Indida Russell esithi: ulwazi olusisiseko, imizekelo, ekwakhiweni

Russell indida yi ezimbini antinomy ganiso ubhalo.

iintlobo ezimbini indida zikaRussell

Eyona ndlela kuxoxwe rhoqo of a ukungaphikisani iiseti logic. Ezinye iseti kubonakala amalungu ngokwawo, kunye nabanye - akukho. Isethi zonke iiseti ngokwaso iseti, ngenxa yoko kuyabonakala ukuba libhekisela ngokwayo. Null okanye ayinanto, ke, kufuneka ube lilungu ngokwayo. Ngoko ke, lo isethi zonke iiseti, njengoko zero ayiqukwanga ibe ngokwayo. kuvela indida xa umbuzo wokuba ingaba isethi ilungu ngokwayo. Oku kungenzeka ukuba kwaye kuphela ukuba ayikho.

Olunye uhlobo indida yintswela malunga iipropati. Ezinye izakhiwo, sibonakala sibhekisele bona, ngoxa abanye. Lo mhlaba ukuba ipropati ngokwayo kwipropati, lo gama i-mhlaba nokuba nekati ayikho. Cinga impahla kokuba ipropati leyo akangowakhe kuye. ukuba usebenza ngokwawo? Kwakhona, naziphi na uqikelelo kufuneka eyahlukileyo. Indida igama imbeko Bertrand Russell (1872-1970), eyafumana ngayo ngo-1901.

indaba

Ivula Russell yenzekile ngexesha umsebenzi wakhe "Principles of Mathematics". Nangona Wasisusa indida ngokuzimeleyo, kukho ubungqina bokuba ezinye zezibalo kunye developers theory iseti, kuquka Ernst Zermelo kunye noDavide Hilbert, babeyazi kwinguqulelo yokuqala ephikisanayo phambi kwakhe. Russell, Noko ke, waba ngowokuqala lowo azixoxwa ngokwenkcukacha indida umsebenzi wakhe ipapashwe, okokuqala wazama ukuqulunqa izisombululo kunye bokuqala ukuqonda ngokupheleleyo intsingiselo yalo. yonke A sahluko "Imigaqo" enikelwe kwingxoxo kwalo mba, kwaye isicelo enikelwe kwithiyori lweentlobo, ecetywayo uRussell njengesisombululo.

Russell wafumanisa "kephofu na ixoki, ahlole theory iseti Cantor ukuba uthi kwagunya lelayiphi iseti incincane kune yeseti subsets yayo. Ubuncinane kule domain kufuneka subsets ezininzi njengoko izinto ezikuye, ukuba omnye yenye isiqalelo ngasinye ukucwangcisa equlethe nesiqalelo kuphela. Ngaphezu koko, Cantor wabonisa ukuba inani lweziqalelo ayikwazi ukuba ilingane nenani subsets. Ukuba kukho inani elifanayo, bekuya kufuneka khona ƒ inkalo ukuba ukubonisa iielementi subsets zabo. Ngelo xesha kukho ubungqina ukuba oku akunakwenzeka. Ezinye izinto zingaboniswa subsets umsebenzi ƒ eziqulathe kubo, ngoxa abanye bangase.

Cinga Isethi izinto musa ngamalungu imifanekiso zabo, apho ukubonisa ƒ. Oku kukodwa elucwangciso olusezantsi lweziqalelo, kwaye ke ngoko, ƒ umsebenzi kuya ukubonisa it on an element ithambeka. Ingxaki kukuba kuphakama umbuzo wokuba ngaba olu element fanele ukuba yenye apho ubonisa ƒ. Oku kunokwenzeka kuphela xa akakulandeli. indida Russell kaThixo kungabonwa umzekelo kumgca efanayo yokuqiqa njengoko, lula kuphela. Yintoni na ngakumbi - alukiweyo okanye subsets iseti? Bekuya kubonakala ukuba kufuneka iiseti ezininzi, njengokuba onke subsets yeeseti ngokwabo. Kodwa ukuba theorem Cantor likaThixo liyinyaniso, ngoko kufuneka subsets ngaphezulu apho. Russell uthathwa nje ukubonisa iiseti bona ndayinikela ndlela kantoriansky kuqwalaselwa iseti zonke ezi zinto, ngaphandle kwe iseti apho zibekwa. Ukubonisa Russell uba iseti zonke iiseti, a non.

