 
Summary: Review of Basic Proof Theory (second edition),
by A. S. Troelstra and H. Schwichtenberg
Jeremy Avigad
January 17, 2001
1 Overview
Beweistheorie, or "Proof Theory," was the phrase that David Hilbert used to de
scribe the program by which he hoped to secure the foundations of mathematics.
Set forth in the early 1920's, his plan was to represent mathematical reason
ing by formal deductive systems, and show, using safe, "finitary," methods, that
such reasoning could never lead to contradiction. This particular goal was shown
by GĻodel to be infeasible. But the more general goal of using formal methods to
explore various aspects of mathematical provability, including the relationship
between classical and constructive methods in mathematics and the strengths
and limitations of various axiomatic frameworks, has proved to be remarkably
robust. Today, these goals represent the traditional, metamathematical branch
of proof theory.
Since Hilbert's time, the subject has expanded in two important respects.
First, it has moved well beyond the study of specifically mathematical reasoning.
Proof theorists now consider a wide range of deductive systems, designed to
model diverse aspects of logical inference; for example, systems of modal logic
