K\u00fcme kavram\u0131<\/b>, i\u00e7erisine ald\u0131\u011f\u0131 nesneler iyi tan\u0131mlan\u0131rsa matematiksel\u00a0olarak tan\u0131ml\u0131 bir kavram haline gelmektedir.\u00a0\u0130yi tan\u0131mlanmam\u0131\u015f bir y\u0131\u011f\u0131n matematiksel anlamda tan\u0131ms\u0131zd\u0131r. \u00d6rnek olarak “Hayvanlar Alemi”, “S\u0131n\u0131ftaki K\u0131zlar Toplulu\u011fu”, “Cebimdeki t\u00fcm paralar” c\u00fcmlelerinde ge\u00e7en\u00a0nesneler a\u00e7\u0131k ve belirgin olduklar\u0131 i\u00e7in her biri birer k\u00fcmeyi tarif etmektedir.<\/p>\nGeorg Ferdinand Ludwig Philipp Cantor<\/h2>\n
3 Mart 1845 do\u011fumlu, Alman matematik\u00e7i\u00a0Georg Ferdinand Ludwig Philipp Cantor,\u00a0K\u00fcmeler kuram\u0131n\u0131 ortaya koyan bilim adam\u0131d\u0131r. K\u00fcmeler aras\u0131nda birebir e\u015flemenin \u00f6nemini ortaya koymu\u015f, “sonsuz k\u00fcme” kavram\u0131na matematiksel bir tan\u0131m getirmi\u015f ve ger\u00e7el say\u0131lar\u0131n sonsuzlu\u011funun do\u011fal say\u0131lar\u0131n sonsuzlu\u011fundan “daha b\u00fcy\u00fck” oldu\u011funu ispatlam\u0131\u015ft\u0131r. Ayr\u0131ca kardinal say\u0131 ve ordinal say\u0131 kavramlar\u0131n\u0131 ortaya atm\u0131\u015f ve bu say\u0131lar\u0131n aritmeti\u011fini tan\u0131mlam\u0131\u015ft\u0131r. Cantor’un bulu\u015flar\u0131n\u0131n matematik ve felsefede \u00f6nemli yeri vard\u0131r.<\/p>\n
Cantor’un “sonsuz\u00f6tesi say\u0131lar” fikri, zaman\u0131n matematik\u00e7ileri taraf\u0131ndan yo\u011fun \u015fekilde ele\u015ftirilmi\u015ftir. Henri Poincar\u00e9, Cantor’un fikirlerini “matemati\u011fi istila eden korkun\u00e7 bir hastal\u0131k” olarak nitelendirmi\u015f, Leopold Kronecker ise Cantor’u “\u015farlatan”l\u0131kla su\u00e7lam\u0131\u015ft\u0131r. Cantor’un 1884’ten hayat\u0131n\u0131n sonuna kadar ya\u015fad\u0131\u011f\u0131 depresyon n\u00f6betlerinin, k\u0131smen bu sald\u0131r\u0131lardan kaynakland\u0131\u011f\u0131 iddia edilmi\u015fse de, n\u00f6betlerin as\u0131l sebebi muhtemelen bipolar bozukluktur.<\/p>\n
G\u00fcn\u00fcm\u00fczde, Cantor’un fikirleri matematik\u00e7ilerin b\u00fcy\u00fck \u00e7o\u011funlu\u011fu taraf\u0131ndan do\u011fru kabul edilmekte ve matematik tarihinin en \u00f6nemli paradigma de\u011fi\u015fimlerinden biri olarak tan\u0131nmaktad\u0131r. David Hilbert, “Cantor’un yaratt\u0131\u011f\u0131 cennetten bizi kimse kovamayacakt\u0131r” diyerek Cantor’un katk\u0131lar\u0131n\u0131n \u00f6nemini vurgulam\u0131\u015ft\u0131r.<\/p>\n
Cantor, Halle \u00dcniversitesi’ndeki meslekda\u015f\u0131 Eduard Heine’nin etkisiyle say\u0131lar kuram\u0131ndan uzakla\u015f\u0131p analizle ilgilenmeye ba\u015flad\u0131. 1870’de, bir fonksiyonun birden fazla trigonometrik seri a\u00e7\u0131l\u0131m\u0131 olamayaca\u011f\u0131n\u0131 kan\u0131tlayarak ad\u0131n\u0131 duyurdu. Cantor’dan \u00f6nce, Heine’nin yan\u0131 s\u0131ra Lejeune Dirichlet, Rudolph Lipschitz ve Bernhard Riemann gibi pek \u00e7ok matematik\u00e7i bu problemle u\u011fra\u015fm\u0131\u015f ama sonuca ula\u015famam\u0131\u015ft\u0131. 