{"id":26542,"date":"2014-04-01T15:29:25","date_gmt":"2014-04-01T12:29:25","guid":{"rendered":"https:\/\/bilimvegelecek.com.tr\/?p=26542"},"modified":"2018-06-07T15:44:29","modified_gmt":"2018-06-07T12:44:29","slug":"boyle-otel-olur-mu","status":"publish","type":"post","link":"https:\/\/bilimvegelecek.com.tr\/index.php\/2014\/04\/01\/boyle-otel-olur-mu","title":{"rendered":"B\u00f6yle otel olur mu?"},"content":{"rendered":"

Bu yaz\u0131da efsanevi Alman matematik\u00e7i David Hilbert taraf\u0131ndan kurgulanm\u0131\u015f, pop\u00fcler matematik merakl\u0131lar\u0131nca iyi bilinen bir hik\u00e2yeden s\u00f6z edece\u011fiz. Hilbert bu hik\u00e2yeyi, vatanda\u015f\u0131 Georg Cantor\u2019un ke\u015ffetti\u011fi, matematik d\u00fcnyas\u0131n\u0131 derinden sarsm\u0131\u015f sonsuz k\u00fcmeler kuram\u0131na giri\u015f yapabilmek i\u00e7in girdi\u011fi derslerde anlat\u0131rm\u0131\u015f.<\/p>\n

Belki \u00e7o\u011fu okura g\u00f6re \u201ctekrar su\u00e7u\u201d i\u015fleyece\u011fim, ama bu m\u00fckemmel hik\u00e2yeden habersiz matematik merakl\u0131s\u0131 bir\u00e7ok gen\u00e7le kar\u015f\u0131la\u015f\u0131yorum. Onlara da ula\u015fmay\u0131 d\u00fc\u015f\u00fcnerek matematiksel heyecan\u0131 y\u00fcksek bu \u00f6yk\u00fcy\u00fc baz\u0131 k\u00fc\u00e7\u00fck kurgu de\u011fi\u015fiklikleriyle kaleme almak istedim.<\/p>\n

Otel Hilbert, sonsuz say\u0131da odas\u0131 olan bir otel! 1, 2, 3, 4\u2026 diye numaraland\u0131r\u0131lm\u0131\u015f, bir ki\u015filik sonsuz say\u0131da odas\u0131 var.<\/p>\n

T\u00fcm odalar\u0131n\u0131n dolu oldu\u011fu bir g\u00fcn, otele bir m\u00fc\u015fteri gelir. Resepsiyondan otelde bo\u015f yer olmad\u0131\u011f\u0131n\u0131 \u00f6\u011frenerek hayal k\u0131r\u0131kl\u0131\u011f\u0131yla \u00e7\u0131k\u0131\u015f kap\u0131s\u0131na do\u011fru y\u00f6neldi\u011fi s\u0131rada lobide oturan David Hilbert\u2019in sesi duyulur:<\/p>\n

– Bir dakika bay\u0131m, sizin i\u00e7in bo\u015f bir oda bulabiliriz.<\/p>\n

– Nas\u0131l? Kalan konuklardan birinden odas\u0131n\u0131 bo\u015faltmas\u0131n\u0131 m\u0131 isteyeceksiniz?<\/p>\n

– Yok, hay\u0131r, \u015f\u00f6yle yapabiliriz: Konuklar\u0131m\u0131zdan odalar\u0131n\u0131 birer kayd\u0131rmalar\u0131n\u0131 rica edece\u011fiz. 1 numaral\u0131 konuk 2 numaral\u0131 odaya, 2 numaral\u0131 konuk 3numaral\u0131 odaya, 3 numaral\u0131 konuk 4 numaral\u0131 odaya ge\u00e7er ve bu \u015fekildeki ta\u015f\u0131nma sonsuza dek yap\u0131l\u0131rsa 1 numaral\u0131 odaya siz ge\u00e7ebilirsiniz.<\/p>\n

– \u0130yi ama, son odada kalan son konu\u011fu d\u0131\u015far\u0131 m\u0131 atacaks\u0131n\u0131z?<\/p>\n

– Otelimizde sonsuz say\u0131da oda oldu\u011funu s\u00f6ylemi\u015ftim, son oda ve son konuk yok maalesef.<\/p>\n

