satatyachi mukhyata

sanch siddhantat, satatyachi mukhyata hii vastavik sankhyanchya sootra _ 1 chya sanchachi mukhyata kinwa "aakar" aahe, jyala kadhikadhi satatya mhanatat. hii eka anant cardinal sankhya aahe aanee sootra _ 2 (lahan akshar frectoor "see") kinwa sootra _ 3 dware darshavili jaate.
vastavik sankhyanche sootra _ 1 heey naisargik sankhyanchya sootra _ 5 peksha adhik sankhyabal asalele aahet. shivay, sootra _ 1 madhye sootra _ 7 chya shakti sanchaitakech ghatak aahet, pratikatmakpane, jarr sootra _ 5 chee mukhyata sootra _ 9 mhanoon darshavili gelee tarr satatyachi mukhyata aahe
george canterne 1874 chya tyachya aganitatechya puravyat heey siddha kele aahe, joou tyachya vividh anantatechya abhootpurv abhyasacha eka bhag aahe. 1891 madhye tyanchya karn yuktivadat nantar asamanata adhik sopya paddhatine namood karnyat aalee. canterne bijective functionschya sandarbhat cardinality paribhashit keleeh done sanchanmadhye samaan cardinality asate jarr, aanee keval jarr tyanchyat bijective function astitvat asel.
konatyahi done vastavik sankhyanmadhye, 'aaa' <'buub', teey ekmekanchya kitihi jawal asale taree, nehmi itar asankhya vastavik sankhya asatat aanee canterne dakhavun dile qee tyaa vastavik sankhyanchya sampoorn sanchatil sankhyanitakach aahet. aahe shabdant, ughade antar ("aaa", "buub") sootra 10 saha samatulya aahe, heey itar anek anant sanchansathi dekhil kharey aahe, jase qee konatyahi "aine"-aayami euclidean space sootra _ 11 (jaaga bharane vakr pahaa). mhanje,
sarvat lahan anant cardinal sankhya sootra _ 9 (eleph-null) aahe. dusare sarvat lahan sootra _ 13 (eleph-vann) aahe. satatyiy gruhitak, jeey thampane sangato qee asey konatehi sanch naheet jyanche mukhyatva sootra _ 9 aanee sootra _ 15 darmyan katekorpane aahe. yaa gruhitkatil satya kinwa khotepana anishchit aahe aanee mothya pramanat vaparlya jaanarya zarmelo-frankel sanch siddhantamadhye nivadichya svayamsiddhatesaha (zdfc) siddha keela jaau shakat nahee.
malmatta.
atulaniyata.
jyorg canterne anant sanchanchya aakaranchi tulana karnyasathi cardinalitychi sankalpana sadar kelee. tyanni prasiddhapane dakhavun dile qee vastavik sankhyancha sanch aganitpane anant aahe. mhanjech, sootra _ 16 heey naisargik sankhyanchya, sootra _ 9 chya, mukhyatepeksha katekorpane mothe aaheh
vyavharat, yacha arth assa aahe qee purnankanpeksha adhik vastavik sankhya aahet. canterne heey vidhan anek vegvegalya prakare siddha kele. yaa vishayavaril adhik mahitisathi, cantercha pahila aganit purava aanee cantercha karn yuktivad pahaa.
mukhya samanata.
cantorche pramey siddha karnyasathi canterchya karn yuktivadatil bhinnata vaparali jaau shakate, jyamadhye asey mhatale aahe qee konatyahi sanchachi mukhyata tyachya shakti sanchapeksha katekorpane qamee aahe. mhanjech, sootra _ 18 (aanee tyamule naisargik sankhyanchya sootra _ 5 chey shakti sanch sootra _ 19 aganit aahe). kharey tarr, sootra _ 19 chee mukhyata sootra _ 16 chya barobarichi aahe heey khalilpramane darshavu shakatoh
cantor-bernstein-schroeder prameyadvare aapan assa nishkarsh kadhato qee
cardinal equality sootra _ 42 heey cardinal ankganitacha vapar karoon darshavile jaau shakateh
cardinal ankganitache niyam vaparoon, heey dekhil darshavu shakato qee
jethe "n" heey konatehi maryadit cardinal ≤2 aahe, aanee
jethe sootra _ 43 heey r chya shakti sanch aanee sootra _ 44 chee mukhyata aahe.
