Should $\mathbb{N}$ contain $0$? [closed]

Let me ask a similar question in rebuttal:

Are finite sets countable?

The answer is simple. It depends on the context. Sometimes it's easier to have finite sets included in the definition of "countable" and sometimes it's easier to have just the finite set, and have "at most countable" for the term which includes finite sets as well.

I will give an argument why $\Bbb N$ should include $0$, though.

One can consider $0$ and $1$ as the basic atoms of the numbers we know. $\Bbb N$ is the set generated by $0,1$ using addition; and then $\Bbb Z$ is generated by adding additive inverses, and $\Bbb Q$ and then $\Bbb R$ and $\Bbb C$.

Of course that if you take your atomic set of numbers to be $\Bbb C$ or something else, then it might as well be redundant, but it's still a reasonable argument. With only some naive set theory, and axioms for addition and multiplication, we can create all the numbers we need! That's an incredible thing. And all just form the assumption that $0$ and $1$ exist.

On the other hand, in analysis it's often more convenient to have $0\notin\Bbb N$. For example when we say that $x^n$ is well defined for every $x\in\Bbb R$ and $n\in\Bbb N$. Or if we often talk about the sequence $\frac1n$, then we find it easier to write $\frac1n$ for $n\in\Bbb N$, rather than adding "...and $n>0$".

They should arguably include zero so that they are a monoid that gives rise to the integers, and all the other numbers, etc, like in Asaf's great response.

In my day to day life though only the cardinality of them matters, as they are really used purely as an index set. In this context it really is purely a matter of aesthetics. I would go back and forth including zero if I want to divide by them or not.

My 2¢: Besides the in my opinion other good arguments why $\mathbb{N}$ should contain $0$, I argue for a purely notational one: since $\mathbb{Z}_+$ is a perfectly good name for the set $\{ 1, 2, 3, \ldots \}$ there is no need for a second name for this set. Hence $0 \in \mathbb{N}$.

Besides problems coming from notation (these are the only ones which distinguish $\mathbb{N}$ from $\omega$, the former used when treated as a monoid and the latter when thought as a ordinal) the reality is this: $0 \in \omega$ and $0 \subset \omega$ are both true ( remember that $0:= \emptyset$), so it's obviously contained, and it belongs to $\omega$ for lots of reasons (transitivity plus the fact that both are cardinals or by definition of $\omega $ as the set of natural numbers). The last is a good definition not given in an intuitive and inexistent way.