Iterated modal operators and Hilbert - Bernays derivability conditions
In this paper, the author analyses some methods for weakening the universality of the Hilbert-Bernays derivability conditions associated with the application of modal logic to the theory of proof, in particular, in the proof of Godel's second incompleteness theorem. The phenomenon of intensional non-equivalence of combinations of modal operators is considered as an explanation of the intensional character of Godel's second theorem. It is shown that the cause of the appearance of deviant concepts of consistency of a formal system is the ambiguous translation of informal mathematical concepts into a formal presentation, as well as the effects of translation of formal theories with different signatures from one to another.
Keywords
derivability conditions,
proof,
intensionality,
incompleteness,
modal operator,
условия выводимости,
доказатель-ство,
интенсиональность,
неполнота,
модальный операторAuthors
| Tselishchev Vitaly V. | Novosibirsk State University; Institute of Philosophy and Law, Siberian Branch of the Russian Academy of Sciences | leitval@gmail.com |
Всего: 1
References
Detlefsen M. Hilbert's Program: An Essay in Mathematical Instrumentalism. Dordrecht : Reidel, 1986.
Auerbach D. How to Say Things with Formalisms // Proof, Logic and Formalization / ed. M. Detlefsen. London : Routledge, 1992. P. 77-93.
Smorynski C. The Development of the Self-Reference: Lob's Theorem // Perspectives on the History of Mathematical Logic / ed. T. Drucker. Berlin : Birkhauser, 1991. P. 110-133.
Edgington D. Williamson on Iterated Attitudes // Philosophical Logic / ed. T. Smiley. Oxford : Oxford University Press, 1998. P. 135-158.
Visser A. The formalization of interpretability // Studia Logica. 1991. Vol. 50 (1). P. 81-105.
Boolos G. The Logic of Provability. Cambridge : Cambridge University Press, 1996.
Williamson T. Iterated Attitudes // Philosophical Logic / ed. T. Smiley. Oxford : Oxford University Press, 1998. P. 85-133.
Boolos G. Godel's Second Incompleteness Theorem Explained in Words of One Syllable // Boolos G. Logic, Logic, and Logic. Cambridge : Harvard University Press, 1998. P. 411-414.
Smith P. An Introduction to Godel's Theorems. Second Edition. Cambridge : Cambridge University Press, 2013.
Kreisel G. Godel's Excursion into Intuitionistic Logic // Godel Remembered / ed. P. Weingartner, L. Schmetterer. Napoli : Bibliopolis, 1987. P. 65-179.
Auerbach D. Intensionality and the Godel Theorem // Philosophical Studies. 1985. Vol. 48. P. 337-351.
Feferman S. Arithmetization of Metamathematics in a General Setting // Fundamenta Mathe-maticae. 1960. Vol. 49. P. 35-92.
Kreisel G. On the Problem of Henkin // Proceedings of Netherland Academy of Science. 1953. Vol. 56. P. 405-406.
Detlefsen M. On Interpreting Godel's Second Incompleteness Theorem // Journal of Philosophical Logic. 1979. Vol. 8. P. 297-315.
