I don't think your definition is complete. We can count a set by mapping its elements to the natural numbers, and then identifying that number which is highest. However, we must have a provision for identifying zero as the highest when the set and mapping are empty.
Math counting doesn't care where you start. You could start -400 if it's useful for the problem
You can name things however you want, but there is a canonical bijection to zero-based ordinals.