Limitative theorems in logic and their philosophical consequences 3501-M49-10
Course of lectures “Limitative Theorems in logic and their philosophical consequences” will focus on the most important metalogical theorems, namely on the famous Gödel’s incompleteness theorems as well as Tarski’s theorem and Loewenheim-Skolem theorem. Their background, principal mathematical techniques, and sketches proofs will be presented. The consequences of the theorem will be studied – for the philosophy of mathematics, and more widely, those indicating limitations of formal methods, of our descriptions of the world, as well as corollaries affecting the philosophy of mind. Also abuses and misapplications of the theorems will be discussed, especially those committed by the philosophers who seem to think that Gödel’s theorem is just a metaphor that can be freely applied.
Much of the course will follow the book (in Polish) by S. Krajewski Gödel’s Theorem and its Philosophical Consequences, IFiSPAN, Warsaw 2003.
More specifically the following topic will be covered:
The background of Gödel’s and Tarski’s theorems
Formalized theories, arithmetization, representability, recursive functions
Models, Loewenheim-Skolem therem, Solem’s paradox
Proofs of Gödel’s theorem
Reception
Consequences of metalogical theorems for Mechanism
Gödel’s Alternative
Philosophical applications of Gödel’s theorems
Metalogic and Postmodernism
Inspirations: limitations of classical paradigm ?
Type of course
Bibliography
1. Stanisław Krajewski Twierdzenie Gödla i jego konsekwencje filozoficzne: od mechanicyzmu do postmodernizmu, IFiSPAN Warszawa 2003.
2. Ernest Nagel, James R. Newman, Gödel’s Proof, 1958. (Twierdzenie Gödla, PWN, Warszawa 1966.)
3. Roger Penrose, Shadows of the mind, Oxford: Oxford University Press 1994. (Cienie umysłu, Zysk i S-ka, Poznań 2000.)
Additional
4. Epstein, Richard L. i Walter A. Carnielli Computability. Computable Functions, Logic, and the Foundations of Mathematics, Wadsworth & Brooks, Pacific Grove 1989
5. Stewart Shapiro, The limits of logic: higher-order logic and the Loewenheim-Skolem theorem, Darmouth Publishing Company, Aldershot 1996.
6. Peter Smith, An Introduction to Gödel’s Theorems, Cambridge University Press 2007.
Supplementary
7. Kurt Gödel: Collected Works, Volume I: Publications 1929-1936, Oxford University Press, New York, Oxford 1986; Volume III, Oxford University Press, 1995.
8. Douglas R. Hofstadter, Gödel, Escher, Bach, an Eternal Golden Braid, Basic Books, NY 1979.
9. Raymond M. Smullyan, Gödel’s Incompleteness Theorems, Oxford University Press, New York, Oxford 1992.
10. Torkel Franzén, Gödel’s Theorem: An Incomplete Guide to its Use and Abuse, A K Peters 2005.
Additional information
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