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Zermelo and the Axiomatic Method

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한국논리학회 (Korean Association For Logic)|2008.08|논리연구 (Korean Journal of Logic)|제11권 제2호|pp. 1-57

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키워드

공리적 방법제르멜로힐버트토대의 심화암묵적 정의집합 개념axiomatic methodZermeloHilbertdeepening the foundationsimplicit definitionthe concept of set
주제분류철학

초록

본 논문은 제르멜로가 집합론을 공리화함에 있어서 힐버트의 공리적 방법을 차용하였다는 널리 무비판적으로 받아들여져 온 가정 자체를 검토하고자 한다. 그들이 공유했다고 가정되는 공리적 방법의 실체가 무엇이고 그것은 과연 어느 시기에 정립된 것인지를 묻는 데서 출발해서 공리적 방법에 관한 제르멜로와 힐버트의 사상이 어떻게 상호작용하며 발전해 나갔는지를 철학적 반성을 통해 규명하려는 것이다. 그 결과 후기 사상에서뿐만 아니라 심지어 전기 사상에 있어서도 제르멜로가 집합론 자체와 공리적 방법에 관하여 힐버트와 상당히 다른 견해를 지녔을 가능성을 확인할 것이다. 이러한 결과는 집합론의 역사와 공리적 방법의 역사, 그리고 나아가서 수학철학 전반에 걸쳐 상당한 함축을 지닐 수밖에 없다고 본다.

Abstract

This article intends to examine the widespread assumption, which has been uncritically accepted, that Zermelo simply adopted Hilbert’s axiomatic method in his axiomatization of set theory. What is essential in that shared axiomatic method? And, exactly when was it established? By philosophical reflection on these questions, we are to uncover how Zermelo’s thought and Hilbert’s thought on the axiomatic method were developed interacting each other. As a consequence, we will note the possibility that Zermelo, in his early as well as late thought, had views about the axiomatic method entirely different from that of Hilbert. Such a result must have far-reaching implications to the history of set theory and the axiomatic method, thereby to the philosophy of mathematics in general.

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