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Infinity, is clearly not finite. But there is some dissent on whether or not zero is finite. I have seen authors use "finite" to indicate the value of $0$ is excluded as well as infinity.

Is there any rule about this?

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There is no rule, and it depends on the context.

  • If you're worried about things being very big, then zero is an OK value to have, and you'd count it as a finite quantity.

  • In other situations, however, you are concerned about whether a quantity $q$ is exactly zero, or whether it is only a finite-precision approximation of it. Thus, you might say that "$q$ is finitely small", but after a while you end up simply saying "$q$ is finite" for that assertion.

It is open to discussion whether this is mathematically correct or a useful shorthand, but the fact is that people do use it and their communications are perfectly clear, so the discussion is pretty moot. Whether zero is 'finite' or not in a given paper should be obvious from the context, and if it is not then the charge against the author should be "ambiguous wording" instead of "incorrect notation".

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