Zero, though

This is a shower thought I will have to research - not a chance in the universe that was I the first to think about this.

Consider natural numbers as a counting device (and not as, say as an assiomatic system, or as points on line). As such, they are naturally attached units of measure - one pear, six aircraft carriers, two days. And, in set theory you can then take the set of sets having two of something as a definition of the number 2 (well, not quite, but almost).

Zero, though. Of course, one can use zero as a counting device: "I have one apple. I will eat it at lunch, and I will then have zero apples"

The notion of attaching measure units to "zero" (quite natural as you go form one apple to none) becomes rather strange as one considers the process of telling apart sets of equal numbered things.

Distinguishing between one elephant and one saxophone player is easy enough (assuming empty the set of saxophone playing elephants). What about telling apart zero apples from zero oranges?

Given an orange, and questioned about apples, I will count to zero, obviously. But, given the same orange, I will count to zero also if questioned about wine bottles, pencils, cars...

And of course,  the notion of "no things" is a gaping void (arh, arh).

So it looks that zero should be dimensionless -  a fact that set theory implicitly acknowledging the empty set as a member and subset of any set, while being unique.