Unit Paradox of Zero
A decade ago, I was reading a primer-book for my daughter, then quite young. The book was a tiny cardboard picture-book, about numbers. There was a page for a pair, for a triple, and so on up to a ten.
One of the pages had a curious sequence at the bottom. It had the numbers 0 to 10; above the 10 were ten flowers; above the 9 were nine marbles; and so on down to two soccer balls above the 2, one teddy-bear above the 1, and above the 0… nothing! Appropriate; but none of what?
Elephants are not mice, yet no elephants is exactly the same as no mice. And also the same as no gorillas, and no Yetis, and no mermaids! None of anything is the same as none of anything else.
Call this the Unit Paradox of Zero: An answer of zero needs no units.
There are no dragons; therefore no dragons are not red; so all dragons are red, by vacuous necessity. By an equally vacuous necessity, all dragons are blue. Nothing red is blue, but since there are no red dragons, that empty set is the same as the set of blue dragons, also empty. Therefore the properties of the elements of any empty set are all indefinite.
Their measures are indefinite too. There are no dragons, nor dragon scales, therefore the average dragon has 0/0 scales; the indefinite ratio.
Therefore zero counts nothing at all, and the nothing that it counts is none of any object, with any properties and all measures.