– M\u00fckemmel! Te\u015fekk\u00fcrler.<\/p>\n

Ertesi g\u00fcn otelde i\u015fler daha da a\u00e7\u0131l\u0131r. Bu kez, otelin \u00f6n\u00fcne sonsuz say\u0131da yolcusu olan bir otob\u00fcs park etmi\u015ftir. Yolcular\u0131 gezdiren rehber resepsiyon g\u00f6revlisinden bo\u015f yer olmad\u0131\u011f\u0131n\u0131 \u00f6\u011frenerek sonsuz say\u0131daki yolcusuna ba\u015fka bir otel bulmay\u0131 d\u00fc\u015f\u00fcnd\u00fc\u011f\u00fc s\u0131rada yine Hilbert\u2019in sesi duyulur:<\/p>\n

– Odalar\u0131m\u0131z dolu, ama yolcular\u0131n\u0131za oda bulabiliriz.<\/p>\n

– Otob\u00fcste sonsuz say\u0131da yolcu var, sonsuz say\u0131da oda a\u00e7abilecek misiniz?<\/p>\n

– Tabii ki, hi\u00e7 sorun de\u011fil. Otelimizdeki konuklar\u0131 i\u00e7inde olduklar\u0131 odan\u0131n numaras\u0131n\u0131n iki kat\u0131 numaral\u0131 odalara ta\u015f\u0131yaca\u011f\u0131z. 1 numaral\u0131 konuk 2 numaral\u0131 odaya, 2 numaral\u0131 konuk 4 numaral\u0131 odaya, 3 numaral\u0131 konuk 6 numaral\u0131 odaya, 4 numaral\u0131 konuk 8 numaral\u0131 odaya ge\u00e7er ve ta\u015f\u0131nma i\u015flemi sonsuza dek yap\u0131l\u0131rsa sonsuz say\u0131daki tek say\u0131l\u0131 (1, 3, 5, 7\u2026) odalar bo\u015f kal\u0131r. Otob\u00fcs\u00fcn 1 numaral\u0131 koltu\u011funda oturan yolcuyu ilk tek say\u0131 olan1 numaral\u0131 odaya, 2 numaral\u0131 koltukta oturan yolcuyu ikinci tek say\u0131 olan 3 numaral\u0131 odaya, 3 numaral\u0131 koltukta oturan yolcuyu \u00fc\u00e7\u00fcnc\u00fc tek say\u0131 olan 5 numaral\u0131 odaya yerle\u015ftirir ve b\u00f6ylece devam edersek otob\u00fcs\u00fcn\u00fczdeki b\u00fct\u00fcn konuklar\u0131 a\u011f\u0131rlayabiliriz.<\/p>\n

– \u00c7ok g\u00fczel, m\u00fcthi\u015f bir bulu\u015f, \u00e7ok te\u015fekk\u00fcrler. Ben yolcular\u0131 \u00e7a\u011f\u0131rmaya gidiyorum.<\/p>\n

\u00dc\u00e7\u00fcnc\u00fc g\u00fcn\u00fcn sabah\u0131nda otel g\u00f6revlileri \u015fa\u015fk\u0131nl\u0131k i\u00e7indedir, otelin \u00f6n\u00fcne sonsuz say\u0131da otob\u00fcs gelmi\u015ftir ve her otob\u00fcste sonsuz say\u0131da yolcu vard\u0131r. Otob\u00fcs 1\u2019in yan\u0131na otob\u00fcs 2, otob\u00fcs 2\u2019nin yan\u0131na otob\u00fcs 3 park eder ve bu \u015fekilde park etmeyi s\u00fcrd\u00fcren sonsuz say\u0131da otob\u00fcsten ucu buca\u011f\u0131 g\u00f6r\u00fcnmeyen bir konvoy olu\u015fmu\u015ftur.<\/p>\n

Otel doludur. Acaba, otob\u00fcslerdeki her yolcu i\u00e7in yeni bir oda bulmak m\u00fcmk\u00fcn m\u00fcd\u00fcr? B\u00fct\u00fcn g\u00f6zlerin \u00fczerine \u00e7evrildi\u011fi David Hilbert gayet sakin, \u201cSorun yok, yeni gelen her yolcuya bir oda bulabiliriz.\u201d diye m\u0131r\u0131ldan\u0131r ve ard\u0131ndan da \u201c\u0130lk yapmam\u0131z gereken \u015fey, \u00f6nceki g\u00fcnk\u00fc gibi sonsuz say\u0131da oday\u0131 bo\u015faltmak olmal\u0131.\u201d diye ekler. Ayn\u0131 y\u00f6ntemi kullan\u0131r, yani yine oteldeki herkesi kendi oda numaras\u0131n\u0131n iki kat\u0131 olan odalara yerle\u015ftirir. B\u00f6ylece sonsuz say\u0131daki tek numaral\u0131 odalar\u0131n t\u00fcm\u00fc bo\u015falm\u0131\u015f olur. \u015eimdi yap\u0131lmas\u0131 gereken tek \u015fey sonsuz say\u0131daki otob\u00fcs kafilesinde bulunan t\u00fcm yolcular\u0131 sayman\u0131n bir yolunu bulmakt\u0131r. B\u00f6ylece ilk yolcu 1 numaral\u0131 odaya, 2. yolcu 3 numaral\u0131 odaya, 3. yolcuyu 5 numaral\u0131 odaya yerle\u015fecek ve b\u00f6yle devam edilirse kafiledeki t\u00fcm yolcular odalar\u0131na yerle\u015fmi\u015f olacak.<\/p>\n

