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?