It depends on whose authority you recognize for a counting system.
Measurement of arbitrary segments of a continuous function are not the same
as countable items.
I do not turn my calendar page at 1:00am, but at 12:00, or 24:00, or 00:00
(at least I would use the latter if I could get my clock to do it.)
Similarly, if I measure a length, I don't put the 1" mark of my ruler at
one end of it.