Introduction to Modern Metamathematics

Andrey Bovykin

COPPE
Federal University of Rio de Janeiro, Brazil

This tutorial will consist of three lectures on metamathematics. Metamathematics is the study of what is possible and what is impossible in mathematics, the study of unprovability, algorithmic undecidability, limits of mathematical methods and "truth".

I will go through the history of metamathematics and explain what people believed in different eras of the history of metamathematics: pre-Godelean history, and at various stages of post-Godelean history.

I will start with four old scenarios that a metamathematical result may follow ("Parallel Worlds", "Insufficient Instruments", "Absence of a Uniform Solution" and "Needed Objects Don't Yet Exist").

Then I will talk about Godel's theorems and modern developments: the Paris-Harrington Principle, Harvey Friedman's machinery, Andreas Weiermann's Phase Transition Programme and my own recent results, some joint with Michiel De Smet and Zachiri McKenzie. I will give some sketches of proofs but will not overload the lectures with technical details. Instead I will concentrate on new and old ideas in modern unprovability theory and explanations about the methods of finding and proving unprovability.

Here are some important questions that will guide us throughout this tutorial. How does one prove that something is unprovable? What are the reasons for unprovability? What are the sources of unprovability in mathematics? Is it possible to argue in one way and get the answer "yes" to a mathematical question and then reason in another, incompatible way and get the answer "no"?

I am planning to make these lectures very simple and accessible to the widest possible audience.

Bibliography