Bu kez Hilbert\u2019in i\u015fi daha zordur. Eline kalemi al\u0131p, a\u015fa\u011f\u0131daki tabloyu yapar. Her yolcuyu listeler. Her bir yolcuya a\/b <\/em>bi\u00e7imindeki bir kesirle bir numara verir.<\/p>\n

\"\"Burada a <\/em>otob\u00fcs\u00fcn numaras\u0131n\u0131, b <\/em>ise yolcunun oturdu\u011fu koltu\u011fun numaras\u0131n\u0131 g\u00f6sterir. \u00d6rne\u011fin 1\/1, ilk otob\u00fcs\u00fcn ilk yolcusunu, 2\/3 ikinci otob\u00fcs\u00fcn \u00fc\u00e7\u00fcnc\u00fc yolcusunu g\u00f6steriyor.<\/p>\n

Hilbert, kafiledeki b\u00fct\u00fcn yolcular\u0131 tabloda g\u00f6r\u00fcld\u00fc\u011f\u00fc gibi zikzak bir yol izleyerek sayabiliyor. Bu y\u00f6ntemle saymaya ba\u015flan\u0131rsa ikinci ki\u015fi birinci \u00a0otob\u00fcs\u00fcn ikinci yolcusu (1\/2), \u00fc\u00e7\u00fcnc\u00fc ki\u015fi ikinci otob\u00fcs\u00fcn birincisi yolcusu (2\/1), d\u00f6rd\u00fcnc\u00fc ki\u015fi \u00fc\u00e7\u00fcnc\u00fc otob\u00fcs\u00fcn birinci yolcusu olur. B\u00f6ylece devam edilirse b\u00fct\u00fcn yolcular a\u015fa\u011f\u0131da g\u00f6sterildi\u011fi gibi say\u0131lm\u0131\u015f olur.<\/p>\n

Bo\u015falt\u0131lm\u0131\u015f odalar: 1 3 5 7 9 \u2026<\/p>\n

Kafiledeki yolcular 1\/1 1\/2 2\/1 3\/1 2\/2 \u2026<\/p>\n

K\u0131ssadan hisse <\/strong><\/p>\n

Otel Hilbert hik\u00e2yesi buraya kadar olan b\u00f6l\u00fcm\u00fcyle bize neyi anlat\u0131yor? Bu soruyu yan\u0131tlamadan \u00f6nce matematiksel sonsuzun fikir babas\u0131 Georg Cantor\u2019un sonsuz k\u00fcmelere \u015fa\u015f\u0131rt\u0131c\u0131 yakla\u015f\u0131m\u0131ndan birka\u00e7 c\u00fcmleyle s\u00f6z edelim.<\/p>\n

Cantor, do\u011fal say\u0131lar k\u00fcmesiyle birebir e\u015flenebilen t\u00fcm sonsuz k\u00fcmelerin (tek tamsay\u0131lar, kareler, asallar gibi) ayn\u0131 b\u00fcy\u00fckl\u00fckte oldu\u011funu saptar. Bu b\u00fcy\u00fckl\u00fc\u011fe, \u0130brani alfabesinin ilk harfine s\u0131f\u0131r alt indisi kullanarak (\u00c00) alef s\u0131f\u0131r ad\u0131n\u0131 verir. Elemanlar\u0131 do\u011fal say\u0131lar k\u00fcmesiyle birebir e\u015flenebilen her bir say\u0131 k\u00fcmesinin eleman say\u0131s\u0131, daha do\u011frusu b\u00fcy\u00fckl\u00fc\u011f\u00fc \u00c00\u2019d\u0131r. E\u011fer sonsuz say\u0131da elemandan olu\u015fan bir k\u00fcmenin elemanlar\u0131n\u0131 do\u011fal say\u0131lar k\u00fcmesinin elemanlar\u0131yla birebir e\u015fleyebiliyorsak bu k\u00fcmenin elemanlar\u0131n\u0131 sayabiliyoruz demektir. Ki bu t\u00fcr k\u00fcmelerin t\u00fcm\u00fcn\u00fcn b\u00fcy\u00fckl\u00fc\u011f\u00fc \u00c00\u2019d\u0131r. Cantor buradan \u201csay\u0131labilir sonsuzluk\u201d kavram\u0131na ula\u015fm\u0131\u015ft\u0131r. Elbette ard\u0131ndan da say\u0131lamaz sonsuzlu\u011fa. Cantor\u2019un bu ad\u0131m\u0131 bir devrimin ba\u015flang\u0131c\u0131 olarak kabul edilir.<\/p>\n

