Are there mathematical contexts where "finite" implicitly means "nonzero?" I recently gave my students in a discrete math class the following problem, a restatement of the heap paradox:

Let's say that zero rocks is not a lot of rocks (surely, 0 is not a lot of rocks) and that if you have a lot of rocks, removing one rock leaves behind a lot of rocks. Prove that no finite number of rocks is a lot of rocks.

A small number of students submitted proofs by induction with the base case starting at one rock rather than zero rocks. We deducted a point for this, saying that this left the case of zero rocks unaccounted for.
Some students replied back to us saying that zero is arguably not a finite number. Some students pointed out this dictionary definition of finite which explicitly excludes 0 as not finite.
My background is in discrete math, and I've never seen zero referred to as not finite. The empty set of zero elements is a finite set, for example. There are no finite groups of size zero, but that's a consequence of the group axioms rather than because 0 isn't finite.
Are there mathematical contexts in which zero is definitively considered to be not finite?
Thanks!

Nobody seems to have picked up on definition 2(a) from the cited dictionary entry:
“capable of being completely counted.”  Students are taught from childhood
(see Wolfram MathWorld, Math/is Fun,
Math Goodies, Purplemath, and For Dummies)
that counting numbers are positive integers. 
If you ask a person, “How many elephants do you have in your pockets?”, most will reply,
“Don’t be silly!  I don’t have any.”, rather than “I have zero elephants in my pockets”,
because it’s just common sense that you can’t count something that doesn’t exist
— and because they’ve been taught that zero isn’t a “counting number”. 
Well, then, if finite means “countable”, then it follows that zero must not be finite.
;-)
