About Rational Numbers

• ⁃ Natural Numbers are defined by iteration (see G. Peano), they have no upper limit by construction so there can not be any number greater than “all” Natural Numbers since there is no “all”.
• ⁃ Rational Numbers are build up in two steps:
• ⁃ first, two independent function are applied on the Natural Numbers, by means of which is build up an ordered couple (.,.) [or ./.]: N → (a,.), N → (a,b) [or N → a/., N → a/b]; where a and b are Natural Numbers. At this step Rational Numbers and ordered couples have exactly the same structure.
• ⁃ then, to the ordered couples are applied rules (or laws) that enrich their meaning (and do not impoverish!). For example, (a,b) can be seen as the sum operation over two numbers, and from that we can deduce that they receive the property of commutation so that (a,b)=(b,a); or, in this instance, they can represent the fraction (and in this case it is used the symbol /) so that we deduce that there is a condition on the second term to be applied: b≠0; and so on. You cannot put them in one-to-one bijection with ℕ simply because they are defined differently; to do that you need to link the two independent functions in some way but doing so you lose the independence of the two members of the ordered couple.