Hikayemize d\u00f6nersek\u2026<\/p>\n

Birinci g\u00fcn: <\/strong>Otelde t\u00fcm odalar doluyken gelen bir m\u00fc\u015fteriye oda a\u00e7\u0131labilir. Matematik sembolleriyle ifade edersek: \u00c00 + 1 = \u00c00 (Yayg\u0131n olarak kullan\u0131lan anlat\u0131m\u0131: sonsuz art\u0131 bir e\u015fittir yine sonsuz)<\/p>\n

\u0130kinci g\u00fcn: <\/strong>Otelde t\u00fcm odalar doluyken say\u0131labilir sonsuzlukta odaya say\u0131labilir sonsuzlukta yolcu yerle\u015febiliyor. Bu c\u00fcmlenin matematiksel ifadesi: \u00c00 + \u00c00 = \u00c00 (Yayg\u0131n anlat\u0131m\u0131: sonsuz art\u0131 sonsuz e\u015fittir yine sonsuz.)<\/p>\n

\u00dc\u00e7\u00fcnc\u00fc g\u00fcn: <\/strong>Otelde t\u00fcm odalar doluyken say\u0131labilir sonsuzluktaki otob\u00fcste bulunan say\u0131labilir sonsuzluktaki yolcuya oda bulunuyor. Matematiksel ifadesi: \u00c00 \u00d7 \u00c00 = \u00c00 (Yayg\u0131n anlat\u0131m\u0131: sonsuz \u00e7arp\u0131 sonsuz e\u015fittir sonsuz)<\/p>\n

Dikkat edilirse sonsuz k\u00fcmelerde yap\u0131lan i\u015flemler aritmetikteki i\u015flemlerden \u00e7ok farkl\u0131d\u0131r. Bu y\u00fczden parantez i\u00e7inde \u201cyayg\u0131n anlat\u0131m\u201d ifadesini kullanarak yazd\u0131\u011f\u0131m\u0131z c\u00fcmleler do\u011fru de\u011fildir; \u00e7\u00fcnk\u00fc bu c\u00fcmleler, aritmetiksel i\u015flem d\u00fcnyas\u0131yla ilgilidir, sonsuz k\u00fcmelerde ge\u00e7ersizdir.<\/p>\n

\u201cG\u00f6r\u00fcyorum, ama inanam\u0131yorum\u201d <\/strong><\/p>\n

\u015eimdi, Cantor\u2019un \u00e7ok \u015fa\u015f\u0131rt\u0131c\u0131 bir ke\u015ffinden s\u00f6z edelim. Koltuklar ve otob\u00fcsler tablosuna bakarsak, a \/ b <\/em>\u015feklinde ifade edilen her ki\u015fi a\/b <\/em>kesriyle g\u00f6sterilebilir. Tablo sa\u011fa ve a\u015fa\u011f\u0131ya do\u011fru sonsuz \u015fekilde geni\u015fletildi\u011finde t\u00fcm pozitif kesirleri kapsayacakt\u0131r. \u00d6rne\u011fin 1789\/668 kesri 1789\u2019uncu sat\u0131r ve 668\u2019inci s\u00fctunda yer al\u0131r. Tabloda her bir otob\u00fcsteki her yolcuyu saymak i\u00e7in kulland\u0131\u011f\u0131m\u0131z zikzak sayma y\u00f6ntemi, pozitif t\u00fcm kesirleri saymak i\u00e7in de kullan\u0131l\u0131r. Dolay\u0131s\u0131yla t\u00fcm pozitif kesirler k\u00fcmesi ve t\u00fcm do\u011fal say\u0131lar k\u00fcmesi ayn\u0131 b\u00fcy\u00fckl\u00fcktedir, yani \u00c00. Sezgilerimiz bize do\u011fal say\u0131lardan daha fazla kesirli say\u0131lar\u0131n oldu\u011funu s\u00f6yler, ama Cantor yan\u0131ld\u0131\u011f\u0131m\u0131z\u0131 g\u00f6stermi\u015ftir. Cantor bile kendi ula\u015ft\u0131\u011f\u0131 sonuca inanamam\u0131\u015f, arkada\u015f\u0131 Dedekind\u2019e yazd\u0131\u011f\u0131 mektupta \u201cG\u00f6r\u00fcyorum, ama inanam\u0131yorum.\u201d c\u00fcmlesini kurmu\u015ftur.<\/p>\n

