Indida zikaRussell: ulwazi oluyisisekelo, izibonelo, ekwakhiweni

Russell indida kuyinto ezimbili kuncikene antinomy okunengqondo.

amafomu amabili indida zikaRussell

Ifomu njalo okuxoxwe kakhulu kuwukuziphikisa ubuhlakani amasethi. Ezinye isethi ibonakala amalungu ngokwabo, nabanye - akukho. Isethi wonke amasethi ngokwaso iqoqo, ngakho-ke kubonakala sengathi libhekisela uqobo. Nalutho Nokho, akufanele kube ilungu uqobo. Ngakho-ke, isethi wonke amasethi, njengoba kuqanda Ayibalwa ku uqobo. kuphakama Indida lapho umbuzo ukuthi isethi ilungu uqobo. Lokhu kungenzeka kuphela uma akusiyo.

Kokunye futhi okuyindida ifomu kuphambene mayelana izakhiwo. Ezinye izakhiwo, kubonakala bazibiza, kanti abanye. Impahla ukuba impahla kukodwa impahla, ngenkathi impahla kungaba ikati akusona. Cabanga impahla kokuba impahla ukuthi ingeyona kuye. uma kusebenza ngokwayo? Nalapha futhi, noma yiziphi ezizathwini kufanele kube okuphambene. Indida owayemuhle wayebizwa ngokuthi behlonipha uBertrand Russell (1872-1970), abasithola ngo-1901.

indaba

Ukuvulwa Russell iphutha ngenkathi umsebenzi wayo "Izimiso kweembalo". Nakuba wathola indida ngokuzimela, kunobufakazi bokuthi ezinye zezibalo kanye Onjiniyela of isethi theory, kuhlanganise u-Ernst Zermelo futhi uDavide Hilbert, babelazi inguqulo yokuqala eziphikisanayo phambi kwakhe. Nokho, uRussell, waba ngowokuqala ngubani idingidwa indida emisebenzini yakhe eshicilelwe, kuqala wazama izixazululo futhi owokuqala bekuqonda ngokugcwele ukubaluleka kwalo. A lonke isahluko "Izimiso" yayizimisele ingxoxo yawo lo magazini, futhi isicelo enikelwe imfundiso yokuziphendukela izinhlobo, okuyinto Russell ezihlongozwayo njengoba isixazululo.

Russell wathola "Okufike kudide nge-ongumqambimanga, acabangela Cantor sika isethi inkolelo yokuthi uthi amandla iyiphi isethi lincane iqoqo subsets yayo. Okungenani kusizinda kufanele kube subsets amaningi njengoba kukhona izakhi kuwo, uma omunye isethi engezansi element ngayinye isethwe equkethe kuphela lesi sici. Ngaphezu kwalokho, Cantor wafakazela ukuthi inani izakhi hhayi kungaba ulingana inani subsets. Uma kwakukhona inani elifanayo, kwakuyodingeka ube khona ƒ isici Uyobonisa izakhi ku subsets yabo. Ngesikhathi esifanayo yini obubonisa ukuthi lokhu akunakwenzeka. Ezinye izinto zingaboniswa subsets umsebenzi ƒ eziqukethe kubo, kuyilapho abanye akufanele.

Cabanga isethi engezansi izakhi okungezona ezakho izithombe zabo, lapho abakubonisa ƒ. Kuyinto ngokwayo isethi engezansi izakhi, futhi ngenxa yalokho, ƒ umsebenzi ngabe silubonisa isici kusizinda. Inkinga wukuthi khona-ke kuphakama umbuzo ngokuqondene nokuthi lesi sici kungokwalabo isethi engezansi kuwo libonisa ƒ. Lokhu kungenzeka kuphela uma ingeyona. indida zikaRussell kungabhekwa njengokuvikela isibonelo line efanayo yokucabanga, lula kuphela. Ngaphezu kwalokho - lishona noma subsets isethi? Kunengqondo ukuphetha ngokuthi kufanele kube khona amasethi ngaphezulu, njengoba bonke subsets we amasethi ngokwabo. Kodwa uma theorem Cantor kuyiqiniso, khona-ke kufanele kube khona subsets ngaphezulu. Russell ithathwa nje ukubonisa lishona ngokwabo futhi kusetshenziswe indlela kantoriansky ucabangela isethi zonke lezi zinto, ngaphandle isethi lapho zibekwa. Ibonisa Russell iba iqoqo wonke amaqoqo, a non.

