"Finite"

In conversation last week, I said to my father that my laptop battery has a “finite number of charge cycles”.

Now, if someone said to me that a battery had fewer than a billion charge cycles, I’d take the speaker to be implicating that it has quite a lot of them, probably between half a billion and a billion. And even besides that implicature, if all my information were that the battery has fewer than a billion charge cycles, then it would seem natural to take a uniform distribution from 0 to 999,999,999 and think that it is extremely likely that it has at least a million charge cycles.

One might think something similar would be the case with saying that the battery has a finite number of charge cycles. After all, that statement is logically equivalent to the statement that it has fewer than ℵ₀ charge cycles, which by analogy should implicate that it has quite a lot of them, or at least give rise to a uniform distribution between 0, inclusive, and ℵ₀, exclusive. But no! To say that it has a finite number of charge cycles seems to implicate something quite different: it implicates that the number is sufficiently limited that running into the limit is a serious possibility.

Actually, this may go beyond implicature. Perhaps outside of specialized domains like mathematics and philosophy, “finite” typically means something like not practically infinite, where “practically infinite” means beyond all practical limitations (e.g., the amount of energy in the sun is practically infinite). Thus, the finite is what has practical limits. (But see also this aberrant usage.)