Practical information You can find the takehome exam here. Please submit your answers by email.
Thursdays, from 2pm to 4pm (s.t., includes practice sessions) Office hours by appointment, in room 234, Ludwigstrasse 31, second floor
You can find the book here. Here's a cool video. Very instructive! (Credits to Nico González Albornoz)
Course description
The course consists in a detailed exposition of Gödel's first incompleteness theorem and its proof. The result establishes that any theory extending elementary arithmetic is incomplete, this is, there will always be a sentence in the language of the theory that the theory neither proves nor refutes. The exposition is carried out in two steps. First, an introduction to firstorder arithmetic is given. Students are presented with the following: firstorder axiomatic theories, the language of firstorder arithmetic, L, firstorder theories of arithmetic Q (Robinson Arithmetic) and PA (for "Peano Arithmetic"), the enumerability and denumerability of sets, the recursivity, semirecursivity, definability in L, and representability in an axiomatic theory formulated in L, of a subset of natural numbers or a function from natural numbers to natural numbers, the standard model of L, arithmetical truth, the arithmetical hierarchy, and (primitive) recursive functions. The second half of the course consists of a presentation of a modern version of Gödel's proof, followed by a reflection on the philosophical consequences of the incompleteness result. Students are introduced to Q's Sigma1completeness, Gödel's coding technique and the arithmetisation of syntax, the diagonalisation lemmata, and Gödel sentences.
Course material
Contents (preliminary)
1. Introduction: What Gödel's Theorems Say12/4 Read: Smith's (2013), Chapter 1.
2. Functions and Enumerations19/4 Read: Smith's (2013), Chapter 2. Exercises: about functions and enumerability (except exercise 5).
3. Effective Computability and Axiomatic Theories: Axiomatisability and Decidability 26/4 Read: Smith's (2013), Chapters 3 and 4.
Exercises: about effective computability (except exercise 5) and formal theories.
4. Expressing and Capturing Numerical Properties
3/5 Read: Smith's (2013), Chapter 5. Exercises: about the language of arithmetic.
5. Subsystems of Firstorder Peano Arithmetic PA
17/5  7/6 Read: Smith's (2013), Chapters 9, 10, 11, 12, and 13. Exercise: read and prepare a summary of chapters 6, 7, and 8 to briefly explain their content in class (note that there are two bank holidays). Exercises: about sufficiently expressive/strong theories, induction, and Robinson Arithmetic. Also, try to prove that BA captures the identity relation, that it is negationcomplete, and principles O1O9 from sec. 11.3 (this homework was given in class). Bonus: evaluate whether Q proves this formula. Exercises: prove Theorems 11.6, 12.1, and 12.3.
6. Primitive Recursive Functions
14/6  21/6
Read: Smith's (2013), Chapter 14. Exercises: complete the proof of claims AE in sec. 14.7, and (iv) examples 16 in sec. 14.8.
7. Capturing Primitive Recursive Functions
25/6
Read: Smith (2013), Chapters 1517.
8. The Arithmetization of Syntax 28/6  5/7
Read: Smith (2013), Chapters 19 and 20.
9. Gödel's First Incompleteness Theorem 12/7 Read: Smith (2013), Chapters 2124.
Preparing for the course An informal introduction to Gödel's Incompleteness Theorems that could be useful to read before the before the course begins: Nagel & Newman's Gödel's Proof
 A more formal introduction can be found here: Frazén's Gödel’s Theorem: An Incomplete Guide to its Use and Abuse
 Here's a funny video (those versed in German turn the sound off)
Bibliography  Boolos, G., Burgess, J. & Jeffrey, R. (2007) Computability and Logic, Cambridge University Press, fifth edition.
Franzén, T. (2005) Gödel’s Theorem: An Incomplete Guide to its Use and Abuse, A. K. Peters.  Gödel, K. (1931) "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I" ("On Formally Undecidable Propositions of Principia Mathematica and Related Systems"), Monatshefte für Mathematik und Physik 38: 173–198. Reprinted in Gödel (1986), pp. 144–195.
 ——(1986) Collected Works. I: Publications 1929–1936, S. Feferman, S. Kleene, G. Moore, R. Solovay, and J. van Heijenoort (eds.), Oxford University Press.
 Nagel, E. & Newman, J. R. (2001) Gödel's Proof, New York University Press, revised edition.
 Smith, P. (2013) An Introduction to Gödel's Theorems, second edition, Cambridge University Press.