iphutha Frege

"Okufike kudide nge-ongumqambimanga" yaba negalelo elikhulu ekuthuthukiseni zomlando inkolelo-mbono yokuziphendukela amasethi. Wabonisa ukuthi umqondo we isethi jikelele kuyinkinga kakhulu. Wabuye wangabaza umqondo wokuthi ngamunye isimo kuchazwe noma isilandiso kuthathwa ukuba khona sebuningini kuphela labo izinto ukwanelisa lesi isimo. Inketho indida mayelana izakhiwo - a ukunweba engokwemvelo ukuba version amasethi - bangabaza sina ngokuqondene nokuthi kungenzeka ukuba ngiphikisane naye mayelana khona Inhloso impahla noma nesikhathi jikelele ngamunye kunqunywa isimo, noma isilandiso.

Ngokushesha kuwukuphikisana nezinkinga emsebenzini we logicians atholakele, zefilosofi kanye zezibalo abenze ezizathwini efanayo. Ngo-1902, uRussell wathola ukuthi okuhlukile indida ungaboniswa uhlelo okunengqondo, ithuthukiswe UMqulu I "Izisekelo izibalo" Gottlob Frege sika, omunye wemisebenzi eyinhloko ku logic of the XIX sekwephuzile - ekuqaleni XX leminyaka. Ngo ifilosofi Frege eziningi waqonda njengenhlangano "isandiso" noma "i-value-Ibanga" umqondo. Imiqondo kukhona esiseduzane nalezo correlates. Kulindeleke kube khona noma yimuphi umbandela onikiwe noma isilandiso. Ngakho, kukhona umqondo isethi, okungadingi ziwela ngaphansi umqondo walo belinganisa. Kukhona isigaba kuchazwe lowo mbono, futhi singaphansi ngokuchaza umqondo walo kuphela uma akusiyo.

Russell ukuba Frege ngalokhu kuphikisana ngoJuni 1902 Ukuxhumana isibe omunye ijabulisa kakhulu futhi sixoxe emlandweni logic. ngokushesha Frege neyaziwayo nemiphumela emibi indida. Nokho, Waphawula ukuthi nguqulo empikiswaneni maqondana izakhiwo ifilosofi yakhe yaxazululwa ngokubheka ahlukanise phakathi imiqondo emazingeni.

umqondo Frege BakaJehova baqonda njengoba aphuma emaphuzwini umsebenzi TRUE. Imiqondo ezingeni lokuqala ukuthatha njengoba agumenti izinjongo imiqondo ezingeni lesibili thathani agumenti kule misebenzi, nokunye. Ngakho, umqondo kungaba sibuthathe uqobo njengephuzu lokuphikisa, futhi indida ngokoMthetho izakhiwo ayikwazi zavela. Noma kunjalo amasethi, ukunwetshwa noma imiqondo Frege waqonda elibiza ngokuthi uhlobo olufanayo okunengqondo njengoba nazo zonke ezinye izinto. Khona-ke kuzo zonke iqoqo kukhona umbuzo noma ngabe ungena ngaphansi nomqondo alichaze.

Lapho Frege, Russell wamukela incwadi yokuqala, ivolumu "Izisekelo izibalo" wesibili kakade kuqedile ukuphrinta. Waphoqwa ukuba ngokushesha ukulungiselela isicelo enikeza impendulo Okufike kudide nge-Russell. Izibonelo Frege equkethwe eziningi izixazululo kungenzeka. Kepha wafinyelela esiphethweni ukuba buthaka nomqondo Abstraction isethi e uhlelo okunengqondo.

Esikhathini sokuqala, kwakunokwenzeka ukuba baphethe ngokuthi into kungokwalabo isethi kuphela uma uwela ngaphakathi umqondo, ichaza ke. Lendlela esibuyekeziwe iphetha ngokuthi into kungokwalabo isethi kuphela uma uwela ngaphakathi lo mqondo ngokuchaza sebuningini, kodwa hhayi usethe embuzweni. kuphakama indida zikaRussell.