imposiso Frege

"Engagqibekiyo ixoki" yaba negalelo elikhulu kuphuhliso yembali thiyori iiseti. Wabonisa ukuba ingqiqo iseti jikelele kuyingxaki kakhulu. Kwakhona wambuza ukuba nombono ukuba imeko okanye isivisa nganye echazwe sigqibe ubukho nobuninzi kuphela ezo zinto ukwanelisa le meko. Option indida malunga iipropati - elulwandiso zendalo iiseti uguqulelo - ndingasathe- kakhulu ukuba ngaba kunokwenzeka ukuphika ubukho Injongo ipropati okanye ezivumelana jikelele nganye kumiselwa meko, okanye nesivisa.

Kungekudala ephikisanayo kunye neengxaki umsebenzi logicians afunyanwa, iingcali zefilosofi kunye zemathematika abaye okucingelwayo efanayo. Ngowe-1902, uRussell wafumanisa ukuba eyongeziweyo lobeko lwe indida ingavezwa kwinkqubo olucwangcisiweyo, eziqulunqwe uMqulu "Isisekelo arithmetic" Gottlob Frege kaThixo, enye imisebenzi ephambili kulandelelwano le XIX kade - ekuqaleni kwenkulungwane XX. Ifilosofi Frege ezininzi iqondwe njenge "ulwandiso" okanye "ixabiso-ibanga" ingqiqo. La magama ke ikufutshane kwezo correlates. Kulindeleke ukuba bube nayiphi na imeko okanye nesivisa. Ngoko ke, kukho ingqiqo iseti, engaquki kuwa phantsi ingqiqo yayo ocacisa. Kukho kwakhona udidi kuchazwa ngulo mbono, kwaye oko kuxhomekeke ekumiseleni ingqiqo yayo kuphela ukuba ayikho.

Russell wabhalela Frege ngale ungquzulwano ngoJuni 1902 Imbalelwano ibe ngomnye 'umdla yaye sithetha ngayo kwimbali ekuqiqeni. Frege walibona nemiphumela emibi indida. Wathi, Noko ke, ukuba inguqulelo imbambano malunga iipropati bulumko sakhe ngokusonjululwa lwahlule phakathi kwala magama kwamanqanaba.

ngcamango Frege aqonda njengoko utshintsho ukusuka ukuxambulisana umsebenzi kwi TRUE. Iikhonsepthi kwinqanaba lokuqala ngokuthatha njengoko iingxoxo izinto ze iikhonsepthi kwinqanaba lesibini uzithabathele, njengoko ndithomalalisa le misebenzi, njalo-njalo. Ngoko ke, lo gama engasoze ngokwayo njenge mpikiswano, kwaye indida ngokwemigaqo iipropati ayikwazi ukwakhiwa. Kodwa litshona, ukwanda okanye iikhonsepthi Frege iqondwe njengesibhekisa uhlobo bokuqiqisa efanayo naleyo zonke ezinye izinto. Ke yonke iseti kukho umbuzo enoba iwela phantsi ingqikelelo usenza.

Xa Frege, uRussell wafumana ileta yokuqala, umthamo yesibini "Isisekelo arithmetic" sele ugqibile print. Wanyanzelwa ukuba ukulungiselela ngokukhawuleza isicelo esinika impendulo indida ka uRussell. Imizekelo Frege eziqulethwe eziliqela izisombululo kunokwenzeka. Kodwa ke, weyiseka ukuba buthathaka imbono yokutsala iseti kwinkqubo olucwangcisiweyo.

