집합과 수리논리

Sets and Mathematical Logic

(교과목 번호: 881.313,   학점형태: 3-3-0)

Je le vois, mais je ne le crois pas! (보고 있지만, 믿을 수가 없답니다…), G. Cantor, 1877.

2013년 1학기

중간고사: 4월 22일(월)     기말고사: 6월 12일(수)

과제 1   강의록 2-3   과제 2  

수업 목표

강의 계획

(Updated: Feb. 25, 2013)

• Week 1: Introduction & Historical Remarks
• Week 2: Naive Set Theory
• Week 3: Axiomatic Set Theory
• Week 4: Relations and Functions
• Week 5: Natural Numbers and Mathematical Induction
• Week 6: Peano Arithmetic, Formal Theory
• Week 7: Logic (Propositional Logic, First Order Logic)
• Week 8: 중간고사 (4월 22일, 월)
• Week 9: Ordinal Numbers
• Week 10: Cardinal Numbers
• Week 11: Axiom of Choice
• Week 12: Geometry, Real Numbers
• Week 13: Turing Machine, P vs. NP
• Week 14: Model Theory, Categories
• Week 15: 기말고사 (6월 12일, 수)

평가 기준

교재라기보다 참고문헌

• K. Devlin, The Joy of Sets, Fundamentals of Contemporary Set Theory, 2nd Ed., Springer, 1979, 1993.
• K. Hrbacek, T. Jech, Introduction to Set Theory, 3rd Ed., CRC press, 1978, 1999.

더 참고문헌

• P. J. Cameron, Sets, Logic and Categories, Springer SUMS, 1998, 2000. (http://www.maths.qmul.ac.uk/~pjc/slc/) 
• H. Enderton, Elements of Set Theory, Academic Press, Inc., 1977.
• E. Mendelson, Introduction to Mathematical Logic, Chapman & Hall/ CRC, 1964, 1997.
• Y. Moschovakis, Notes on Set Theory, 2nd Ed., Springer UTM, 1993, 2006. 
• 김명환, 김홍종, 현대수학입문: 힐베르트 문제를 중심으로, 경문사, 2000. 
• K. Podieks, http://www.ltn.lv/~podnieks/gt.html   

더 읽을거리

• T. Gowers, Ed., The Princeton Companion to Mathematics, Princeton Univ. Press, 2008. 
• G. Priest, LOGIC: A Very Short Introduction, Oxford Univ. Press, 2000.
• A. Doxiadis, C. Papadimitriou, A. Papadatos, A. di Donna, Logicomix: An Epic Search for Truth, 2009. (로지코믹스, 전대호 옮김, RHK, 2011)
• Lewis Carroll (Charles Lutwidge Dodgson), Alice's Adventures in Wonderland, 1865; Through the Looking-Glass, and What Alice Found There, 1871. (cf. M. Gardner, The Annotated Alice: The Definitive Edition, W.W. Norton & Company, 1999; ISBN 0-393-04847-0. 최인자 옮김, 마틴 가드너의 앨리스 깊이 읽기 - 이상한 나라의 앨리스, 거울 나라의 앨리스, 북폴리오, 2005)

