What differences are there between the set of natural numbers with zero included vs excluded, ignoring broader mathematical context?

Question

My apologies for the somewhat vague title, my question is more specific but very difficult to condense to title length.

There's a significant amount of discourse regarding whether or not the set of 'Natural Numbers' should include the number zero (such as Is $0$ a natural number?)

(For the record, in my view, the 'truth' of the matter is that these are just two different sets which should have two different names, and the current situation is an unfortunate consequence of the ossified state of mathematical language)

However, most of this discourse seems to revolve around applications of the natural numbers in a more general context, e.g. "the harmonic sequence 1/n is defined for any natural number n [if zero is excluded]". Of course, that statement has no meaning in the context of the natural numbers alone, no matter how you define them.

That there are differences between these two sets, $\mathbb{N}_0:=\mathbb{N}\cup\left\{0\right\}$ and $\mathbb{N}_1:=\mathbb{N}\setminus\left\{0\right\}$, even within their own context, is of course obvious- statements like $\exists n\in \mathbb{N} | \forall m\in\mathbb{N},n+m=m$ are true for $\mathbb{N}_0$ but not $\mathbb{N}_1$.

Are there any non-trivial differences between $\mathbb{N}_0$ and $\mathbb{N}_1$- that is, statements, possible to construct using only the language of natural numbers, about one which are not true of the other and which aren't obviously equivalent to '$\mathbb{N}_0$ has zero in it'?

Answer 1

Credit to Freakish for this answer, though I've rephrased it somewhat to more explicitly match the intention of my question.

In short, I made the assertion that the natural numbers including zero ($\mathbb{N_0}$ in my notation) and the natural numbers excluding zero ($\mathbb{N_1}$ in my notation) had obvious differences between them in their own context, all of which that I was aware of could be very easily restated as '$\mathbb{N_0}$ has zero in it', and asked about the existence of less obvious differences that didn't rely on the context of different mathematical objects, like rational numbers.

In fact, I'd mistakenly done just that in my assertion that $\mathbb{N}_0$ containing zero was a meaningful difference in the first place, though in a non-obvious way.

As pointed out in the comments, my attempt at crafting a statement true about $\mathbb{N_0}$ that was false for $\mathbb{N_1}$ relied on the definition of '$+$', which is not axiomatic in the construction of either set, but instead defined in terms of the set's first/smallest member, defined by axiom as '$0$' in $\mathbb{N}_0$ and as '$1$' in $\mathbb{N}_1$, and an axomatically defined successor function, $S(n)$, which generates the other members of the set (and can be understood as the increment function in both sets).

For $\mathbb{N}_0$, '$+_0$' can be defined recursively as $n+_0 0 = n$, and $n+_0S(m)=S(n+_0m)$ to result in standard addition, giving $0$ it's unique property of being the additive identity and formalizing that $S(n) = S(n+_00) = n+_0S(0) = n+_01$.

Using this same definition for '$+_1$' in $\mathbb{N}_1$ however does not produce an operation equivalent to addition however, with it instead being the case that $n+_11 = n$ and $S(n) = n+_12$. Using their respective definitions of '$+$', the statement $\exists n \in \mathbb{N} | \forall m \in \mathbb{N}, n+m=m$ is true for both sets now. There is in fact no difference between $\mathbb{N}_0$ and $\mathbb{N}_1$ except for those imposed by external context, such as by describing them as a subset of $\mathbb{R}$ or applying them as a model.