Philosophy 1885: Syllabus
If you wish, you can download the syllabus as a PDF. Please note, however, that the syllabus is approximate. We will proceed at whatever pace works for this particular group, and if enough of us get interested in some topic not on the particular route we are taking, we may digress for a bit. The PDF version of the syllabus will probably not be kept up to date. You should check this website for updates as we proceed.
The forms in which the readings are available are explained
- 25 January
- 27 January–15 February
Richard Heck, "Formal Background for the Incompleteness and Undefinability Theorems" (PDF).
This will be review for some, but for the rest it will get us a general sense for what Gödel's theorem says and how it is proved.
If you're having a hard time understanding the diagonal lemma, then read this informal account, as well.
- 22 February–8 March
Kurt Gödel, "On Formally Undecidable Propositions of Principia Mathematica and Related Systems I", in Collected Works v. 1, ed. S. Feferman, J. Dawson, and S. Kleene (Oxford: Oxford University Press, 1986), pp. 144-95. (DJVU)
Only the odd pages are in the DJVU; the even pages from this edition are in German. There are also inexpensive books that have this paper in them.
There are some reading notes to help you get through the first couple sections.
- 10–24 March
Alfred Tarski, Andrzej Mostowski, and Raphael Robinson, Undecidable Theories, chapters I and II.
- For Chapter I, focus on sections I.4 and I.5. You will need to read the earlier sections for terminology, but you do not need to worry about all the details. We will not discuss section I.6, so you can skip or skim it, if you wish.
- We will discuss most of Chapter II, but you can again skip or skim section II.6.
Optional: Julia Robinson, "General Recursive Functions", Proceedings of the American Mathematical Society
1 (1950), pp. 703-18
James P. Jones and John C. Shepardson, "Variants of Robinson's Essentially Undecidable Theory R", Archiv for mathematische Logik
23 (1983), pp. 61-4.
- 7 April–14 April
George Boolos, The Logic of Provability (Cambridge: Cambridge University Press, 1993, Ch. 2. (DJVU)
Optional: Martin Löb, "Solution of a Problem of Leon Henkin", Journal of Symbolic Logic
20 (1955), pp. 115-8. (JSTOR
Robert G. Jeroslow, "Redundancies in the Hilbert-Bernays Derivability Conditions for Gödel's Second Incompleteness Theorem", Journal of Symbolic Logic
38 (1973), pp. 359-67. (JSTOR
I encourage everyone to read the former paper, which is not too difficult. The latter is recommended only for those who are otherwise having an easy time with this material and are looking for a challenge.
- 17 April–??
Solomon Feferman, "Arithmetization of Metamathematics in a General Setting", Fundamenta Mathematicae 49 (1960), pp. 35-92. (PDF)
- 17 May, 2pm
Final Exam Due
The following papers are ones we probably will not have time to read, but I will list them here for students who wish to pursue these issues on their own.
- Alex Wilkie and Jeff Paris, "On the Scheme of Induction for Bounded Arithmetic Formulas", Annals of Pure and Applied Logic 35 (1987), pp. 261-302. (Science Direct, DJVU)
- Pavel Pudlák, "Cuts, Consistency Statements and Interpretations", Journal of Symbolic Logic 50 (1985), pp. 423-41. (JSTOR, DJVU)
- Andrzej Grzegorczyk, "Undecidability Without Arithmetization", Studia Logica 79 (2005), pp. 163-230. (JSTOR, DJVU)
- Albert Visser, "Can We Make the Second Incompleteness Theorem Coordinate Free?" Journal of Logic and Computation 21 (2009), pp. 543-60. (Oxford Journals, DJVU)