Kwimeko yokuqala, kwenzeka ukugqiba ukuba into fanele ukuya iseti ukuba kwaye kuphela ukuba iwela ngaphakathi ingqiqo, sichaza oko. Le nkqubo ehlaziyiweyo sinokuqonda ukuba into fanele ukuya iseti ukuba kwaye kuphela ukuba singena phantsi ingcamango ukuchaza ezininzi kuThixo, kodwa kubeka mbuzo. kuvela indida Russell kaThixo.

Isisombululo, Noko ke, engonelisekanga ngokupheleleyo Frege. Nantsi ke isizathu. kweminyaka eliqela kamva, uhlobo ngxuba le ntswela Kufunyenwe inkqubo ehlaziyiweyo. Kodwa phambi kokuba oku kwenzeke, Frege ashiya izigqibo zakhe yaye kubonakala ukuba bafikelele kwisigqibo sokuba indlela yakhe nje nocalu, nokuba logic kuya kufuneka wenze ngaphandle yeeseti.

Kanti abanye baye ecetywayo, izisombululo ezinye gqitha ngempumelelo ngakumbi. Ezi kuxoxwa ngezantsi.

Imfundiso kweentlobo

Kwaqondwa ngaphezu Frege yaba impendulo eyaneleyo zango kwethiyori iseti kwi uguqulelo zavela ngenxa kwiipropati. impendulo Frege kaThixo kwakhokelwa isisombululo kakhulu ngazo rhoqo kule ndlela imangalisayo. Oku kusekelwe yokuba mihlaba zixhomekeke iintlobo ezahlukeneyo yaye yintoni uhlobo izakhiwo akazange iyafana izinto ukuba libhekisela.

Ngaloo ndlela, nditsho umbuzo, ingaba lo mhlaba esebenzayo ngokwayo. ulwimi acwangciswe, leyo esahlula iziqalelo ze uluhlu enjalo, esebenzisa imfundiso kweentlobo. Nangona sele isetyenziswa yi Frege, okokuqala ucaciswe ngokupheleleyo kwaye esivakalayo uRussell kwi kwiSihlomelo kwi "umgaqo". Ithiyori kweentlobo kwaba kokupheleleyo ngaphezu umahluko amanqanaba Frege. Abize iipropati nje kuphela iindidi ezahlukeneyo ekuqiqeni, kodwa wabeka. uhlobo theory yokusombulula le ntswela kwi indida ka uRussell hlobo.

Ukuze ube philosophically eyaneleyo, ukwamkelwa thiyori iindidi zeepropati kufuna ukuphuhliswa thiyori yohlobo mihlaba ukuze akwazi ukucacisa ukuba kutheni ingenakusetyenziswa kulophulo ngokwabo. Ekuqaleni, iyavakala ukuba nesivisa impahla yakhe. Lo mhlaba yokuba ukuzeyisa ungubani, oko kuya kubonakala, kukwayinto entle self-nesazisi. Lo mhlaba kubonakala ukuba mnandi nice. Ngendlela efanayo, kubonakala ukuba, kubonakala zobuxoki ukuthi ipropati lokuba ikati ikati.

Noko ke, izazi ezahlukeneyo sigwetyelwe ukwahlulwa kweentlobo ezahlukeneyo. Russell wada wancama iinkcazo abohlukileyo ngamaxesha ahlukileyo kwiphupha lakhe. Kuba inxenye yayo, ngenxa yeso sizathu ukuba ukwahlulwa iingqiqo ezahlukeneyo kumanqanaba Frege avela imfundiso yakhe concepts ezihluthiyo. Concepts njengomsebenzi, eneneni, aziphelelanga. Ukubonelela ixabiso, kufuneka ingxabano. Awukwazi ingqiqo nje omnye ukuya nesivisa ingqiqo uhlobo olufanayo, kuba usalindele ingxabano yayo. Umzekelo, nangona kunokwenzeka ukuba singathatha ingcambu yalama cala ingcambu esikweri yenani, awukwazi nje ukusebenzisa i ingcambu umsebenzi kwi ingcambu umsebenzi kwaye ufumane yoko.