Otelde karga\u015fa, matematikte devrim! <\/strong><\/p>\n

Otel Hilbert\u2019e d\u00f6nerek Cantor\u2019un \u00e7ok \u015fa\u015f\u0131rt\u0131c\u0131 bir ke\u015ffinden daha s\u00f6z edelim. Bu ke\u015fif matematiksel akl\u0131n en g\u00fczel meyvelerinden biridir. Bir\u00e7ok matematik\u00e7inin g\u00fcvenli d\u00fcnyalar\u0131n\u0131 sarsm\u0131\u015f, kimilerince matematiksel sapk\u0131nl\u0131k olarak nitelendirilmi\u015f, ama matematikte \u00e7\u0131\u011f\u0131r a\u00e7\u0131c\u0131 sonu\u00e7lara yol a\u00e7m\u0131\u015ft\u0131r.<\/p>\n

Bu kez, otelin t\u00fcm odalar\u0131 bo\u015ftur ve otel y\u00f6netimi sonsuz say\u0131da m\u00fc\u015fterinin geldi\u011fi g\u00fcnlerin \u00f6zlemiyle i\u015flerin yoluna sokulmas\u0131 i\u00e7in ne yap\u0131lmas\u0131 gerekti\u011fini kara kara d\u00fc\u015f\u00fcnmektedir. Hilbert\u2019in \u201cBen bu konuda bir \u015fey yapamam, ama istedi\u011finiz say\u0131da m\u00fc\u015fteri bulun, odalara nas\u0131l yerle\u015ftirece\u011finizi s\u00f6yleyeyim.\u201d dedi\u011fi s\u0131rada m\u00fckemmel bir haber gelir. Otelin \u00f6n\u00fcne yine sonsuz say\u0131da otob\u00fcs park etmi\u015f ve her bir otob\u00fcste sonsuz say\u0131da yolcu vard\u0131r. Bu sefer, her bir yolcunun elinde 0 ve 1 say\u0131lar\u0131 aras\u0131nda bulunan sonsuz geni\u015flikteki t\u00fcm ondal\u0131k say\u0131lar\u0131n yaz\u0131l\u0131 oldu\u011fu kartlar bulunmaktad\u0131r. Bu kartlardaki ondal\u0131k say\u0131lar\u0131n t\u00fcm\u00fc birbirinden farkl\u0131 olup, (0, 1) aral\u0131\u011f\u0131ndaki t\u00fcm sonsuz geni\u015flikteki ondal\u0131k say\u0131lar\u0131 kapsamaktad\u0131r. Tabii ki yolcular\u0131n ellerindeki kartlar\u0131n uzunluklar\u0131 da sonsuz geni\u015fliktedir. Elbette, geni\u015fli\u011fi sonlu olan ondal\u0131k say\u0131lar\u0131n yaz\u0131l\u0131 oldu\u011fu kartlar\u0131 ta\u015f\u0131yan yolcular da olabilir. Ama bu kartlardaki say\u0131lar sa\u011f taraflar\u0131na sonsuz say\u0131da s\u0131f\u0131r ekleyerek geni\u015fli\u011fi sonsuz olan ondal\u0131k say\u0131lara d\u00f6n\u00fc\u015ft\u00fcr\u00fclebilir. B\u00f6ylece (0,1) aral\u0131\u011f\u0131ndaki t\u00fcm ondal\u0131k say\u0131lar kartlarda yaz\u0131l\u0131d\u0131r.<\/p>\n

Yolcular\u0131 gezdiren rehber resepsiyona gelip otelde kendilerini a\u011f\u0131rlaman\u0131n bir yolu olup olmad\u0131\u011f\u0131n\u0131 sorar. Otel y\u00f6netimi bu sorunu Hilbert\u2019in kolayl\u0131kla \u00e7\u00f6zece\u011fini d\u00fc\u015f\u00fcnerek rehbere olumlu yan\u0131t verir ve yolcular yava\u015f yava\u015f otele gelmeye ba\u015flar.<\/p>\n

