tias lub teeb ntawm cov zauv tiag tiag muaj ib tug muaj kuab thiab irrational, thiab uncountable infinite. Yuav ua li cas ib tug paradox, Cantor muaj pov thawj hais tias cov teeb ntawm tag nrho algebraic xov tooj muaj li ntau yam ntsiab raws li tus txheej ntawm tag nrho cov zauv, thiab hais tias transcendental xov tooj uas yog tsis algebraic, uas yog ib tug subset ntawm irrational xov tooj yog uncountable thiab chaw pib lawv tus xov tooj yog ntau dua qhov zauv thiab yuav tsum raug xam tias yog li mus ib txhis.

Tw thiab los ntawm UFW

Tab sis txoj hauj lwm Cantor, nyob rau hauv uas nws ntej muab tso rau pem hauv ntej qhov kev tshwm sim, tsis yog luam tawm nyob rau hauv "Krell" magazine raws li ib tug ntawm cov reviewers, Kronecker twb txwv. Tab sis tom qab cov kev pab ntawm lub Dedekind nws twb luam tawm nyob rau hauv 1874 nyob rau hauv lub title "Tus yam ntxwv ntawm tag nrho cov tiag tiag algebraic xov tooj."

Science thiab koj tus kheej lub neej

Nyob rau hauv tib lub xyoo, thaum lub sij hawm lub honeymoon nrog nws tus poj niam, Valli Gutman nyob rau hauv Interlaken, Switzerland, Cantor tau ntsib Dedekind uas zoo commented rau nws cov tshiab kev tshawb xav. George nyiaj hli yog me me, tab sis nrog cov nyiaj rau nws tus txiv, leej twg tuag nyob rau hauv 1863, nws tau los ua rau nws tus poj niam thiab cov me nyuam tsib lub tsev. Muaj ntau ntawm nws tej hauj lwm tau luam tawm nyob rau hauv Sweden nyob rau hauv lub tshiab phau ntawv journal feeb Mathematica, lub editor thiab tus tsim ntawm uas yog Gosta Mittag-Leffler, cov thawj kom paub txog cov txuj ci ntawm cov German mathematician.

Kev sib txuas lus nrog rau cov metaphysics

Ziag Cantor yog kiag li tshiab kev kawm ntawm kev tshawb fawb txog kev ua lej infinite (e.g., ib theem zuj zus 1, 2, 3 ,. D., Thiab ntau poob lawm), uas yog kev lom zem ntau nyob rau ntawm ib-rau-ib tug xov xwm. Singer kev loj hlob ntawm txoj kev tshiab ntawm qhov chaw kawm lus nug txog cov rooj saab laaj thiab infinity rau thiaj tau qiv nws kev tshawb fawb xyaw.

Thaum nws sib cav hais tias infinite xov tooj tiag tiag tshwm sim, nws tig mus rau lub ancient thiab medieval philosophy xav txog tiag tiag thiab tej zaum infinity, raws li zoo raws li thaum ntxov kev cai dab qhuas kev kawm ntawv, uas cov niam txiv muab rau nws. Nyob rau hauv 1883, nyob rau hauv nws phau ntawv "Fundamentals ntawm lub dav kev tshawb xav ntawm poob lawm" Kantor ua ke nws tswvyim ntawm tus metaphysics ntawm Plato.

Kronecker kuj, uas khav tias "muaj" tsuas zauv ( "Vajtswv tsim lub zauv, tus so - cov ua hauj lwm ntawm tus txiv neej"), rau ntau xyoo xav tseg nws cov lus thiab tiv thaiv nws lub sij hawm mus rau lub tsev kawm ntawv ntawm Berlin.

transfinite xov tooj

Nyob rau hauv 1895-97 gg. Georg Cantor siab tsim nws lub tswv yim ntawm cov rooj saab laaj thiab infinity, xws li ib qho kev txom theem zuj zus thiab cardinal xov tooj, nyob rau hauv nws nto moo tshaj plaws ua hauj lwm, luam tawm nyob rau hauv lub title "Tej Hauj rau lub hom phiaj ntawm transfinite xov tooj" (1915). Qhov no ua hauj lwm nrog nws conception, uas nws coj ib tug ua qauv hais tias ib tug infinite set peev xwm ua rau nyob rau hauv ib tug ib tug-rau-ib tug xov xwm nrog ib tug ntawm nws cov subsets.

