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This seems to be (from what I've heard from various math people of various statures) a heated debate. One of my previous professors proclaimed very strongly that $0$ is not a natural number. Another recently said the same. But then there are people who say it should be. One very good logical reasoning given was that, if the natural numbers are supposed to be the "counting" numbers, then $0$ is useful in the sense that it denotes the lack of something.
I do realize that the natural numbers were defined prior to the discovery of the concept of "zero", but that shouldn't be a reason why we can't formally agree that it should be a natural number.
And for the cases where using $0$ breaks things, there's no reason why calculations can't explicitly omit $0$.