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A historical study of de Zolt's axiom

졸트 공리의 역사적 고찰

  • Jo, Kyeonghee (Division of Liberal Arts and Sciences, Mokpo National Maritime Univ.)
  • Received : 2017.04.09
  • Accepted : 2017.09.30
  • Published : 2017.10.31

Abstract

De Zolt's axiom which is a precise formulation of Euclid's Common Notion 5, "the whole is greater than the part", for the notion of 'content' holds in any Hilbert plane. In this article, we study the history of de Zolt's axiom which has its origin in Euclid's Common Notions, and introduce an example of a plane geometry in which de Zolt's axiom does not hold. We show that there is no area function in this geometry and every square is equidecomposable with a square which is properly contained in the first one. From this we also show that there are two equidecomposable rectangles which have the same base and do not have the same altitude, and there is a rectangle which is equicomplementable with an emptyset.

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