Nokho ikhambi, onganelisekile ngokuphelele Frege. Futhi lokhu kwaba isizathu. Ngemva kweminyaka eminingana, ifomu ngaphezulu eyinkimbinkimbi ukuphikisana Kuye kwatholakala uhlelo olubukeziwe. Kodwa ngisho ngaphambi kokuba lokhu kwenzeke, Frege ishiywe izinqumo zakhe, nokuthi kubonakale ukuza ufinyelele esiphethweni sokuthi le ndlela yakhe yamane lingaphumeleli, futhi yikuphi kuyodingeka wenze ngaphandle we amasethi.

Futhi abanye baye baphathwa ezihlongozwayo, kuqhathaniswa kuphumelele kakhulu okunye izixazululo. Zazo zichaziwe ngezansi.

Imfundiso yokuziphendukela izinhlobo

Kwaphawulwa ngenhla ukuthi Frege kwaba yimpendulo enele zango yesethi theory kule nguqulo zavela kuzakhiwo. Frege asabela kwandulelwe ukugunyazwa okwenziwe isixazululo njalo ngawo kakhulu lolu hlobo indida. Kusekelwe yokuthi izakhiwo angaphansi izinhlobo ezahlukene futhi yini uhlobo impahla akawona efanayo izinto ukuze ngalo libhekisela.

Ngakho, kuphakama ngisho umbuzo, kungakhathaliseki impahla kusebenza uqobo. ulimi enengqondo, ohlukanisa izingxenye yokwehlayo enjalo, esebenzisa inkolelo-mbono yokuziphendukela izinhlobo. Nakuba isivele esetshenziswa Frege, okokuqala kuchazwe ngokuphelele futhi kugomela Russell kule Annex "isimiso". Imfundiso yokuziphendukela izinhlobo kwaba egcwele ngaphandle kokuhlukanisa amazinga Frege. Yena wabe izakhiwo akuzona kuphela izinhlobo ezahlukene logic, kodwa futhi wabeka. thayipha theory ukuxazulula ukuphikisana kule Okufike kudide nge-Russell elandelayo.

Ukuze kube ezanele philosophically, ukwamukelwa kwenkolelo-inkolelo-mbono yokuziphendukela izinhlobo izakhiwo kudinga ukuthuthukiswa inkolelo-mbono yokuziphendukela ubunjalo izakhiwo ukuze wakwazi ukuchaza ukuthi kungani abakwazi ozofakwa ngokwabo. Uma uthi nhlá, kunengqondo umenziwa awabo. Impahla ukuzikhathalela-ungubani, kubonakala, liphinde self-ungubani. Impahla Kubonakala sengathi kujabulise enhle. Ngendlela efanayo, ngokusobala, kubonakala Kungamampunge ukuthi impahla ekubeni ikati kuyinto ikati.

Noma kunjalo, izazi zefilosofi ahlukahlukene walungisiswa division of izinhlobo ezahlukene. Russell yaze yanikeza izincazelo abahlukene ngezikhathi ezehlukene emsebenzini wakhe. Ngasohlangothini lwalo, inkolelo zokwehlukanisa imiqondo ezahlukene amazinga Frege livela kakhulu inkolelo yakhe yokuziphendukela imiqondo unsaturated. Concepts njengoba umsebenzi, empeleni aziphelele. Ukuhlinzeka i-value, badinga impikiswano. Awukwazi umqondo owodwa nje umenziwa umqondo yohlobo olufanayo, ngoba namanje kusadinga-agumenti yayo. Ngokwesibonelo, nakuba kungenzeka ukuthatha impande skwele impande skwele inombolo, awukwazi nje usebenzise impande umsebenzi isikwele isikwele impande umsebenzi bese uthole umphumela.

