@HarryGindi: I don't want to index by arbitrary complex numbers, but by natural numbers (seen as particular complex numbers). Just wanted to say I prefer not to index by "pure" natural numbers as defined through inductive sets. For example, I prefer to use the digit 0 for the complex number (0R,0R) and not for the empty set (0R stands for the zero defined as a dedekind cut or a cauchy sequence). This is because complex numbers and their operations are much more used in practice than "pure" natural numbers with addition and multiplication alone.

@KonradSwanepoel: In the second definition you can avoid the dots by requiring a set X (intuitively made of x_1,...,x_n) prior to forming the n-tuple (x_1,...,x_n). So an n-tuple is just a function from the set of numbers from 1 to n (both included) to X. No dots at all. No need at all to name the elements of the n-tuple.

I would use ordered pairs and sets of ordered pairs in a "temporary" manner, only to define the various number systems up to the complex numbers, and then I would "forget" them in favour of n-tuples (where n is a complex number) and sets of n-tuples. Also, as you correctly pointed out, there are at least five versions of the number 3, but four of them are "temporary", and only the latest one is the "real", useful one. But I think in set theory the procedure to define something is at least as important as the thing you define, so you cannot really skip that step.