Week-long Tutorial: Gödel's Incompleteness Theorems
5th UniLog Summer School
The goal of this tutorial is to introduce, review, and discuss selected concepts that play a key role in Gödel's two theorems on the incompleteness of consistent formal systems of arithmetic (= G1) and their inability to prove their own consistency (= G2). Each lecture will fall into two parts. A first shorter part will briefly review the traditional text book approach to Gödel's theorems and be accessible to everyone with a modest background in logic. As such, these parts double as either a self-contained introduction to or a refresher course in the incompleteness theorems and their proofs. A subsequent second part of each lecture will then introduce and review more recent and more advanced work pertinent to the lecture's topic (as indicated by its title). As such, the second part will require greater fluency in the language and the techniques of mathematical logic.
Lecture 1: Diagonalization, self-reference, and paradox. We start out with Gödel's proof for the fixed-point theorem and end with certain generalization in the framework of category theory.
Lecture 2: Models: weak and non-standard. We start out with some basic model-theoretic considerations of incompleteness and end with the role of cuts for proving G2.
Lecture 3: Provable closures and interpretability. We start out with the roles provable closure under modus ponens and provable Σ1-completeness play for proving G2 and end with the question of whether G2 should better be framed in terms of interpretability of theories.
In the spirit of universal logic, some mention of non-classical alternatives will be made throughout the lectures.
Background readings (free internet resources only):
1. Peter Smith, Gödel without (too many) tears.
2. Petr Hájek & Pavel Pudlák, Metamathematics of First-Order Arithmetic.
3. Per Lindström, Aspects of Incompleteness.
Tutorial readings (will be made available in June 2015): http://users.ipfw.edu/buldtb/goedel/index.html.
Bernd Buldt (2015).
Week-long Tutorial: Gödel's Incompleteness Theorems. Presented at 5th UniLog Summer School, Istanbul.