Countable set is a set for which we have injective function to set of natural numbers. Countable sets has its own structure, you may union, substrate, and intersect it. You may form more complicated operations like Cartesian products.
On the image side, thous operations in most natural setting are reflected by simple operations on natural numbers. However it is not a part of countability definition, there is no homomorphism of structures required in definition. On the other side, presented examples of such injective functions, or even bijections, usually behaves like homomorphisms, transporting arithmetic structure into set elements counting operations.
Is multiplication of image set of natural numbers important in any meaning for countability? Or they are not related at all? For example is it possible to define countability in Presssburger arithmetic with the same meaning as for Peano setting?
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