The Hilbert program

Please enjoy this free excerpt from Lectures on the Philosophy of Mathematics. This essay appears in chapter 7, focused on Gödel’s incompleteness theorems.

Mathematical logic, as a subject, truly comes of age with Kurt Gödel's incompleteness theorems, which show that for every sufficiently strong formal system in mathematics, there will be true statements that are not provable in that system, and furthermore, in particular, no such system can prove its own consistency. The theorems are technically sophisticated while also engaged simultaneously with deeply philosophical issues concerning the fundamental nature and limitations of mathematical reasoning. Such a fusion of mathematical sophistication with philosophical concerns has become characteristic of the subject of mathematical logic—I find it one of the great pleasures of the subject. The incompleteness phenomenon identified by Gödel is now a core consideration in essentially all serious contemporary understanding of mathematical foundations.

In order to appreciate the significance of his achievement, let us try to imagine mathematical life and the philosophy of mathematics prior to Gödel. Placing ourselves in that time, what would have been our hopes and goals in the foundations of mathematics? By the early part of the twentieth century, the rigorous axiomatic method in mathematics had found enormous success, helping to clarify mathematical ideas in diverse mathematical subjects, from geometry to analysis to algebra. We might naturally have had the goal (or at least the hope) of completing this process, to find a complete axiomatization of the most fundamental truths of mathematics. Perhaps we would have hoped to discover the ultimate foundational axioms—the bedrock principles—that were themselves manifestly true and also encapsulated such deductive power that with them, we could in principle settle every question within their arena. What a mathematical dream that would be.

Meanwhile, troubling antinomies—contradictions, to be blunt—had arisen on the mathematical frontiers in some of the newly proposed mathematical realms, especially in the naive account of set theory, which exhibited enormous promise as a unifying foundational theory. Set theory had just begun to provide a unified foundation for mathematics, a way to view all mathematics as taking place in a single arena under a single theory. Such a unification allowed us to view mathematics as a coherent whole, enabling us sensibly, for example, to apply theorems from one part of mathematics when working in another; but the antinomies were alarming. How intolerable it