Gödel's incompleteness theorems

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…e even deigns to mention Godel's Second incompleteness theorem, the one about axiomatic systems being unable to prove their own consistency. Now if you were to take up a better book than Manin's, say Smullyan's Godel's Incompleteness Theorems , and study the proof of the second incompleteness theorem, you'd find out that its proof requires a syntactic proof of the first theorem, the proof that Manin says isn't important. This is because proving the second theorem entails formalizing the proof of the first theorem inside the axiomatic system itself, and you can't do that with a proof that uses semantic arguments and talks about true and false theorems (why? because of Tarski's undefinability of truth!). Think of that what you will.