Reading group: Nagel and Newman, Gödel’s Proof

I've noticed that half the physicists and engineers I work with cite Gödel and the other half wince when they do. I would like to understand the theorem properly, and I suspect I'm not alone.

Ernest Nagel and James R. Newman's Gödel's Proof (1958) is a short book that walks through the argument without assuming much mathematics. There is a revised edition from 2001, edited with a foreword by Douglas Hofstadter.

Plan, three weeks:

1. The opening chapters: the problem of consistency, and Hilbert's programme of proving mathematics consistent by strictly finite methods.
2. The middle chapters: systematic codification of formal logic, and the idea of mapping statements about a system into the system itself, which leads to Gödel numbering.
3. The proof itself, and the closing reflections on what follows.

I'd like us to keep two questions in view. What exactly does the theorem say, stated carefully? And where do popular uses of it go beyond what it says?