• 더더 읽을거리
  • 김종명, 수리철학의 변화와 수학교육관, 한국수학사학회지 14 (2001), 83– 100.
  • 이승온, 김혁수, 박진원, 이병식, 직관주의 논리, 한국수학사학회지 12 (1999), 32–44.
  • 정계섭, 수학과 실제, 한국수학사학회지 14 (2001), 47–60.
  • 정주희, 수리 논리학, 대한수학회 소식 제 64 호 (1999 년 3 월), 23–29.
  • 정주희, 수리논리와 집합론 입문, 경문사, 2012.
  • P. Benacerraf, H. Putnam, Eds., Philosophy of Mathematics, 2nd Ed., 1964, 1983. (박세희 옮김, 수학의 철학, 아카넷, 2002.)
  • G. Birkhoff, S. Mac Lane, A Survey of Modern Algebra, Macmillan, 1946.
  • L. E. J. Brouwer, Intuitionism and formalism, Bull. Amer. Math. Soc. (New Series) 37 (2000), 55–64. (Reprinted from Bull. Amer. Math. Soc. 20 (1913), 81–96.)
  • N. Bourbaki, Theory of Sets, Hermann, 1968.
  • G. Cantor, Contribution to the foundation of the theory of Transfinite Numbers, Dover, 1955.
  • J. Casti, Five Golden Rules, John Wiley and Sons, Inc., 1996.
  • G. Chaitin, The Unknowable, http://www.umcs.maine.edu/~chaitin/ unknowable/. (Book Review: P. Raatikainen, Exploring Randomness and The Unknowable, Notices of the Amer. Math. Soc., 48 (2001), 992–996.)
  • J.-P. Changeux, A. Connes, Conversations on Mind, Matter, and Mathematics, Princeton Univ. Press, 1989.
  • P. Cohen, Set Theory and the Continuum Hypothesis, W. A. Benjamin, Inc., 1966.
  • A. Connes, A. Lichnerowicz, M. P. Schüt zenberger, Triangle of Thoughts, Amer. Math. Soc. 2001.
  • J. H. Conway, R. K. Guy, The book of Numbers, Copernicus, 1996.
  • H. Coxeter, Introduction to GEOMETRY, John Wiley and Sons, Inc., 1961, 1969.
  • M. Davis, The universal computer, W. W. Norton & Company, 2000. (Book Review : Logical Dilemas : The Life and Work of Kurt Gödel and Gödel : A Life of Logic, Notices Amer. Math. Soc. 48 (2001), 807–813.)
  • J. Dawson, Jr., Gödel and the Limits of Logic, Scientific American, June 1999, 68–73.
  • K. Devlin, Mathematics: The New Golden Age, Penguin Group Ltd., 1988 (허민 옮김, 수학: 새로운 황금시대, 경문사, 1995, 1999)
  • J. Dieudonné, A panorama of pure Mathematics, Academic Press, 1982.
  • ___, Mathematics—The Music of Reason, Springer-Verlag, 1992.
  • H. Eves, An Introduction to the History of Mathematics, Sixth ed., Saunders College Publishing, 1990. (이우영, 신항균 역, 경문사, 1996.)
  • R. Feynman, Lectures on Computation, Addison Wesley, 1996.
  • M. Gardner, “Nothing”, “More Ado About Nothing”, “Everything”, Mathematical Magic Show, Math. Assoc. Amer., 1989.
  • ___, Mathematical Carnival, Math. Assoc. Amer., 1989.
  • ___, Penrose Tiles to Trapdoor Ciphers . . . , Math. Assoc. Amer., 1989.
  • L. Gillman, Two Classical Surprise Concerning the Axiom of Choice and the Continuum Hypothesis, Monthly 109 (2002), Math. Assoc. Amer., 544--553. ☆☆☆
  • R. Goodstein, On the restricted ordinal theorem, J. Symbolic Logic 9 (1944), 33–41.
  • P. Halmos, Naive Set Theory, Van Nostrand Reinhold Company, 1960.
  • H. Herrlich, G. Strecker, Category Theory, Allyn and Bacon Inc., Boston, 1973.
  • D. R. Hofstadter, Gödel, Escher, Bach : An Eternal Golden Braid, Basic Books Inc., 1979.
  • R. Honsberger, Ingenuity in Mathematics, Math. Assoc. Amer., 1970.
  • E. Kamke, Theory of Sets, Dover, 1950.
  • I. Kaplanski, Set Theory and Metric Spaces, Allyn and Bacon, 1972.
  • J. Kelley, General Topology, Van Nostrand-Reinhold Company, 1955. (Appendix).
  • L. Kirby, J. Paris, Accessible independence results for Peano arithmetic, Bull. London Math. Soc. 14 (1982), 285–93.
  • V. Klee, S. Wagon, Old and New UNSOLVED PROBLEMS in Plane Geometry and Number Theory, Math. Assoc. Amer., 1991.
  • E. Kramer, The Nature and Growth of Modern Mathematics, Princeton Univ. Press, 1970.
  • P. Maddy, Naturalism in Mathematics, Clarendon Press, 1997.
  • E. Mendelson, Introduction to Mathematical Logic, Chapman & Hall/ CRC, 1964, 1997.
  • R. Penrose, The Emperor’s New Mind, Oxford Univ. Press, 1989. , Shadows of the Mind, Oxford Univ. Press, 1994.
  • R. Penrose, M. Longair, The large, the small and the human mind, Cambridge Univ. Press, 2000.
  • C. Pinter, Set Theory, Addison-Wesley Publishing Co. 1971.
  • K. Podnieks, Around Gödel’s Theorem, http://www.ltn.lv/~podnieks/
  • H. Poincaré, Science et methode, Paris, 1908. , Dernierés Pensés, 1913. (김성숙, 김형보 옮김, 포앙카레의 만년의 사상–수학과 과학, 교우사, 2002.)
  • E. Regis, Who Got Einstein’s Office, Addison-Wesley, 1987.
  • C. Reid, Hilbert, 1996. (이일해 옮김, 현대 수학의 아버지 힐베르트, 사이언스북스, 2005.)
  • R. Rucker, Infinity and the Mind, Princeton Univ. Press, 1995.
  • D. Schnabel, Review of ‘Goodbye, Descartes’ by K. Devlin, Amer. Math. Monthly 105 (1998), 870–873.
  • M. Spivak, A comprehensive introduction to DIFFERENTIAL GEOMETRY, Vol. I, Publish or Perish, Inc., Berkeley, 1979. (Appendix)
  • R. Smullyan, The Tao is Silent, 1977.
  • ___, What Is the Name of This Book? The Riddle of Dracula and Other Logical Puzzles, 1978.
  • ___, Gödel’s Incompleteness Theorems, 1992.
  • ___, Set Theory and the Continuum Problem, 1996.
  • B. L. Van der Waerden, Algebra 1, Frederick Ungar Publishing Co., Inc., 1970.
  • W. Woodin, The Continuum Hypothesis, Part I, Notices Amer. Math. Soc., 48 (2001), 567–576; Part II, 681–690.