Tus me tshaj transfinite cardinal tooj nws meant lub hwj chim ntawm tej teeb, uas yuav tsum tau muab tso rau hauv ib tug-rau-ib tug xov xwm nrog lub ntuj tus xov tooj. Kantor piav nws aleph-zero. Loj transfinite plurality Alef-xws ib tug, ob tug los yog Aleph-t. D. Nws tsim ntxiv xam ordinals, uas yog zoo xws li cov finite xam. Yog li, nws tau enriched lub tswvyim ntawm infinity.

Lub txheem nws ntsib, thiab lub sij hawm nws coj mus xyuas kom meej tias nws lub tswv yim twb tag nrho txais, piav lub complexities ntawm lub revaluation ntawm cov ancient lo lus nug ntawm yog dab tsi yog tus naj npawb. Kantor pom tias ib co ntawm cov ntsiab lus ntawm cov kab muaj ib tug ntau dua muaj peev xwm tshaj Aleph-zero. Qhov no coj mus rau lub zoo-paub cov teeb meem ntawm cov pab hypothesis - tsis muaj cardinals ntawm aleph-xoom thiab tsis muaj hwj chim cov ntsiab lus nyob rau hauv txoj kab. Qhov teeb meem no nyob rau hauv thawj thiab thib ob ib nrab ntawm lub xyoo pua 20th yog ntawm kuj zoo kawg txaus siab thiab tau kawm los ntawm ntau mathematicians, nyob rau hauv Vol. H. Kurt Gödel thiab Paul Cohen.

kev nyuaj siab

Biography Georga Kantora 1884 twb txom nyem los ntawm nws incipient kev puas hlwb, tiam sis nws tseem mus ua hauj lwm nquag. Nyob rau hauv 1897 nws tau pab los tuav lub thawj International Congress of thom nug nyob rau hauv Zurich. Tej vim hais tias nws txwv lub Kronecker, nws feem ntau sympathized nrog tus tub budding mathematicians thiab sim mus nrhiav ib txoj kev los cawm lawv ntawm kev ua phem los ntawm cov xib fwb uas xav tias raug teeb meem los ntawm tswv yim tshiab.

paub

Thaum lub lem ntawm lub xyoo pua nws ua hauj lwm tau siab lees paub tias raws li ib lub hauv paus rau lub hom phiaj ntawm kev khiav dej num, tsom xam thiab topology. Nyob rau hauv tas li ntawd, Kantora Georga phau ntawv tau txais kev pab raws li ib tug impetus rau ntxiv txoj kev loj hlob ntawm lub formalist thiab intuitionist tsev kawm ntawv ntawm cov zajlus kom cov me nyuam yaus ntawm kev kawm txog zauv. Qhov no tau ho hloov lub system ntawm kev qhia thiab kev yog feem ntau txuam nrog cov "tshiab lej."

Nyob rau hauv 1911, Cantor yog cov caw tuaj rau lub koob tsheej ntawm lub 500th hnub tseem ceeb ntawm lub University of St Andrews nyob rau hauv Scotland. Nws mus muaj hoping kom tau raws li Bertrand Russell, uas nyob rau hauv nws nyuam qhuav luam tawm ua hauj lwm Principia Mathematica pheej xa mus rau lub German mathematician, tab sis hais tias tsis tshwm sim. University muab Cantor ib tug mus saib xyuas degree, tab sis vim kuv muaj mob nws yog tsis mus txais qhov puav pheej nyob rau hauv tus neeg.

Singer so nyob rau hauv 1913 thiab nyob rau hauv cov neeg txom nyem thiab tshaib thaum lub sij hawm thawj lub ntiaj teb ua tsov ua rog. Celebrations nyob rau hauv Honor ntawm nws 70th xyoos nyob rau hauv 1915 tau tso tseg vim hais tias ua tsov ua rog, tab sis ib tug me me kab li kev cai nyob rau hauv nws lub tsev. Nws tuag rau 06.01.1918, nyob rau hauv Galle, nyob rau hauv ib tug puas siab ntsws hauv tsev kho mob, qhov chaw uas nws tau siv lub xeem xyoo ntawm nws lub neej.

Georg Cantor: A Biography. tsev neeg

Lub yim hli ntuj 9, 1874, lub German mathematician sib yuav Valli Gutman. Qhov ob peb muaj 4 tug tub thiab 2 tug ntxhais. Tus me nyuam kawg yug nyob rau hauv 1886 nyob rau hauv Cantor muas ib lub tsev tshiab. Txhawb lub tsev neeg nws pab tau nws tus txiv lub legacy. Cov kev kho mob ntawm Cantor heev cuam tshuam rau txoj kev tuag ntawm nws tus tub nyob rau hauv 1899 - vim nws yeej tsis tshuav lub kev nyuaj siab.