1870-72 aras\u0131nda Cantor trigonometrik serilere ili\u015fkin bir dizi makale yay\u0131mlad\u0131, ve 1872’de S\u0131rad\u0131\u015f\u0131 Profes\u00f6r \u00fcnvan\u0131n\u0131 kazand\u0131. Ayn\u0131 sene yaz\u0131\u015fmaya ba\u015flad\u0131\u011f\u0131 meslekda\u015f\u0131 Richard Dedekind, ger\u00e7el say\u0131lar\u0131 “Dedekind kesitleri” olarak tan\u0131mlad\u0131\u011f\u0131 me\u015fhur makalesinde, Cantor’un trigonometrik seri makalelerinden birini referans olarak g\u00f6sterdi.<\/p>\n
Cantor 1873’te rasyonel say\u0131lar\u0131n do\u011fal say\u0131larla birebir e\u015flenebildi\u011fini kan\u0131tlad\u0131. Ayn\u0131 y\u0131l, cebirsel say\u0131lar\u0131n da say\u0131labilir oldu\u011funu kan\u0131tlad\u0131. 1874’te ise ger\u00e7el say\u0131lar\u0131n tamam\u0131n\u0131n say\u0131labilir olmad\u0131\u011f\u0131n\u0131<\/i> g\u00f6sterdi. B\u00f6ylece ger\u00e7el say\u0131lar\u0131n \u00e7ok k\u00fc\u00e7\u00fck bir k\u0131sm\u0131n\u0131n cebirsel oldu\u011fu, neredeyse tamam\u0131n\u0131n a\u015fk\u0131n say\u0131lar oldu\u011fu ortaya \u00e7\u0131kt\u0131.<\/p>\n
Cantor bundan sonra, boyut say\u0131lar\u0131 farkl\u0131 olan k\u00fcmelerin, birebir e\u015flenip e\u015flenemeyece\u011fini ara\u015ft\u0131rmaya ba\u015flad\u0131. 1877’de buldu\u011fu sonu\u00e7 olduk\u00e7a \u015fa\u015f\u0131rt\u0131c\u0131yd\u0131: Bir birim uzunlu\u011funda bir do\u011fru par\u00e7as\u0131n\u0131n \u00fczerindeki noktalar, p<\/i> boyutlu uzay\u0131n t\u00fcm noktalar\u0131yla birebir e\u015flenebiliyordu. Arkada\u015f\u0131 Dedekind’e bu sonu\u00e7tan bahsederken “Je le vois, mais je ne le crois pas!” (“G\u00f6r\u00fcyorum, ama inanm\u0131yorum!”) diye yazd\u0131.<\/p>\n
1878’te yazd\u0131\u011f\u0131 bir makalede, birebir e\u015fleme, say\u0131labilirlik ve boyut kavramlar\u0131na a\u00e7\u0131kl\u0131k getirdi.<\/p>\n
1879 ve 1884 aras\u0131nda yay\u0131mlad\u0131\u011f\u0131 alt\u0131 makaleyle, k\u00fcmeler kuram\u0131n\u0131n temellerini att\u0131, “sonsuz\u00f6tesi” (kardinal ve ordinal) say\u0131lar fikrini anlatt\u0131. Bu makaleleri yay\u0131mlayan Mathematische Annalen<\/i> dergisinin edit\u00f6rleri, asl\u0131nda b\u00fcy\u00fck bir cesaret \u00f6rne\u011fi sergiliyorlard\u0131, \u00e7\u00fcnk\u00fc Cantor’un fikirleri, Kronecker’un ba\u015f\u0131n\u0131 \u00e7ekti\u011fi bir grup n\u00fcfuzlu matematik\u00e7i taraf\u0131ndan \u015fiddetle ele\u015ftiriliyor ve hatal\u0131 bir d\u00fc\u015f\u00fcnce \u015fekli olarak yorumlan\u0131yordu. Bu kuvvetli muhalefetin fark\u0131nda olan Cantor, makalelerinde ele\u015ftirilere uzun uzun cevap vermeye \u00f6zen g\u00f6steriyordu.<\/p>\n
