On Fri, 5 Apr 2013, Dmitry A. Kazakov wrote:

>> Numbers, naturals, integers, rationals and reals, are semantically exactly
>> same except for encoding [representation] and constraints.
>
> A quite common misunderstanding. Structure such as field is not same as a
> subset. To see the difference between R and Z consider the following
> predicate:
>
>   forall x in S exists y such as x=1/y  [multiplicative inverse]
>
> This is true for S=R and false for S=Z.

Fair enough!

But the same is true for N and Z: every in Z has an additive inverse, but 
not every number in N. If the non-existence of a multiplicative inverse 
would justify different root types for Z and R, why should the 
non-existence of an additive inverse not justify different root types for 
N and Z (Universal_Positive versus Universal_Integer).

As it turns out, the fact that Naturals and Positives have the same 
representation as Integers, while Float has a different one, matters more 
than any "mathematical structure" ...





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