Hilbert lobidedir ve elinde ka\u011f\u0131t kalem baz\u0131 hesaplar yapmaktad\u0131r. Bu arada otob\u00fcslerdeki m\u00fc\u015fteriler ak\u0131n ak\u0131n otele geliyordur. Otel y\u00f6netimi hen\u00fcz Hilbert\u2019ten olumlu bir yan\u0131t alamam\u0131\u015ft\u0131r. Hangi odaya gideceklerini bilemeyen yolcular\u0131n g\u00f6sterdikleri tepkiler b\u00fcy\u00fck bir karga\u015faya neden olur.<\/p>\n

Hilbert aya\u011fa kalkarak, bu kez, gelen t\u00fcm m\u00fc\u015fterilerin otele yerle\u015ftirilmesinin m\u00fcmk\u00fcn olmad\u0131\u011f\u0131n\u0131 s\u00f6yler. Otel y\u00f6neticileri, hi\u00e7 beklemedikleri bu s\u00f6z \u00fczerine \u015fa\u015fk\u0131nl\u0131k i\u00e7inde Hilbert\u2019ten biraz daha \u00e7al\u0131\u015fmas\u0131n\u0131 ve bir y\u00f6ntem bulmas\u0131n\u0131 rica ederler. Y\u00f6neticilerin bir t\u00fcrl\u00fc akl\u0131 alm\u0131yordur, \u00e7\u00fcnk\u00fc daha \u00f6nce de sonsuz say\u0131daki otob\u00fcsle sonsuz say\u0131da yolcu gelmi\u015ftir, bug\u00fcn de. Ama bu kez durum farkl\u0131d\u0131r, (0, 1) aral\u0131\u011f\u0131ndaki her bir ondal\u0131k say\u0131y\u0131 listeleyerek saymak pek kolay de\u011fildir.<\/p>\n

Kahraman\u0131m\u0131z Hilbert, bu i\u015flemin m\u00fcmk\u00fcn olmayaca\u011f\u0131n\u0131 \u015f\u00f6yle a\u00e7\u0131klar: (0,1) aral\u0131\u011f\u0131ndaki t\u00fcm ondal\u0131k say\u0131lar\u0131 i\u00e7erece\u011fini varsayarak yap\u0131lacak her bir sonsuz listede olmayan, ama (0,1) aral\u0131\u011f\u0131nda bulunan bir say\u0131 mutlaka vard\u0131r. Bunu kan\u0131tlayabilirim. Oda bulma umuduyla otelde bulunan yolcular, otel \u00e7al\u0131\u015fanlar\u0131 ve y\u00f6neticiler h\u00fcsran duygusuyla Hilbert\u2019in etraf\u0131nda toplanm\u0131\u015flard\u0131r.<\/p>\n

Hilbert ilk i\u015f olarak lobideki yolcular\u0131n elindeki kartlar\u0131n numaralar\u0131ndan olu\u015fan bir liste yapar ve bu listenin d\u0131\u015far\u0131daki sonsuz say\u0131da yolcunun kartlar\u0131ndaki numaralarla sonsuza deks\u00fcrd\u00fcr\u00fclebilece\u011fini s\u00f6yleyerek, a\u015fa\u011f\u0131daki listeyi k\u00e2\u011f\u0131da yazar. (Geli\u015fig\u00fczel se\u00e7ilmi\u015f bu numaralar\u0131n sonsuz geni\u015flikte olduklar\u0131n\u0131 biliyoruz.)<\/p>\n

Oda 1\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0,5738949\u2026
\nOda 2 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0,2023246\u2026
\nOda 3 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0,4218654\u2026
\nOda 4 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0,0398716\u2026
\nOda 5 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0,7770175\u2026
\nOda 6 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0,3469679\u2026
\nOda 7 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0,1875696\u2026
\nOda\u2026\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0<\/p>\n

Hilbert\u2019in amac\u0131 (0, 1) aral\u0131\u011f\u0131nda olup listede bulunmayan ondal\u0131k say\u0131y\u0131 bulmakt\u0131r. Bunu da a\u015fa\u011f\u0131daki y\u00f6ntemi kullanarak yapar. \u0130lk olarak 1\u2019inci odadaki ondal\u0131k say\u0131n\u0131n virg\u00fclden sonraki ilk rakam\u0131n\u0131, daha sonra 2\u2019inci odadaki say\u0131n\u0131n virg\u00fclden sonraki ikinci rakam\u0131n\u0131, sonras\u0131nda da 3\u2019\u00fcnc\u00fc odadaki say\u0131n\u0131n virg\u00fclden sonraki \u00fc\u00e7\u00fcnc\u00fc rakam\u0131n\u0131 alarak bu \u015fekilde devam eder ve yine sonsuz geni\u015flikte yeni bir ondal\u0131k say\u0131 elde eder. Bu say\u0131y\u0131 a\u015fa\u011f\u0131daki listede koyu renkte g\u00f6sterilmi\u015f \u00e7apraz rakamlardan olu\u015fturmu\u015ftur.<\/p>\n

