[auggierose]

> 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.

[qsort]

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.

[ColinWright]

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.

[thaumasiotes]

> I believe that in France (for example) zero is regarded as both positive and negative.

That would cause all kinds of problems, so I'd be pretty surprised if it turned out to be true.

I note that this is the heading of the relevant wikipedia page:

> Un nombre négatif est un nombre réel qui est inférieur à zéro, comme −3 ou −π.

( https://fr.wikipedia.org/wiki/Nombre_n%C3%A9gatif )

It'd be hard to be more explicit that zéro is not a negative number.

[ColinWright]

If you're going to quote wikipedia:

> "Zéro est le seul nombre qui est à la fois réel, positif, négatif et imaginaire pur."

From: https://fr.wikipedia.org/wiki/Z%C3%A9ro#Propri.C3.A9t.C3.A9s...

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).

================

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…)

-- https://mathstodon.xyz/@antoinechambertloir/1153275891164575...

[dooglius]

Presumably, GP only worked on the problems in English and someone else would translate it appropriately.

[thaumasiotes]

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.

[auggierose]

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.

[auggierose]

On the other hand, even for writing a perfectly fine natural number like "10", you need the zero... Maybe it is just ℕ and ℤ after all.

And round we go.

[thaumasiotes]

What do you use for the negative integers?

[auggierose]

I very rarely use just the negative integers, so I don't need a symbol for it.

[gjm11]

I never write ℕ, for exactly this reason. I write ℤ with a subscript ">0" or ">=0". Doesn't take up much more space, and completely unambiguous.

[rossant]

I didn't know that. In French textbooks, I believe ℕ always includes 0. I didn't even know that not including it was another possible convention.

[sureglymop]

Well, in any textbook I've read they at least defined ℕ in the beginning and then used e.g ℕ₀ to include 0 or ℕ⁺ to not include it.