# “PM及相关系统的形式不可判定命题”（1）- 哥德尔不完全性定理的历史背景与内容 精选

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“PM及相关系统的形式不可判定命题”（1）- 哥德尔不完全性定理的历史背景与内容

“命题IX：在命题VI中言及的所有形式系统中，都存在有受限谓词演算的不可判定问题（亦即，受限谓词演算的逻辑式，其普遍有效性以及其反例的存在性都不可证）。”

“命题XI：如果c是一个给定的递归且一致的逻辑式类，则表达“c是一致的”之内容的命题逻辑式不是c-可证的；特别地，P的一致性在P中不可证，在假设P是一致的前提下（如果不是，那么当然，任何言明都是可证的）。”

[1] K. Gödel, “über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I,” Monatshefte für Mathematik Physik, Vol. 38, pp. 173–198, 1931. (The summary of the results of this work, published in Anzeiger der Akad. D. Wiss. In Wien (math.-naturw. Kl.) 1930, No. 19.)
English Translation: B. Meltzer (Translation) and R. B. Braithwaite (Introduction), K. Gödel, “On formally undecidable propositions of Principia Mathematica and related systems I,” Basic Books, 1962, Dover Publications, 1992.

[2] K. Gödel (1930b, 1931, and 1932b), “Some metamathematical results on completeness and consistency, On formally undecidable propositions of Principia Mathematica and related systems I, and On completeness and consistency,” in J. Van Heijenoort (Translation, Ed.), “Frege and Gödel: Two Fundamental Texts in Mathematical Logic,” pp. 83-108, Harvard University Press, 1970.

[3] K. Gödel, “über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I,” “On formally undecidable propositions of Principia Mathematica and related systems I,” Repringted and translated (by J. Van Heijenoort) in S. Feferman, et al. (Eds.), “Kurt Gödel: Collected Works,” Volume I, Publications 1929-1936, pp. 144-195, Oxford University Press, 1986.

[4] D. Hilbert, “Grundlagen der Geometrie,” Teubner, 1899 (in German).

[5] D. Hilbert, “Mathematische Probleme,” Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Math.-Phys. Klasse, pp. 253–297，1900 (in German) (Lecture Delivered before the International Congress of Mathematicians at Paris in 1900)

[6] D. Hilbert,“über die Grundlagen der Logik und der Arithmetik”, in Verhandlungen des dritten Internationalen Mathematiker-Kongresses in Heidelberg, 1904 (in German).

[7] 王宪钧, “数理逻辑引论 第三篇 数理逻辑发展简述”，北京大学出版社，1982, 1998.

[8] R. Zach, “Hilbert’s Program,” Stanford Encyclopedia of Philosophy, 2003,2019.

[9] K. Gödel, “über die Vollständigkeit des Logikkalküls,” “On the completeness of the calculus of logic,”Doctoral Dissertation, University of Vienna, 1929, Repringted and translated in S. Feferman, et al. (Eds.), “Kurt Gödel: Collected Works,” Volume I, Publications 1929-1936, pp. 60-101, Oxford University Press, 1986.

[10] J. Dawson, “Logical Dilemmas: The Life and Work of Kurt Gödel,”  A. K. Peters, 1997, 2005. 中译(唐璐译):“哥德尔：逻辑的困境”，2009.

[11] M. Davis, “Engines of Logic – Mathematics and the Origin of the Computer,” W.W. Norton, 2000. 中译(张卜天译):“逻辑的引擎”，2005.

[12] A. N. Whitehead and B. A. W. Russell, “Principia Mathematica,” Vol.1, 1910(1 ed.), 1925(2 ed.), Vol.2, 1912(1 ed.), 1927(2 ed.), Vol.3, 1913(1 ed.), 1927(2 ed.), Cambridge University Press.

[13] B. Linsky, “Principia Mathematica,” Stanford Encyclopedia of Philosophy, 1996,2021.

[14] J. B. Rosser, “Extensions of some theorems of Gödel and Church,” Journal of Symbolic Logic， Vol. 1，No. 3, pp. 87-91, 1936.

[15] E. Mendelson, “Introduction to Mathematical Logic,” Chapman & Hall, 2015 (6th Edition).

[16] D. Hilbert and P. Bernays,“Grundlagen der Mathematik”, Vol. 1, 1934, Vol. 2, 1939 (in German).

[17] 程京德，“哥德尔不完全性定理的原始陈述”，微信公众号“数理逻辑与哲学逻辑”，2023年3月13日。

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