0,5738949\u2026
\n0,2023246\u2026
\n0,4218654\u2026
\n0,0398716\u2026
\n0,7770175\u2026
\n0,3469679\u2026
\n0,1875696\u2026<\/p>\n

Bu say\u0131 0,5018176\u2026 olur.<\/p>\n

Hilbert, izleyenlerin \u015fa\u015fk\u0131n bak\u0131\u015flar\u0131 aras\u0131nda \u015fu s\u00f6zleri s\u00f6yler: Yapt\u0131\u011f\u0131m listede bulunmayan say\u0131y\u0131 olu\u015fturmak i\u00e7in 0,5018176\u2026 say\u0131s\u0131n\u0131n rakamlar\u0131n\u0131 de\u011fi\u015ftirmem gerekiyor. Bunu da her bir rakama 1 ekleyerek yapabiliriz. B\u00f6ylece \u015fu say\u0131y\u0131 elde ederiz: 0,6129287\u2026<\/p>\n

Alk\u0131\u015flar ba\u015flar. Yap\u0131lan listede bulunmayan say\u0131n\u0131n 0,6129287\u2026 oldu\u011funu g\u00f6renler co\u015fkuyla Hilbert\u2019i alk\u0131\u015fl\u0131yordur. Hilbert s\u00f6zlerini s\u00fcrd\u00fcr\u00fcr: 0,6129287\u2026 say\u0131s\u0131 yapt\u0131\u011f\u0131m listeye ait de\u011fildir. Bu say\u0131, 1\u2019inci odadaki numara olamaz, \u00e7\u00fcnk\u00fc ilk basama\u011f\u0131ndaki rakam 1\u2019inci odadaki numaran\u0131n ilk basama\u011f\u0131ndan farkl\u0131d\u0131r. 2\u2019inci odada da olamaz, \u00e7\u00fcnk\u00fc ikinci basama\u011f\u0131 2\u2019inci odadaki numaran\u0131n ikinci basama\u011f\u0131ndan farkl\u0131d\u0131r. Bunu bu \u015fekilde sonsuza dek s\u00fcrd\u00fcrd\u00fc\u011f\u00fcm\u00fczde listede bulunmayan sonsuz geni\u015flikteki bir ondal\u0131k say\u0131n\u0131n mutlaka bulunaca\u011f\u0131 sonucuna ula\u015f\u0131r\u0131z. Pek\u00e2l\u00e2, listede ilk yedi basama\u011f\u0131 0,6129287 olan bir say\u0131 bulanabilir, ama bu say\u0131 listede olsa da bizim olu\u015fturdu\u011fumuz sonsuz geni\u015flikteki say\u0131 bu say\u0131dan en az bir basamak bile olsa daha sonraki basamaklarda farkl\u0131 olacakt\u0131r. Bu y\u00fczden, gelen m\u00fc\u015fterileri sonsuza dek odalara g\u00f6ndermeye devam etsek bile elindeki kartta bizim olu\u015fturdu\u011fumuz 0,6129287\u2026 say\u0131s\u0131n\u0131 ta\u015f\u0131yan m\u00fc\u015fteri i\u00e7in bir oda bulamay\u0131z.<\/p>\n

Yolcular\u0131 gezdiren rehber, otel y\u00f6neticileri, gelen yolcular\u0131n odalara yerle\u015femeyecek olmas\u0131ndan dolay\u0131 \u00fczg\u00fcnd\u00fcrler, ama \u00f6te yandan b\u00fcy\u00fck bir olaya tan\u0131k olduklar\u0131 i\u00e7in \u015fansl\u0131 olduklar\u0131n\u0131 d\u00fc\u015f\u00fcnerek da\u011f\u0131l\u0131rlar.<\/p>\n

