> The set ℕ is the set of positive integers, not including 0.
Hell yeah!
I've agonised over this quite a lot over the decades. Not including 0 is more intuitive, but including 0 is more convenient. Of course, both approaches are correct. My main reason for not including 0 is that I hate seeing sequences numbered starting with 0.
I used to write and review problems for math competitions. This is why we avoided saying "natural numbers". We used "nonnegative integers" or "positive integers" instead.
You need to be careful about this ... I believe that in France (for example) zero is regarded as both positive and negative. So in France:
Non-negative integers: 1, 2, 3, 4, 5, ...
Positive integers: 0, 1, 2, 3, 4, 5, ...
Similarly, for some countries "Whole Numbers" is equivalent to all the integers, while in other countries it's the set { 0, 1, 2, 3, 4, ... } while in still other countries it's { 1, 2, 3, 4, ... }
There is no approach that uses "natural language" and is universal, and being aware of this is both frustrating and useful. Whether it is important is up to the individual.
It's hard to be more explicit that it is considered both.
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Added in edit
In speaking with a French colleague, he says that "inférieur" often means "less-than-or-equal-to" rather than "strictly-less-than", so the passage you quote would still imply that 0 is negative (and most likely also positive).
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Second edit:
> In France, "positive" means "supérieur à 0", and "supérieur à " means "greater than or equal to". Similarly, "négative" means "inférieur à 0", that is "less than or equal to 0".
> (We have the similar reaction towards the anglosaxon world and the introduction of nonnegative…)
From a technical perspective you frequently need 0 in there.
From a pure convenience perspective, it doesn't make sense to assign ℕ to the positive integers when they're already called ℤ⁺. Now you have two convenient names for the smaller set and none for the larger set.
By convenience I mean "convenient from a technical perspective", and yes, you often need 0 in there.
Your other argument doesn't make much sense. I learnt both in school and at university ℕ, ℕ₀, and ℤ as THE symbols for the natural numbers, the natural numbers including 0, and the whole numbers.
Fuck convenience. ℕ, ℕ₀, and ℤ it is :-) It is just so much prettier (ℤ⁺ is a really ugly symbol for such a nice set). It is actually also not inconvenient if you don't use static types.
Hell yeah!
I've agonised over this quite a lot over the decades. Not including 0 is more intuitive, but including 0 is more convenient. Of course, both approaches are correct. My main reason for not including 0 is that I hate seeing sequences numbered starting with 0.