Most of this book is about positive results: certain logical problems can in principle be automated. Here we consider the limits of automation, showing that algorithms in the usual sense cannot exist for certain logical problems. In particular we show that pure first-order logic is not decidable, and that the theory of natural numbers with addition and multiplication is, in a precise sense, nowhere near decidable. In making our way to these results, we prove Gödel's famous first incompleteness theorem.

Hilbert's programme

The idea of mechanizing reasoning fascinated people long before computers. Specific questions about the scope and limits of mechanization were investigated systematically in the early part of the twentieth century, largely due to the influence of Hilbert's programme to place mathematics on firm foundations. To appreciate the full cultural significance of the results that follow, it's worth examining the intellectual ferment over the foundations of mathematics that made these questions so significant at the time.

At various points in history, mathematicians have become concerned over apparent problems in the accepted foundations of their subject. For example, the Pythagoreans tried to base mathematics just on the rational numbers, and so were disturbed by the discovery that √2 must be irrational. Subsequently, the apparently self-contradictory treatment of infinitesimals in Newton and Leibniz's calculus disturbed many (Berkeley 1734), as later did the use of complex numbers and the discovery of non-Euclidean geometries. Later still, when the theory of infinite sets began to be pursued for its own sake and generalized, mainly by Cantor, renewed foundational worries appeared.