Reading a little more closely, the axioms appear to be agnostic to the interpretation of the first element; it is only when one defines the operation of addition that any difference emerges. The distinction arises from the choice of definition of the addition operation. As such, it seems that this article is making untrue statements about the Peano axioms (which do not define an addition operation, only a successor function). This seems to be a significant misrepresentation; I'd appreciate comment from those more familiar with working more formally in this area. —Quondum