K\u0131ssadan Hisse<\/strong><\/p>\n

Otel Hilbert\u2019te sonsuz say\u0131da oda var, ama yine de (0,1) aral\u0131\u011f\u0131ndaki sonsuz say\u0131daki ondal\u0131k say\u0131yla numaraland\u0131r\u0131lm\u0131\u015f, sonsuz say\u0131daki yolcuyu odalara yerle\u015ftiremiyoruz, hep a\u00e7\u0131kta kalan birileri oluyor. Oysa daha \u00f6nce yine sonsuz say\u0131da yolcuyu ayn\u0131 otele yerle\u015ftirebiliyorduk. Bunun nedeni, b\u00fcy\u00fckl\u00fckleri ayn\u0131 olmayan iki farkl\u0131 sonsuzlukla kar\u015f\u0131 kar\u015f\u0131ya olmam\u0131zd\u0131r. Hik\u00e2yenin son b\u00f6l\u00fcm\u00fc bize, do\u011fal say\u0131lar\u0131n \u201csonsuzlu\u011fundan\u201d daha b\u00fcy\u00fck sonsuzluklar\u0131n oldu\u011funu anlat\u0131r. Otel Hilbert\u2019in odalar\u0131 do\u011fal say\u0131lar k\u00fcmesinin elemanlar\u0131yla numaraland\u0131r\u0131lm\u0131\u015ft\u0131r, (0,1) aral\u0131\u011f\u0131nda bulunan sonsuz geni\u015flikteki ondal\u0131k say\u0131lar, daha do\u011frusu reel say\u0131lar oda numaralar\u0131yla birebir e\u015flenemez. Cantor, hik\u00e2ye kahraman\u0131m\u0131z Hilbert\u2019in kulland\u0131\u011f\u0131 \u201c\u00e7apraz y\u00f6ntemiyle\u201d do\u011fal say\u0131lar k\u00fcmesinden daha b\u00fcy\u00fck sonsuzluklar\u0131n oldu\u011funu g\u00f6stermi\u015f, sonras\u0131nda da \u201csonsuzun\u201d tek de\u011fil \u00e7ok oldu\u011funu ve baz\u0131 sonsuzlar\u0131n baz\u0131lar\u0131ndan daha b\u00fcy\u00fck oldu\u011funu ke\u015ffetmi\u015ftir. Sonsuz k\u00fcmeleri say\u0131labilir ve say\u0131lamaz olarak ikiye ay\u0131rm\u0131\u015ft\u0131r. (0,1) aral\u0131\u011f\u0131ndaki reel say\u0131lar k\u00fcmesinin \u201csay\u0131lamaz\u201d, do\u011fal say\u0131lar k\u00fcmesinin ise \u201csay\u0131labilir\u201d oldu\u011funu g\u00f6stermi\u015ftir.<\/p>\n

Ba\u015flang\u0131\u00e7 ad\u0131mlar\u0131n\u0131 anlatmaya \u00e7al\u0131\u015ft\u0131\u011f\u0131m\u0131z Cantor\u2019un bu ke\u015ffi, 19. Y\u00fczy\u0131l matemati\u011fine damgas\u0131n\u0131 vurmu\u015f, sonras\u0131nda da devrim etkisi yaratm\u0131\u015ft\u0131r. Sonsuzluk art\u0131k g\u00f6z\u00fcm\u00fcz\u00fcn \u00f6n\u00fcndedir, \u00fcstelik derecelendirilmi\u015f olarak.<\/p>\n

Kaynak<\/strong><\/p>\n

1) Bellos, A, Alex Say\u0131lar Diyar\u0131nda<\/em>, \u00c7ev. K\u00f6ksal G\u00fclerkaya, Pegasus Yay\u0131nevi, 2012.<\/p>\n","protected":false},"excerpt":{"rendered":"

Bu yaz\u0131da efsanevi Alman matematik\u00e7i David Hilbert taraf\u0131ndan kurgulanm\u0131\u015f, pop\u00fcler matematik merakl\u0131lar\u0131nca iyi bilinen bir hik\u00e2yeden s\u00f6z edece\u011fiz. Hilbert bu hik\u00e2yeyi, vatanda\u015f\u0131 Georg Cantor\u2019un ke\u015ffetti\u011fi, matematik d\u00fcnyas\u0131n\u0131 derinden sarsm\u0131\u015f sonsuz k\u00fcmeler kuram\u0131na giri\u015f yapabilmek i\u00e7in girdi\u011fi derslerde anlat\u0131rm\u0131\u015f. Belki \u00e7o\u011fu okura g\u00f6re \u201ctekrar su\u00e7u\u201d i\u015fleyece\u011fim, ama bu m\u00fckemmel hik\u00e2yeden habersiz matematik merakl\u0131s\u0131 bir\u00e7ok gen\u00e7le kar\u015f\u0131la\u015f\u0131yorum. […]<\/p>\n","protected":false},"author":375,"featured_media":26544,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[159,25,514],"tags":[208,1530],"class_list":["post-26542","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-122-sayi","category-matematik","category-matematik-sohbetleri","tag-matematik","tag-sayilar"],"acf":[],"aioseo_notices":[],"aioseo_head":"\n\t\t\n\t\n\t\n\t\n\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t