Mayelana conservatism izakhiwo

Elinye ikhambi esingenzeka indida izakhiwo izakhiwo kuphikisa khona ngaphansi kwanoma yiziphi izimo unikezwa, noma isilandiso eyakheke kahle. Yiqiniso, uma othile eschews izakhiwo elifanayo ezintweni eziphilayo kokubili izakhi umgomo futhi ezimele wonkana, uma sithatha nominalism indida kungagwenywa ngokuphelele.

Nokho, ukuxazulula antinomy Akudingeki ukuba kube ukwaziswa kakhulu ngakho. Logic lesezingeni lelisetulu izinhlelo athuthukile Frege futhi Russell, aqukethe lokho okubizwa ngokuthi isimiso lomqondo, owawuthi ngamunye amafomula evulekile kungakhathaliseki ukuthi eziyinkimbinkimbi ukhona njengengxenye impahla noma umqondo isibonelo, kuphela labo izinto ehambelana ifomula. Ayesebenza izimfanelo zonke isethi okungenzeka izimo noma isilandiso, kungakhathaliseki ukuthi eziyinkimbinkimbi ayenjalo.

Noma kunjalo, kungenzeka ukuthatha azinabo ngaphezulu Metaphysics izakhiwo, enika ilungelo ukuba khona umgomo izakhiwo ezilula, kuhlanganise, ngokwesibonelo, ezifana umbala obomvu, ukuqina umusa nokunye. D. Ungase uvumele ngisho lezi izakhiwo zisebenze ngokwabo, ezinjengomusa can ube nomusa.

Futhi isimo esifanayo ngezimfanelo eziyinkimbinkimbi kungenziwa sinqatshiwe, isibonelo, "izakhiwo" njengokuthi nesikhombisa-amakhanda, kube-eyabhalwa-amanzi nokunye okunjalo. D. Kulokhu, akukho isimo predetermined singahlangabezani impahla, kuqondwe njengelakhiwe eceleni isici, okuyinto ine impahla yaso ekhona. Kanjalo umuntu angakwazi ukuphika ukuthi ukhona izakhiwo ezilula abe-impahla-ukuthi-non-isicelo lokuzilondolozela futhi ugweme indida ngokusebenzisa olandelanayo ngaphezulu izakhiwo elifanayo ezintweni eziphilayo.

indida zikaRussell: isixazululo

Ngenhla yaphawula ukuthi ekupheleni kokuphila kwakhe Frege ngokuphelele logic of amasethi. Yiqiniso, omunye ikhambi antinomy ngesimo amasethi: uyaphika okulula khona izici eziningana ezinjengolwazi iyonke. Ngaphezu kwalokho, kunezinye izinqumo ethandwa, izisekelo zazo kukhonjisiwe ngezansi.

Imfundiso for izinhlobo eziningi

Njengoba kushiwo ngaphambili, Russell wadlalela umbono more ephelele izinhlobo, abayohlanganyela hhayi kuphela izindawo noma imiqondo ezahlukene, kodwa futhi wabeka. Russell sethi eyabiwe ku sebuningini amayunithi ehlukile, sebuningini amasethi lezinto ahlukene, njll amasethi lezinto awazange kubhekwe, futhi sebuningini amasethi - .. Imisa. Baningi abakaze bajabulele uhlobo, ikuvumela ukuthi ube njengoba ilungu uqobo. Ngakho-ke akukho iqoqo wonke amaqoqo ukuthi awawona amalungu yayo, ngoba iyiphi isethi imibuzo mayelana kungaba njengelungu, kuyisipho uhlobo ukuhlukumeza. Nalapha futhi, le nkinga lapha ukuchaza Metaphysics amasethi ukuchaza izisekelo yefilosofi the division zibe izinhlobo.

babhekane nezinga elikhulu

Ngo-1937, V. Kuayn unikezele esinye isixazululo, ngendlela efanayo-mbono yokuziphendukela izinhlobo. ulwazi oluyisisekelo mayelana kulokho.

Ukwehlukanisa isici amasethi nabanye. Made ukuze nokucabanga lokuthola sebuningini ayilungile noma lutho ngaso sonke isikhathi. Imisa kungenziwa kuphela enikeziwe lapho belinganisa izimo zabo akuzona uhlobo ukuhlukumeza. Ngakho, i-Quine, inkulumo ethi "x akuyona ilungu x" iyona isitatimende okunenjongo akusho khona iqoqo zonke izakhi x wanelise kulesi simo.

