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My main interest is physics. However, being self-taught in mathematics for the most part, my proofs tend to be more intuitive than it is acceptable. Yet, I recognize my inaptitude in rigor, and I would like to proceed in practicing my writing proofs in a more mathematical way. Having no formal instruction, and due to my lack of experience in writing rigorous proofs, the mistake that I make is primarily attacking a problem directly, without being able to formulate a sketch, which implies a, in principle, sloppy result.

I want, if possible, to start generalizing each problem in translating a given theorem in set-theoretical language, and form a sketch of proof in providing the proof for the generalization of the theorem. What are your experiences?

Example of an elementary result:

Theorem. If $a$ and $p$ are natural numbers such that $a^p-1$ is prime, then $a=2$ or $p=1$.

How would that be described in formal, set-theoretical language, and proved from there?

(I understand that the proof is itself trivial, but bear with me.)

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Writing rigorous proofs does not mean formulating it in terms of set theory and proving it from the axioms of ZFC. For one thing, the explicit statement in the language of set theory would be nigh unrecognisable to the vast majority of mathematicians... –  Zhen Lin Mar 17 '13 at 20:22
    
My purpose would of course be to flesh it out based on the particular requirements of the case. –  MikhailSchmokloff Mar 17 '13 at 20:24
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@user67170 There's a difference between rigour and formalism. What you're asking about is formalism, but that's not what you want, trust me. I'm sure someone will expand on this in an answer in a better way that I could put it, in any case I recommend you read this to get a taste of what you're actually looking for. –  Git Gud Mar 17 '13 at 20:25
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Please don't attempt it! It is a sure recipe for not doing any real mathematics ever again. Oh yes, Set Theory is real mathematics. And real Set Theory people do mathematics like anybody else, carefully but not fully formally. –  André Nicolas Mar 17 '13 at 21:15

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