Ngeemilo conservatism

Enye Isisombululo singaba iipropati kephofu iipropati kuphika ubukho phantsi nayiphi na imiqathango enikwe, okanye isivisa ibunjwe-kakuhle. Kakade ke, ukuba umntu eschews iimpawu nobuntu zombini izinto njengenjongo ezizimeleyo liphela, ukuba sithabatha nominalism kephofu unako ayekwe ngokuphelele.

Noko ke, ukuba ukusombulula antinomy mfuneko yokuba ide ibe kakhulu kangako. Logic iinkqubo ukuze eliphezulu kuphuhliswa Frege uRussell, ziqulathe oko kuthiwa umgaqo sengqiqo, njengako apho ngamnye fomyula open kungakhathaliseki ukuba esimbaxa ikhona njengenxalenye wepropati okanye ingqiqo umzekelo, kuphela ezo izinto angqinelane ifomula. Baya isicelo iziphumo iseti ngazo zonke iimeko okanye izivisa, enoba ukuntsonkotha ngayo.

Noko ke, kungenzeka ukuba i iipropati ezikhona ngqongqo, enika ilungelo ubukho kwinjongo kwiipropati ezilula, kuquka, umzekelo, ezifana nombala obomvu, ukuqina, ububele njalo njalo. D. Unga bayekeni aba mihlaba kubo ngokwabo, ezifana ububele ube nobubele.

Kunye nesimo efanayo iimpawu ezinzima na waphika, umzekelo, "iipropati" ezifana lineentloko-elinesixhenxe, abe-ebhalwe phantsi-kwamanzi kunye ezifana. D. Kulo mzekelo, akukho imeko kwangaphambili oko ukuhlangabezana ipropati, njengemvume ngokwahlukileyo element, nto leyo eye iimpawu zalo okhoyo. Ngaloo ndlela umntu onokuthi akayiboni ubukho iipropati ezilula kuba-mhlaba-oko-non-isicelo-to-self kwaye uphephe kephofu ngokusebenzisa iipropati angqingqwa ngakumbi ezintweni.

indida Russell esithi: isisombululo

Ngaphezu kwaye kwaqatshelwa ukuba ekupheleni kobomi bakhe Frege bamshiya ngokupheleleyo ingqiqo iiseti. Kakade ke, omnye isisombululo antinomy ngohlobo iiseti: wayekhanyela elula ubukho izinto ezifana iyonke. Ukongeza, kukho ezinye iinketho ezithandwayo, iziseko zazo ziboniswa apha ngezantsi.

Le mbono iintlobo ezininzi

Njengoko kuchaziwe ngaphambili, uRussell wadlalela kwingcamango epheleleyo ngakumbi iintlobo, ababeza ukwabelana kuphela iipropati okanye iikhonsepthi iintlobo ezahlukeneyo, kodwa wabeka. Russell utitshala ebekwe phezu nobuninzi kweeyunithi ezahlukeneyo, kukho ezininzi iiseti zezinto ezahlukeneyo, njalo njalo iiseti izinto zange ingqalelo, kunye nobuninzi leeseti - .. Imisa. A lot of neze uhlobo, ikuvumela ukuba njengelungu ngokwayo. Ngoko ke akukho isethi zonke iiseti ukuba amalungu yaso, ngenxa yokuba nayiphi iseti na imibuzo malunga nokuba loo nto njengelungu, ngokwaso uhlobo ukungathobeli. Kwakhona, umba apha ukuchaza iiseti ezikhona ukuchaza iziseko zeentanda-bulumko zokwahlula ngeentlobo.

ezithile ngokwezinga

Ngowe-1937, V. Kuayn uye wanikela enye indlela yokusombulula, ngendlela efanayo kwithiyori kweentlobo. ulwazi olusisiseko malunga ngalo.

Ukwahlula iiseti elementi kunye nabanye. Made ukuze Kwingcingane lokufumana Ukuba kwisininzi usoloko azichanekanga okanye nantsingiselo. Iiseti kubonelelwa kuphela xa uchaza iimeko zabo akukho uhlobo ukungathobeli. Ngoko ke, ngokuba Quine, ibinzana elithi "x elingelilo ilungu x" mazwi lebyi akuthethi kuthi ubukho iseti zonke izinto x kwanelisa le meko.

