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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    $\begingroup$ With the definition of finiteness/infiniteness that comes from set theory, zero (i.e. the empty set) is indeed finite. Now, I do not know which literature you are referring to and what is the definition of infinity given there, however I find somewhat unorthodox to consider zero as not finite (or in some sense different from the other natural numbers). $\endgroup$ – yuggib May 27 '15 at 10:52
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    $\begingroup$ I think i have read this expression often meaning explicitly not zero even more than not infinite. Example from Shankar discussing a particle scattered at a not infinite potential barrier:"Thus there is a finite probability The correspondence holds for finite transformations as well, for these may be viewed as a sequence of infinitesimal transformations.for finding the particle in the region where its kinetic energy E0 — Vo is negative" $\endgroup$ – pindakaas May 27 '15 at 11:01
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    $\begingroup$ It seems to be used like that in order to differentiate from infinitesstimal quantities. Example Shankar: "The correspondence holds for finite transformations as well, for these may be viewed as a sequence of infinitesimal transformations." $\endgroup$ – pindakaas May 27 '15 at 11:01
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    $\begingroup$ No there is no rule. I have been guilty of using finite to mean non-zero, and while I probably shouldn't do this it's ofetn a convenient shorthand. It's normally obvious from the context if the author is (ab)using the term finite to mean non-zero. $\endgroup$ – John Rennie May 27 '15 at 11:02
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    $\begingroup$ @pindakaas Zero is a natural number (finite) and not an infinitesimal quantity. I agree that finite can mean not infinite and not-infinitesimal, however saying finite for "non-zero" is far from being a good choice (and it is just two characters longer so not really a shorthand :-P) $\endgroup$ – yuggib May 27 '15 at 11:14

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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