한국수학사학회지 (Journal for History of Mathematics)

한국수학사학회 (The Korean Society for History of Mathematics)
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A reconstruction of the G$\ddot{o}$del's proof of the consistency of GCH and AC with the axioms of Zermelo-Fraenkel set theory

  • 최창순 (전북대학교 윤리교육과)
  • Choi, Chang-Soon (Department of Ethics Education, Chonbuk National University)
  • 투고 : 2011.04.07
  • 심사 : 2011.07.25
  • 발행 : 2011.08.31
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초록

NBG의 공리들을 충족시키는 모델로서의 집합 V 를 도입하고 그것의 요소들을 sets라 부르고 그것의 부분집합들을 classes라 부른다. 일반연속체가설 (GCH) 와 선택공리 (AC) 가 ZF 집합론과 무모순이라는 것에 대한 괴델의 증명을 그 이후 나온 Mostowski-Shepherdson mapping 정리, Tarski-Vaught 정리 및 Montague-Levy 정리의 반사원리들, NBG가 ZF의 보존적 확장이라는 정리 등을 이용하여 재구성해 본다.

Starting from a collection V as a model which satisfies the axioms of NBG, we call the elements of V as sets and the subcollections of V as classes. We reconstruct the G$\ddot{o}$del's proof of the consistency of GCH and AC with the axioms of Zermelo-Fraenkel set theory by using Mostowski-Shepherdson mapping theorem, reflection principles in Tarski-Vaught theorem and Montague-Levy theorem and the fact that NBG is a conservative extension of ZF.

키워드

참고문헌

  1. Devlin, K. (1984), Constructibility, Springer-Verlag.
  2. Flannagan, T. B. (1976),"A new finitary proof of a theorem of Mostowski"in Sets and Classes, edited by Muller, G. H., North Holland Publishing Company.
  3. Godel, K. (1939), "Consistency proof for the generalized continuum hypothesis", Proceedings of the National Academy of Sciences, U.S.A. 25, 220-224. https://doi.org/10.1073/pnas.25.4.220
  4. Godel, K. (1940), The consistency proof of the axiom of choice and of the generalized continuum hypothesis with the axioms of set theory, Princeton UP.
  5. Godel, K. (1944),"Russell's mathematical logic", in Schilpp, Paul A.(ed) The philosophy of Bertrand Russell 3rd ed. (1951), 123-153.
  6. Godel, K. (1947),"What is Cantor's continuum problem?", American Mathematical Monthly 54, 515-525. https://doi.org/10.2307/2304666
  7. Jech, T. (2002), Set Theory, Springer-Verlag.
  8. Mendelson, E. (2010), Introduction to Mathematical Logic, CRC Press.
  9. Montague, R. (1961),"Semantic closure and non-finite axiomatizability", Infinitistic Methods, Pergamon, 45-69.
  10. Levy, A. (1979), Basic set theory, Springer-Verlag.
  11. Sierpinski, W.(1947),"L'hypothese generalisee du continu et l'axiome du choix", Fundamenta Mathematicae 34, 1-5. https://doi.org/10.4064/fm-34-1-1-5
  12. Shoenfield, J. R. (1967), Mathematical Logic, Addition-Wesley.
  13. Wang, H. (1974), From mathematics to philosophy, Humanities Press.