I always define $\mathbb{N}$ to include $0$ but some authors don't. Since the elements of $\mathbb{N}$ are used for counting, shouldn't $0\in\mathbb{N}$? $0$ is the number of cows in a classroom for example. Moreover, $0\in\mathbb{N}$ is a consequence of the Peano axioms and in fact the digit $0$ is used in writing integers ($10$, $205$) so why on earth would anyone define $\mathbb{N}=\{1;2;3;\cdots\}$ in the twentieth and 21th century? Is he a babylonian?
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