I've just started P.Halmos' book "Finite Vector Spaces", and at the first chapter, after defining the axioms of a field, there is an exercide that asks: if we consider the non negative integers, can we redefine addition and multiplication so that they form a field? Can't we, since $\mathbb{N}$ is countable just like $\mathbb{Q}$ that is a field, use some sort of coding to represent the field $\mathbb{Q}$ with it's operations with the natural numbers?
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$\begingroup$Yes. In general, if you have two sets that are in bijection with each other, and one of them has any sort of extra structure (e.g. is a group, a field, a vector space), then you can use the bijection to define the corresponding structure on the other set. This is sometimes referred to as "transport of structure."
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