Kule nkqubo iseti sikhoyo avulekile ifomula A ukuba kwaye kuphela ukuba wakubo, t. E. Ukuba yezahlukileyo babelwa integers ezakhayo ezifana ukuba isenzeko ngasinye uphawu nobuninzi elandulela ngayo variable yabelwe unit isabelo incinane variable, alandelayo emva kwakhe. indida Le iibhloko kaRussell, ekubeni Ifomula esetyenzisiweyo ukumisela iseti ngxaki, kukho enye phambi kwaye emva kophawu ubulungu bume eyenza kube unstratified.

Kodwa kusafuneka ukufumanisa ukuba ingaba inkqubo onesiphumo, leyo Quine ebizwa "Isisekelo New ekuqiqeni kwemathematika" kuhambelana.

ukungafunwa

Indlela ehlukileyo sidlalwa kumhlaba kwithiyori Zermelo - Fraenkel (uyifuna). Apha kwakhona, umise umda kubukho iiseti. Endaweni yoko, yiya "top-phantsi" of uRussell kunye Frege, ekuqaleni wacinga ukuba onke amagama, iimpawu, okanye imiqathango zibonisa ubukho iseti yezinto zonke nalo mhlaba okanye ukuhlangabezana i imeko enjalo, xa uyifuna-theory, yonke into iqala "ukusuka ezantsi ukuya phezulu."

izinto ngabanye iseti engenanto kwaye zenze iseti. Ngoko ke, ngokungafaniyo nenkqubo yangaphambili uRussell Frege FIT asisiso ukuya iseti jikelele equka zonke izinto yaye zonke iiseti. Uyifuna ubeka imida engqongqo kubukho iiseti. Lunokubakho kuphela ezo kwananwe ngayo Aristotle ngokucacileyo okanye enokuthi kwenziwa ngokusebenzisa iinkqubo iterative kunye like. D.

Emva koko, endaweni ingqiqo okungxalwe iseti obonisa ocacisa ukuba into ethile idityanisiwe iseti ukuba kwaye kuphela ukuba luyahlangabezana iimeko umgaqo yokuzahlula esetyenzisiweyo DF, ngokwahlukana okanye "yokulungisa". Kunokuthatha ubukho yeseti izinto zonke izinto ngaphandle kokukhetha ukwanelisa imeko ethile, kuba iseti ngamnye okhoyo Aussonderung ibonisa ubukho elucwangciso olusezantsi lwe zonke izinto kwiseti original oyifaneleyo imeko.

Emva koko kuza Umgaqo yokutsala: ukuba iseti A ikhona, ngoko, ngokuba bonke x e A, x omelwe ukwenziwa yenye A, oyifaneleyo imeko ukuba kwaye kuphela ukuba x ukuzifezekisa C. imeko Le ndlela usombulula indida Russell, ekubeni ngeke ithathe nje oko kukuthi, iiseti zonke iiseti ukuba amalungu ngokwabo.

Ukuba kakhulu leeseti, ungakhetha okanye alabe libe iiseti, wona ngokwawo, kunye nabo abangengabo ezinjalo, kodwa ekubeni akukho iseti jikelele thina simelwe isethi zonke iiseti. Ngaphandle kokuthabatha ingxaki icwangcisa uRussell ukumphikisa ayikwazi kunokuveliswa ubungqina.

nezinye izisombululo

Ukongezelela koko, kuye kwaba Izandiso ezilandelayo okanye utshintsho ezi izisombululo, ezifana ifolokhwe-uhlobo ingcamango "Principles of Mathematics" inkqubo ukwanda "logic zemathematika" Quine, kwakunye nophuhliso ezininzi zakutshanje kwithiyori iiseti, wenza Bernays, Gödel kunye von kunikezelo. Umbuzo wokuba ingaba impendulo indida ezingenakunyibilika Bertrand Russell wafumana, Kusekho nto-mpikiswano.