May\u0131s 1884’te ilk a\u011f\u0131r depresyon n\u00f6betini ge\u00e7iren Cantor, birka\u00e7 hafta i\u00e7inde kendini toparlad\u0131ysa da matemati\u011fe d\u00f6nmek i\u00e7in yeterli \u00f6zg\u00fcveni bulamad\u0131\u011f\u0131ndan, felsefe ve edebiyatla ilgilenmeye ba\u015flad\u0131. Sonsuzluk ve k\u00fcmeler hakk\u0131nda kendi geli\u015ftirdi\u011fi fikirlerin felsefi ve teolojik sonu\u00e7lar\u0131yla ilgileniyor, ve bu konuda pek \u00e7ok filozofla yaz\u0131\u015f\u0131yordu. Bu yaz\u0131\u015fmalar\u0131n bir k\u0131sm\u0131n\u0131 1888’de yay\u0131mlad\u0131.<\/p>\n
Cantor, son \u00f6nemli makalesini 1895 ve 1897’de iki k\u0131s\u0131m halinde yay\u0131mlad\u0131. Bu makalede, k\u00fcmeler kuram\u0131yla ilgili bug\u00fcn al\u0131\u015f\u0131k oldu\u011fumuz baz\u0131 kavramlar\u0131 (altk\u00fcmeler gibi) tan\u0131ml\u0131yor, kardinal ve ordinal aritmeti\u011fi tekrar g\u00f6zden ge\u00e7iriyordu. Cantor bu makalesinde s\u00fcreklilik hipotezinin de bir kan\u0131t\u0131n\u0131 sunmak istemi\u015f, ama \u00e7ok u\u011fra\u015ft\u0131\u011f\u0131 halde kan\u0131t\u0131 bulamam\u0131\u015ft\u0131.<\/p>\n
6 Ocak 1918 de ge\u00e7irdi\u011fi bir kalp krizi sonucunda\u00a0\u00f6ld\u00fc.<\/p>\n","protected":false},"excerpt":{"rendered":"
K\u00fcme kavram\u0131, i\u00e7erisine ald\u0131\u011f\u0131 nesneler iyi tan\u0131mlan\u0131rsa matematiksel\u00a0olarak tan\u0131ml\u0131 bir kavram haline gelmektedir.\u00a0\u0130yi tan\u0131mlanmam\u0131\u015f bir y\u0131\u011f\u0131n matematiksel anlamda tan\u0131ms\u0131zd\u0131r. \u00d6rnek olarak “Hayvanlar Alemi”, “S\u0131n\u0131ftaki K\u0131zlar Toplulu\u011fu”, “Cebimdeki t\u00fcm paralar” c\u00fcmlelerinde…<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[92],"tags":[2670,2665,2666,2662,2661,2664,2663,2669,910,2667,2668],"class_list":["post-6297","post","type-post","status-publish","format-standard","hentry","category-bilim","tag-altkumeler","tag-cantor","tag-georg-ferdinand-ludwig-philipp-cantor","tag-kume","tag-kumeler","tag-kumeler-kavrami","tag-kumeler-kurami","tag-kumeler-kuraminin-kurucusu","tag-matematik","tag-sonsuz-otesi-sayilar","tag-trigonometrik-seriler"],"_links":{"self":[{"href":"https:\/\/www.ilkkimbuldu.com\/index.php?rest_route=\/wp\/v2\/posts\/6297"}],"collection":[{"href":"https:\/\/www.ilkkimbuldu.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.ilkkimbuldu.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.ilkkimbuldu.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.ilkkimbuldu.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=6297"}],"version-history":[{"count":1,"href":"https:\/\/www.ilkkimbuldu.com\/index.php?rest_route=\/wp\/v2\/posts\/6297\/revisions"}],"predecessor-version":[{"id":6298,"href":"https:\/\/www.ilkkimbuldu.com\/index.php?rest_route=\/wp\/v2\/posts\/6297\/revisions\/6298"}],"wp:attachment":[{"href":"https:\/\/www.ilkkimbuldu.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=6297"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.ilkkimbuldu.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=6297"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.ilkkimbuldu.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=6297"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}