> In some sense that's defendable (that what could call natural numbers is
> a cultural question or historical, like looking at what Peano did nor
> did not define).
> On the other hand, one normally does not just deals with the numbers as
> such, one does something with it (like comparing them or calculating
> with them). If one takes the reservoir of numbers (in decimal notation,
> {0,1,2,3 .....} indeed it is irrelelvant where to start, 0,1, or
> 23. Also if one does nothing else than comparing them (that would be
> considering them as "ordinals", one has one single smallest number, and
> again it's it's irrelevant if that's ``called'' nor notated $0$, "zero"
> or "1", or "23".
> Now, if one starts doing simple calculations (addition, multiplication),
> the natural numbers including 0 is simply more "elegant" or ``richer''
> than without. One has laws like n+0 = n, n*0=0 (one then says, 0 is a
> neutral element wrt. +, there is terminology for all than, and it's
> simply that N with 0 has nicer ``algebraic'' characteristics than
> without. It's quite analogous to the choice between defining lists as to
> include the empty list '() as a ``natural'' list, or insist on that
> ``natural'' lists must have 1 or more elements.
> Foundational folks can elaborate on that analogy between lists and nats,
> but as you say, in both cases they favor to include 0 to nats and the
> empty list to lists (and there are more examples) and it's favored for
> good reasons (at least to them).
That is the first time I remember that on this list, questions of the
foundation of mathematics are discussed 😉
Back to the point, maybe I am too conservative, but I would include 0
within the natural numbers, and the example I started with, needs to
cover that case (student marks range between 0 to 10 both included), so
sorting should work (for me) in that case.
I don't see, so far any benefit for not considering 0 in that sorting
process.
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