History and philosophy of maths · Ambitious · Dissertation

What Gödel really proved: a maths EPQ idea

Why it works as an EPQ

Widely misquoted results; explaining them correctly and judging the popular claims is demanding but rewarding.

Scope and difficulty

Ambitious. Ambitious. Explain the idea of the proof (coding statements as numbers), not the full proof.

The maths

Builds on these A Level topics: Proof.

You would learn:

• Formal systems and axioms
• Gödel numbering
• Self-reference

Learning maths beyond the course is expected in a maths EPQ, but you must understand and explain everything you use.

One possible plan

1. Explain formal systems with a small example.
2. Explain Gödel numbering and the self-referential sentence.
3. State both theorems carefully.
4. Test popular claims ('maths is unreliable', 'minds beat machines') against what was proved.

Pitfalls

• Overclaiming what the theorems say.
• Reproducing a proof you cannot explain.

Where to start reading

• Gödel's Proof (Ernest Nagel and James Newman)
• Gödel, Escher, Bach (Douglas Hofstadter)

Prefer books, lecture notes, journal articles and official data to a single website, and record every source as you read (how to reference).

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All history and philosophy of maths ideas · all 93 ideas

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