(FWIW, the "natural numbers" in fact does not include 0: if you include 0 you get the "whole numbers". I otherwise agree with you that 0 isn't at all strange.)
I did not know that "whole numbers" refer to nonnegative integers too.
> The whole numbers were synonymous with the integers up until the early 1950s. In the late 1950s, as part of the New Math movement, American elementary school teachers began teaching that whole numbers referred to the natural numbers, excluding negative numbers, while integer included the negative numbers. The whole numbers remain ambiguous to the present day.
> In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers 0, 1, 2, 3, ..., while others start with 1, defining them as the positive integers 1, 2, 3, ... . Some authors acknowledge both definitions whenever convenient. Sometimes, the whole numbers are the natural numbers plus zero. In other cases, the whole numbers refer to all of the integers, including negative integers. The counting numbers are another term for the natural numbers, particularly in primary school education, and are ambiguous as well although typically start at 1.
The definition of natural numbers used in any sort of formalized mathematics includes zero. Without zero, you don't have induction (and ℕ wouldn't form a monoid).
