For the first sixty pages Nagel and Newman combine history — Euclid, Hilbert and attempts to formalise mathematics, Russell and Whitehead's Principia Mathematica — with an explanation of the necessary preliminaries — the problem of consistency, absolute proofs of consistency, the codification of formal logic, the difference between mathematics and meta-mathematics (reasoning about mathematics), and the key idea that "meta-mathematical statements about a formalized arithmetical calculus can ... be represented by arithmetical formulas within the calculus".
In thirty pages they then present the proof itself: Gödel numbering, arithmetizing meta-mathematics, and the actual construction. This is a drastic simplification of the full proof, obviously, but it is enough to, as they put it, "afford the reader glimpses of the ascent and of the crowning structure". They close with some brief comments on the broader philosophical implications of Gödel's Theorem.
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