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Edited question: In response to Qiaochu Yuan's comment asking me to narrow down the question,the edited question is as follows: When did mathematicians first think of axiomatically building set theory with the level of rigour seen nowadays?(I am asking this because I learnt of the Peano's axioms and was amazed at how little needs to be assumed to build the number system)

Original question: I am interested to know when and under what circumstances mathematicians first thought of axiomatically building the number system and defining binary operations like addition and subtraction or for that matter the theory of sets ?

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    @plm: Not really. It's easier to confuse people who already know something than people who know nothing. You should read [this](http://www.dpmms.cam.ac.uk/~piers/maths%20and%20whig%20historiography%202010.pdf).2012-05-07

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