K\u00fcmeler kuram\u0131n\u0131n, dolay\u0131s\u0131yla modern matemati\u011fin babas\u0131 Georg Cantor’dur (1845-1918). Cantor, Berlin \u00dcniversitesi’nde Kummer’in \u00f6\u011frencisi olarak 1869’da say\u0131lar kuram\u0131nda tezini bitirdikten sonra, meslek hayat\u0131n\u0131n sonuna kadar \u00e7al\u0131\u015faca\u011f\u0131 Halle \u00dcniversitesi’nde i\u015fe ba\u015flam\u0131\u015ft\u0131r.<\/p>\n
Profesyonel matematik\u00e7ili\u011fin ilk y\u0131llar\u0131nda, ayn\u0131 \u00fciversiteden E.Heine’nin Cantor’a sordu\u011fu bir soru Cantor’un ya\u015fam\u0131nda matemati\u011fin de seyrini de\u011fi\u015ftirecekti. Soru \u015fuydu: [0,2pi] aral\u0131\u011f\u0131nda toplam\u0131 s\u0131f\u0131r olan bir trigonometrik serinin katsay\u0131lar\u0131n\u0131n hepsi s\u0131f\u0131r m\u0131d\u0131r?<\/p>\n
Cantor bu soruyla u\u011fra\u015f\u0131rken ger\u00e7el say\u0131lar\u0131n o g\u00fcne kadar fark edilmeyen bir \u00f6zelli\u011finin fark\u0131na var\u0131r: Rasyonel say\u0131larla irrasyonel say\u0131lar ayn\u0131 \u00e7oklukta de\u011fildir. Ba\u015fka bir ifadeyle, rasyonel say\u0131lar k\u00fcmesiyle irrasyonel say\u0131lar k\u00fcmesi aras\u0131nda, her ikisi de sonsuz olmas\u0131na kar\u015f\u0131n, bir e\u015fle\u015fme yoktur. O halde bu iki k\u00fcmenin sonsuzluklar\u0131 ayn\u0131 de\u011fildir. B\u00f6ylelikle ortaya k\u00fcme kavram\u0131 ve k\u00fcmelerin, i\u00e7erdikleri eleman “\u00e7oklu\u011fu” a\u00e7\u0131s\u0131ndan s\u0131n\u0131fland\u0131r\u0131lmas\u0131 sorunu \u00e7\u0131kt\u0131. Bu son kavram “sonsuzun” tek de\u011fil, \u00e7ok oldu\u011funu s\u00f6ylemektedir. Bu da \u00e7ok tepki \u00e7ekecekti. <\/p>\n
Tarih boyunca, Zeno’dan ba\u015flayarak, g\u00fcn\u00fcm\u00fcze kadar, sonsuzluk kavram\u0131 ve d\u00fc\u015f\u00fcncesi insanlar\u0131 rahats\u0131z etmi\u015ftir. Aristo’dan Cantor’a kadar ge\u00e7en zaman diliminde “sonsuz” anlay\u0131\u015f\u0131, temelde Aristo’nun g\u00f6r\u00fc\u015f\u00fc olan \u015fu anlay\u0131\u015ft\u0131r: Sonsuz, ufuk \u00e7izgisi gibi, var olmayan ama konu\u015fma kolayl\u0131\u011f\u0131 sa\u011flad\u0131\u011f\u0131 i\u00e7inkulland\u0131\u011f\u0131m\u0131z bir kavramd\u0131r. Bu kavram\u0131 “s\u0131n\u0131rs\u0131zl\u0131k” kavram\u0131 yerine kullan\u0131r\u0131z; bir \u015fey, \u00e7o\u011falarak ya da b\u00fcy\u00fcyerek, \u00f6nceden belirleyece\u011fimiz bir \u00e7oklu\u011fun ya da b\u00fcy\u00fckl\u00fc\u011f\u00fcn \u00f6tesine ge\u00e7me potansiyeline sahipse, o \u015feye “sonsuza gidiyor” deriz. Ba\u015fka bir deyimle, Aristo’nun sonsuz anlay\u0131\u015f\u0131 “potansiyel sonsuz” anlay\u0131\u015f\u0131d\u0131r.<\/p>\n
Cantor’a g\u00f6re ise “sonsuz” tek ba\u015f\u0131na anlaml\u0131 bir s\u00f6zc\u00fck de\u011fildir. Anlaml\u0131 olan “sonsuz k\u00fcme” kavram\u0131d\u0131r. Sonsuz k\u00fcmeler de var olan nesnelerdir. Burada “sonsuz k\u00fcme” deyimi, “b\u00fcy\u00fckanne” gibi, b\u00f6l\u00fcnmez bir terim olarak anla\u015f\u0131lmal\u0131d\u0131r. O halde k\u00fcmeler \u00f6nce sonlu-sonsuz diye ikiye ayr\u0131lacak; sonra da sonsuz k\u00fcmeler, kendi aralar\u0131nda, sonsuzluklar\u0131na g\u00f6re \u00e7e\u015fitli s\u0131n\u0131flara ayr\u0131lacaklard\u0131r. B\u00f6ylelikle ortaya say\u0131s\u0131z “sonsuz k\u00fcme” s\u0131n\u0131flar\u0131 \u00e7\u0131kacakt\u0131r. Bu da \u00e7ok \u00e7e\u015fitli “sonsuzl\u011fun” oldu\u011fu anlam\u0131na gelmektedir. <\/p>\n
Cantor’un bu sonsuzluk anlay\u0131\u015f\u0131, Leopold Kronecker ve Henri Poincar\u00e9 gibi bir \u00e7ok \u00fcnl\u00fc matematik\u00e7i taraf\u0131ndan tepkiyle kar\u015f\u0131land\u0131. Bunun sonucu olarak da, matematik\u00e7iler, “sonsuzu” Cantor gibi alg\u0131layanlar ve Aristo gibi alg\u0131layanlar olmak \u00fczere iki gruba ayr\u0131ld\u0131lar.<\/p>\n","protected":false},"excerpt":{"rendered":"
K\u00fcmeler kuram\u0131n\u0131n, dolay\u0131s\u0131yla modern matemati\u011fin babas\u0131 Georg Cantor’dur (1845-1918). Cantor, Berlin \u00dcniversitesi’nde Kummer’in \u00f6\u011frencisi olarak 1869’da say\u0131lar kuram\u0131nda tezini bitirdikten sonra, meslek hayat\u0131n\u0131n sonuna kadar \u00e7al\u0131\u015faca\u011f\u0131 Halle \u00dcniversitesi’nde i\u015fe ba\u015flam\u0131\u015ft\u0131r. Profesyonel matematik\u00e7ili\u011fin ilk y\u0131llar\u0131nda, ayn\u0131 \u00fciversiteden E.Heine’nin Cantor’a sordu\u011fu bir soru Cantor’un ya\u015fam\u0131nda matemati\u011fin de seyrini de\u011fi\u015ftirecekti. Soru \u015fuydu: [0,2pi] aral\u0131\u011f\u0131nda toplam\u0131 s\u0131f\u0131r olan bir …<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1404,1403],"tags":[2132,7269,7268],"class_list":["post-3070","post","type-post","status-publish","format-standard","hentry","category-matematik-odevleri","category-odevler","tag-aristo","tag-georg-cantor","tag-modern-matematik-cagi"],"_links":{"self":[{"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/posts\/3070","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/comments?post=3070"}],"version-history":[{"count":0,"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/posts\/3070\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/media?parent=3070"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/categories?post=3070"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/tags?post=3070"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}