Cantor'un Kümeler Kuramının felsefi sonuçları

This thesis is devoted to examining Georg Cantor's understanding of infinity and his philosophy of mathematics. Even though Aristotle differentiated the concept of infinity as potential infinite and actual infinite, he argued against the existence of actual infinity and accepted only the existence of potential infinity. With the effect of this distinction, the impossibility of actual infinity was regarded as the fundamental principle in the history of the concept of infinity. Cantor was the first thinker to attempt to refute Aristotle's arguments by introducing a new understanding of infinity that has one of the greatest impacts on its development in mathematics. Cantor mathematically demonstrated that there would not be any one-to-one correspondence between the set of natural numbers and the set of real numbers. This result implies that there must be at least two different sizes of infinite sets, namely the set of real numbers and the set of natural numbers. Based on the concept of a well-ordered set, Cantor not only showed the way how to count infinite sets but also assigned numbers to differentiate the different sizes of infinite sets. Thus, transfinite numbers and their arithmetic are introduced into mathematics. After examining the distinction between potential infinite and actual infinite in both Aristotle's framework and Cantor's framework, the existence of mathematical objects in the Cantorian framework will be shown. Keywords: Potential infinity, actual infinity, well-ordered sets, transfinite numbers, transfinite arithmetic