What Gödel Discovered (2020) (stopa.io)

The article explains, in a programmer-friendly way, how Kurt Gödel’s 1931 work showed that any sufficiently powerful formal system for arithmetic (like Russell and Whitehead’s “Principia Mathematica”) cannot be both complete and consistent. It walks through the motivation from earlier foundational efforts and Russell’s paradox, then sketches Gödel’s method of encoding logic and proofs into numbers using Gödel numbering, leading to unprovable statements. The overall point is that mathematics cannot be fully captured by a single, purely mechanical set of rules.