Kulesi simiso iqoqo ekhona ezinye ovulekile ifomula A uma futhi kuphela uma wakubo, t. E. Uma eziguquguqukayo babelwa integers esihle sengqondo, ngakho ngamunye nesenzakalo nobuntu sebuningini esandulele ke variable inikezwa isabelo iyunithi lincane variable, ukumlandela. indida Lokhu kuvimbela uRussell, njengoba ifomula kuncuma isethi inkinga, kukhona okufanayo ngaphambi nangemva variable ubulungu uphawu okwenza kube unstratified.

Kodwa-ke nokho ukuthi noma ngabe uhlelo okuholela, okuyinto Quine ngokuthi "Izikhwama okusha logic zezibalo" kuyavumelana.

Ukwenqabela

Indlela ehlukile ngokuphelele esithathwayo imfundiso yokuziphendukela Zermelo - Fraenkel (ZF). Nalapha futhi, uzibekele ukuba khona amasethi. Kunalokho, iya kulowo "phezulu" of Russell Frege, ngubani ekuqaleni wayecabanga ukuthi zonke imiqondo, izakhiwo, noma izimo kungase kusikisele khona iqoqo zonke izinto nale impahla noma ukuhlangabezana isimo esinjalo, e ZF-theory, konke kuqala "kusukela phansi kuze."

izakhi ngalinye isethi ezingenalutho bese bakha isethi. Ngakho-ke, ngokungafani izinhlelo ngaphambili futhi Russell Frege FIT ingeyona isethi jikelele okuhlanganisa zonke izakhi ngisho nazo zonke amasethi. ZF ubeka imingcele eqinile ngokuba khona amasethi. Kwangathi khona kuphela labo okuyiwona ngokucacile ongoti kwezempilo abathi zihlobene noma ingakleliswa esebenzisa izinqubo iterative nokunye okunjalo. D.

Khona-ke, esikhundleni umqondo Abstraction isethi ukuzikhohlisa othi isici esithile likhona isethi kuphela uma ihlangabezana nemibandela e isimiso ngokwahlukana esetshenziswa DF, ukuhlala ngokwahlukana noma "ukuhlunga". Esikhundleni sicabangela khona iqoqo zonke izakhi okuyizinto abantu ngothi lwabo ukwanelisa isimo esithile, ngayinye isethi ekhona Aussonderung kubonisa ukuba khona isethi engezansi zonke izakhi isethi yokuqala okuyinto kwanelisa isimo.

Bese kufika isimiso Abstraction: uma isethi A ukhona, khona-ke, kuzo zonke x e A, x kungokwalabo isethi engezansi A, okuyinto kwanelisa isimo kuphela uma x esuthisa C. isimo Lendlelanchubo kuxazululwa indida Russell, kusukela ngeke nje simane siyishalazele lokucabanga okungukuthi, isethi wonke amasethi ukuthi awawona amalungu ngokwabo.

Ukuba okuningi amasethi, ungakhetha noma ulwahlukanise abe amasethi, okuyizinto ngokwabo, nalabo abangekho ezinjalo, kodwa njengoba kungekho isethi jikelele thina engaboshiwe iqoqo wonke amaqoqo. Ngaphandle uma kucatshangwa ukuthi inkinga usetha Russell ukungqubuzana ayikwazi efakazelwe.

nezinye izixazululo

Ngaphezu kwalokho, kuye kwaba izandiso okwalandela noma ukulungiswa kwalezi izixazululo, ezifana ngemfoloko-uhlobo umbono 'Izimiso kweembalo "uhlelo ukunwetshwa" logic zezibalo "Quine, kanye ngaphezulu tintfutfuko imfundiso yokuziphendukela amasethi, wenza Bernays, Gödel futhi von Neumann. Umbuzo ka-kungakhathaliseki indlela abantu abasabela ngayo indida zingaxazululeki uBertrand Russell wathola, namanje kuyindaba yokukhathalelwa mpikiswano.