Is zero even a number?
Charles Seife's "Zero: The Biography of a Dangerous Idea" (http://www.amazon.com/Zero-Biography-Dangerous-Idea-ebook/dp...)
Robert Kaplan's "The Nothing That Is: A Natural History of Zero" (http://www.amazon.com/Nothing-that-Natural-History-ebook/dp/...)
I would expect any useful system of arithmetic, offering addition and subtraction, to contain some concept of "nothing". Because if a is something that might be subtracted from anything else including itself, we have the result that a - a is something we should be able to talk about, namely nothing.
If you accept a sufficiently strong set theory as a foundation of mathematics, for example the Zermelo Fraenkel set theory including the axiom of choice (ZFC), it is possible to use the finite ordinal numbers - which are actually sets - as the natural numbers.
From the ZFC axioms we have the existence of the empty set, often denoted by a pair of empty brackets {}.
We can then define numbers by using the empty set:
0 := {}
1 := {0} = {{}}
2 := {0,1} = {{},{{}}}
... and so on. By this definition zero is a number.
On the other hand, by defining numbers on the basis of set theory we seem to accept that numbers are sets. Some people will have very different opinions about this matter.
If you prefer something else than set theory, you should still be able to construct natural numbers satisfying the Peano axioms. In other words, any reasonable axiomatization of mathematics will give you the naturals.