TITLE: Why is zero important?
QUESTION [8 upvotes]: I am not sure whether this question is more appropriate here or in
theoretical computer science. I leave it to the wisdom of moderators.
On the computer science site I came across the following (possibly
unintended) question:

is the empty string a necessary concept?

This reminded me of the fact that the ancient Greeks did good work in
arithmetics without having zero, syntactically (as a place holder, as
in positional number systems) or semantically.
Having zero (resp. the empty string) will of course simplify analysis
and reasoning about integers (resp. strings and formal languages), and
the way definitions and theorems can be stated.
However, I came to wonder whether it changes in any significant way
the kind of results that can be proved, or whether we get essentially
the same theories.
In the case of arithmetics, I guess this would mean that 0 is replaced
by 1 in the nine Peano's axioms, and that for addition, the axiom
$a+0=a$ is replaced by $a+1=S(a)$. Indeed, it has to give the same
theory as the original axiom $a+0=a$ can be proved from the new one if
we add a new integer 0 such that $S(0)=1$. But I am not sure things
are always that simple, and the lack of zero could cause considerable
complexity in expressing other concepts.
So my question is whether this issue has been formally analyzed, or
whether it even makes sense of asking that question. Would the power
of mathematics be impaired (other than in expressive perspicuity) if
the concept of zero were somehow not available (assume that some
church will deal harshly with you if you even only hint at the
possibility of the concept)?
In the case of the empty string, another user asserted that not having
the empty string would change some properties of strings, such as
prefixing. I asserted that it is only a matter of consistent
definitions, and that the only change is in expressiveness (which is
far from negligible).
Is there any text discussing formally or informally this issue? I
would guess it is important, at least for epistemology and history of sciences.

REPLY [4 votes]: Here's my take on the matter.
Let $X$ denote a set. Then we certainly want to allow empty subsets of $X$. Otherwise, the intersection of two subsets of $X$ needn't exist; furthermore, the maximum subset of $X$ won't have a complement. Basically, getting rid of empty subsets would destroy all that nice symmetry of the powerset lattice.
Therefore, empty subsets are really nice. But, what is a subset? In category theory, a subset of a set $X$ would be defined as an injective function $f : A \rightarrow X$ modulo some largely irrelevant details. So if we want every set to have an empty subset, then there had better be an empty set $\emptyset$ for the domain of $f$. Therefore, the empty set $\emptyset$ is really important.
Finally, since natural numbers are primarily used to count the number of elements in a finite set, it would be better to include $0$ in our list of natural numbers.
There's many other reasons to include $0$ in our list, but I'll leave it here for now.