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What axiomatic system is most commonly used by modern mathematicians to describe the properties of the integers and the rationals?

Properties like

$a+0=a$,

$a*1=a$,

$a+b=b+a$,

Also given these axioms, can I show that a contradiction cant be derived from them, such as $0=1$.. etc

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    Ring Theory and its branch Field Theory2012-12-24
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    I don't want to learn a whole new subject, I just want some confidence in that, that the expressions I manipulate daily are consistant with each other.2012-12-24
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    see: http://math.stackexchange.com/questions/264361/axioms-for-sets-of-numbers/264367#2643672012-12-24
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    Yes, I appreciated aspects of your answer there, that is why, I am only looking for axioms for the integers/rationals, because as you said, I can use those to construct most of the other systems.2012-12-24

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