paryayi spashtikaran.
pratyek vastavik sankhyecha kiman eka anant dashansh vistar asato. udaharanarth,
konatyahi paristhitit, ankanchi sankhya mojata yete kaaran tee naisargik sankhyanchya sootra _ 45 chya sanchashi eka-eka patravyavaharat thewali jaau shakatat. yamule, samaja, pahilya, shambharvya kinwa π chya dashalakshvya ankabaddal bolane samajanyasarkhe tharate. naisargik sankhyanna cardinality sootra _ 46 asalyane pratyek vastavik sankhyechya vistarat sootra _ 9 ank asatat.
pratyek vastavik sankhya purnank bhag aanee dashansh apurnankat vibhagali jaau shakate, tyamule aaplyala milateh
jithe aamhi hii vastusthiti vaparali
dusarikade, jarr aapan sootra _ 48 chaa sootra _ 49 madhye nakasha banavala aanee keval 3 kinwa 7 asalele dashansh apurnank heey vastavik sankhyancha keval eka bhag aahet asey manale, tarr aaplyala milate.
aanee asha prakare
beth sankhya.
beth sankhyancha kram sootra _ 50 aanee sootra _ 51 nishchit karoon paribhashit keela jaato. mhanoon sootra _ 16 hii dusari beth sankhya aahe, beth-ekah
tisari beth sankhya, beth-too, hii sootra _ 28 (i.eae. vastavik reshechya serve upasanchancha sanch):
satatyiy gruhitak.
prasiddha satatya gruhitak asey thampane sangate qee sootra _ 16 hii dusari alf sankhya, sootra _ 13 dekhil aahe. aahe shabdant, satatya gruhitak asey sangate qee konatehi nishchit sootra _ 56 nahee jyachi mukhyata sootra _ 9 aanee sootra _ 16 darmyan katekorpane aahe.
curt godel aanee paul cohain yanni darshavilyapramane, heey vidhan aata zarmelo-frankel sanch siddhantachya swasiddhanpasun swatantra asalyache dnyat aahe, jyat nivadichya swasiddhantacha (zdfc) samavesh aahe. mhanjech, gruhitak aanee tyache nishedh heey donhi yaa svayamsiddhateshi susangat aahet. kharey tarr, pratyek shunyetar naisargik sankhya "aine" sathee, samanata sootra _ 16 = sootra _ 60 heey zdfcpasoon swatantra aahe (cases sootra _ 61 heey satatya gruhitak aahe). bahutek itar aalefsasathihi hetch kharey aahe, jaree kahee prakarananmadhye, konigchya prameyadvare saharthatechya aadharavar samanata nakarli jaau shakate (udaa.jee., sootra _ 62). visheshtah, sootra _ 63 heey ektar sootra _ 13 kinwa sootra _ 65 asoo shakate, jethe sootra _ 66 heey pahile aganit kramvari aahe, mhanoon teey ektar uttaradhikari cardinal kinwa maryada cardinal aanee ektar niyamit cardinal kinwa ekvachan cardinal asoo shakate.
satatyachya mukhytesah sanch.
ganitat abhyas kelelya anek sanchanmadhye sootra _ 16 chya barobarichi mukhyata asate. kahee samanya udaharane khalilpramane aaheth
adhik cardinalities sanch.
sootra _ 16 peksha jaast cardinality asalelya sanchanmadhye khaleel goshtincha samavesh hotoh
yaa sarvanche cardinality sootra _ 85 (beth too) aahe.

